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J3$? ifr-, Ti~~^ 95-oo f7y SYMMETRY bonss Symmetry is one of the ideas by which man through the ages has tried to comprehend PRESTON POLYTECHNIC and create order, beauty, and perfection. LIBR^^SuLEARNING RESOURCES SERVICE

Starting from the somewhat vague notion This book must be returned drt of before the date last stamped of symmetry = harmony of proportions, this book gradually develops first the geometric

concept of symmetry in its several forms as bilateral, translator^, rotational, ornamental and crystallographic symmetry, and -5. jun. m finally rises to the general abstract mathe-

matical idea underlying all these special .41 DEC 1989 forms. Professor Weyl on the one hand displays the great variety of applications of the principle of symmetry in the arts, in inorganic and organic nature, and on the other hand \\. he clarifies step by step the philosophical- mathematical significance of the idea of V symmetry. The latter purpose makes it necessary for the reader to confront the notions and theories of symmetry and relativity, while a wealth of illustrations supporting the text help to accomplish the former. This book is semi-popular in character. K m 1995 It does not shun mathematics, but detailed treatment of most of the problems it deals with is beyond its scope. The late Professor Weyl was world PI FASF RPTIIRN TO PAMDIK TNinTTflTPD famous for his contributions to mathematics and the philosophy of science. In this book his penetrating mathematical insight illumines and transforms the worlds 30107 2 of nature and of art. 001 374 963 SYMMETRY

4 01788 701.17

SYMMETRY

BY HERMANN WEYL

PRINCETON UNIVERSITY PRESS, 1952

PRINCETON, NEW JERSEY 5"Z0O6g"S0 5bo uj ^u? PREFACE

AND BIBLIOGRAPHICAL REMARKS

Starting from the somewhat vague notion of symmetry = harmony of proportions, these four lectures gradually develop first the

geometric concept of symmetry in its several forms, as bilateral, translatory, rotational, ornamental and crystallographic symmetry,

etc., and finally rise to the general idea underlying all these special forms, namely that of invariance of a configuration of elements under a group of automorphic transforma-

tions. I aim at two things: on the one hand to display the great variety of applications of the principle of symmetry in the arts, in inorganic and organic nature, on the other hand to clarify step by step the philosophico- mathematical significance of the idea of sym-

metry. The latter purpose makes it necessary to confront the notions and theories of symmetry and relativity, while numerous illustrations supporting the text help to accomplish the former. As readers of this book I had a wider circle in mind than that of learned specialists. It does not shun mathematics (that would defeat

its purpose), but detailed treatment of most

of the problems it deals with, in particular complete mathematical treatment, is beyond

its scope. To the lectures, which reproduce in slightly modified version the Louis Clark Vanuxem Lectures given by the author at University in February 1951, two Copyright 1952, by Princeton University Press Princeton appendices containing mathematical proofs Second Printing i960 have been added. Third Printing 1962 Other books in the field, as for instance Fourth Printing 1965 F. M. Jaeger's classical Lectures on the principle Fifth Printing 196(3

Printed in the United States of America of symmetry and its applications in natural science (Amsterdam and London, 1917), or the CONTENTS much smaller and more recent booklet by

Jacque Nicolle, La symetrie et ses applications (Paris, Albin Michel, 1950) cover only part of the material, though in a more detailed Bilateral symmetry fashion. Symmetry is but a side-issue in D'Arcy Thompson's magnificent work On Translatory, rotational, and related growth and form (New edition, Cambridge, symmetries 41 Engl., and New York, 1948). Andreas Speiser's Theorie der Gruppen von endlicher Ornamental symmetry 83 Ordnung (3. Aufl. Berlin, 1937) and other mathematical publications by the same author are impor- Crystals. The general 119 tant for the synopsis of the aesthetic and idea of symmetry mathematical aspects of the subject. Jay Hambidge's Dynamic symmetry (Yale Uni- APPENDICES versity Press, 1920) has little more than the name in common with the present book. Its A. Determination of all finite groups closest relative is perhaps the July 1949 num- 149 of proper rotations in 3-space ber on symmetry of the German periodical 155 Studium Generale (Vol. 2, pp. 203-278: quoted B. Inclusion of improper rotations as Studium Generale). Acknowledgments 157 A complete list of sources for the illustrations is to be found at the end of the book. Index ,61 To the Princeton University Press and its editors I wish to express warm thanks for the inward and outward care they have lavished on this little volume; to the authorities of Princeton University no less sincere thanks for the opportunity they gave me to deliver this swan song on the eve of my retirement from the Institute for Advanced Study. Hermann Weyl Zurich December 1951 BILATERAL SYMMETRY BILATERAL SYMMETRY

If i am not mistaken the word symmetry is used in our everyday language in two mean- ings. In the one sense symmetric means something like well-proportioned, well-balanced, and symmetry denotes that sort of con- cordance of several parts by which they inte- grate into a whole. Beauty is bound up with symmetry. Thus Polykleitos, who wrote a book on proportion and whom the ancients praised for the harmonious perfection of his sculptures, uses the word, and Diirer follows him in setting down a canon of proportions for the human figure. l In this sense the idea is by no means restricted to spatial objects; the synonym "harmony" points more toward its acoustical and musical than its geometric applications. Ebenmass is a good German equivalent for the Greek symmetry; for like this it carries also the connotation of "middle

1 Diirer, Vier Biicher von menschlicher Proportion, 1528. To be exact, Diirer himself does not use the word symmetry, but the "authorized" Latin translation by his friend Joachim Camerarius (1532) bears the title

De symmetria parlium. To Polykleitos the statement is ascribed (^epl fie\oiroiiKv, iv, 2) that "the employment of a great many numbers would almost engender correctness in sculpture." See also Herbert Senk, Au sujet de l'expression ovunirpia dans Diodore i, 98, 5-9, in Chronique d'EgypU 26 (1951), pp. 63-66. Vitruvius defines: "Symmetry results from propor-

tion . . . Proportion is the commensuration of the various constituent parts with the whole." For a more elaborate modern attempt in the same direction see George David Birkhoff, Aesthetic measure, Cam- bridge, Mass., Harvard University Press 1933, and the lectures by the same author on "A mathematical theory of aesthetics and its applications to poetry and music," Rice Institute Pamphlet, Vol. 19 (July, 1932), pp. 189-342. measure," the mean toward which the of space upon itself, S: p-> />', that carries virtuous should strive in their actions accord- the arbitrary point p into this its mirror image ing to Aristotle's Nicomachean Ethics, and p' with respect to E. A mapping is defined which Galen in De temperamentis describes whenever a rule is established by which every as that state of mind which is equally removed p' point p is associated with an image . An- from both extremes: ovp-ntTpov oirtp knarkpov other example: a rotation around a perpen- Tcbv anpoiv airkxtt- dicular axis, say by 30°, carries each point p The image of the balance provides a of space into a point p' and thus defines a natural link to the second sense in which the mapping. A figure has rotational symmetry word symmetry is used in modern times: around an axis / if it is carried into itself by bilateral symmetry, the symmetry of left and all rotations around /. Bilateral symmetry right, which is so conspicuous in the structure appears thus as the first case of a geometric of the higher animals, especially the human concept of symmetry that refers to such body. Now this bilateral symmetry is a operations as reflections or rotations. Be- strictly geometric and, in contrast to the cause of their complete rotational symmetry, vague notion of symmetry discussed before, the circle in the plane, the sphere in space an absolutely precise concept. A body, a were considered by the Pythagoreans the spatial configuration, is symmetric with re- most perfect geometric figures, and Aristotle spect to a given plane E if it is carried into ascribed spherical shape to the celestial bodies itself by reflection in E. Take any line / per- because any other would detract from their pendicular to E and any point p on /: there heavenly perfection. It is in this tradition exists one and only one point p' on / which that a modern poet 2 addresses the Divine has the same distance from E but lies on the Being as "Thou great symmetry": other side. The point p' coincides with p only

if/? is on E. Reflection in E is that mapping God, Thou great symmetry, Who put a biting lust in me From whence my sorrows spring,

For all the frittered days That I have spent in shapeless ways

p- Give me one perfect thing. ? * Symmetry, as wide or as narrow as you may

define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection. The course these lectures will take is as follows. First I will discuss bilateral symmetry in some detail and its role in art as

FIG. 1 E Reflection in B, 2 Anna Wickham, "Envoi," from The contemplative quarry, Harcourt, Brace and Co., 1921.

4 well as organic and inorganic nature. Then we shall generalize this concept gradually, in the direction indicated by our example of

rotational symmetry, first staying within the confines of geometry, but then going beyond these limits through the process of mathematical abstraction along a road that will finally lead us to a mathematical idea of

great generality, the Platonic idea as it were behind all the special appearances and applications of symmetry. To a certain degree

this scheme is typical for all theoretic knowledge: We begin with some general but vague

principle (symmetry in the first sense), then find an important case where we can give that notion a concrete precise meaning (bilateral symmetry), and from that case we gradually rise again to generality, guided more by mathematical construction and abstraction than by the mirages of philosophy;

and if we are lucky we end up with an idea no less universal than the one from which we

started. Gone may be much of its emotional

appeal, but it has the same or even greater unifying power in the realm of thought and

is exact instead of vague.

I open the discussion on bilateral symmetry by using this noble Greek sculpture from the fourth century b.c, the statue of a

praying boy (Fig. 2), to let you feel as in a symbol the great significance of this type of

symmetry both for life and art. One may ask whether the aesthetic value of symmetry de-

pends on its vital value: Did the artist discover the symmetry with which nature according to some inherent law has endowed

its creatures, and then copied and perfected what nature presented but in imperfect realizations; or has the aesthetic value of symmetry an independent source? I am in-

FIG. 2 clined to think with Plato that the mathe-

matical idea is the common origin of both: the mathematical laws governing nature are the origin of symmetry in nature, the intuitive realization of the idea in the creative

artist's mind its origin in art; although I am ready to admit that in the arts the fact of the

bilateral symmetry of the human body in its outward appearance has acted as an additional stimulus.

Of all ancient peoples the Sumerians seem

to have been particularly fond of strict bilateral or heraldic symmetry. A typical design on the famous silver vase of King En- FIG. 4 temena, who ruled in the city of Lagash around 2700 B.C., shows a lion-headed eagle with spread wings enface, each of whose claws FIG. 3 grips a stag in side view, which in its turn is

tiS8BB8EBBEgBfflEESBM jagBSBSBBBMBMBgai frontally attacked by a lion (the stags in the upper design are replaced by goats in the

lower) (Fig. 3). Extension of the exact symmetry of the eagle to the other beasts obviously enforces their duplication. Not much

later the eagle is given two heads facing in either direction, the formal principle of symmetry thus completely overwhelming the imitative principle of truth to nature. This heraldic design can then be followed to Persia, Syria, later to Byzantium, and anyone who lived before the First World War will remember the double-headed eagle in the coats-of-arms of Czarist Russia and the Austro- Hungarian monarchy. Look now at this Sumerian picture (Fig. 4). The two eagle-headed men are nearly but not quite symmetric; why not? In plane

geometry reflection in a vertical line / can also be brought about by rotating the plane

in space around the axis / by 180°. If you look at their arms you would say these two

8 monsters arise from each other by such rotation; the overlappings depicting their position in space prevent the plane picture from having bilateral symmetry. Yet the artist aimed at that symmetry by giving both figures a half turn toward the observer and also by the arrangement of feet and wings: the drooping wing is the right one in the

left figure, the left one in the right figure.

FIG. 6

FIG- 7

legs. In Christian times one may see an

FIG analogy in certain representations of the Eucharist as on this Byzantine paten (Fig. The designs on the cylindrical Babylonian 5), where two symmetric Christs are facing

seal stones are frequendy ruled by heraldic the disciples. But here symmetry is not

symmetry. I remember seeing in the collec- complete and has clearly more than formal tion of my former colleague, the late Ernst significance, for Christ on one side breaks Herzfeld, samples where for symmetry's sake the bread, on the other pours the wine. not the head, but the lower bull-shaped part Between Sumeria and Byzantium let me of a god's body, rendered in profile, was insert Persia: These enameled sphinxes (Fig. doubled and given four instead of two hind 6) are from Darius' palace in Susa built in

10 11 the days of Marathon. Crossing the Aegean we find these floor patterns (Fig. 7) at the Megaron in Tiryns, late helladic about 1200

B.C. Who believes strongly in historic continuity and dependence will trace the grace-

ful designs of marine life, dolphin and octopus, back to the Minoan culture of Crete, the heraldic symmetry to oriental, in the last instance Sumerian, influence. Skipping

thousands of years we still see the same influences at work in this plaque (Fig. 8) from the altar enclosure in the dome of Torcello, Italy, eleventh century a.d. The peacocks drinking from a pine well among vine leaves are an ancient Christian symbol of immor- FIG. 9 tality, the structural heraldic symmetry is oriental.

For in contrast to the orient, occidental

art, like life itself, is inclined to mitigate, to FIG. 8 loosen, to modify, even to break strict sym-

metry. But seldom is asymmetry merely the absence of symmetry. Even in asymmetric designs one feels symmetry as the norm from which one deviates under the influence of forces of non-formal character. I think the riders from the famous Etruscan Tomb of the Triclinium at Corneto (Fig. 9) provide a good

example. I have already mentioned representations of the Eucharist with Christ dupli- cated handing out bread and wine. The central group, Mary flanked by two angels, in this mosaic of the Lord's Ascension (Fig. 10) in the cathedral at Monreale, Sicily (twelfth century), has almost perfect symmetry. [The band ornament's above and below the mosaic will demand our attention in the second lecture.] The principle of

symmetry is somewhat less strictly observed in an earlier mosaic from San Apollinare in

13 FIG. 11

Mother, to the left John the Baptist. You may also think of Mary and John the Evangelist on both sides of the cross in crucifixions as examples of broken symmetry.

FIG. 12

FIG. 10

Ravenna (Fig. 11), showing Christ sur- rounded by an angelic guard of honor. For instance Mary in the Monreale mosaic raises both hands symmetrically, in the orans ges- ture; here only the right hands are raised. Asymmetry has made further inroads in the next picture (Fig. 12), a Byzantine relief

ikon from San Marco, Venice. It is a Deesis, and, of course, the two figures praying for

mercy as the Lord is about to pronounce the

last judgment cannot be mirror images of each other; for to the right stands his Virgin

14 Clearly we touch ground here where the mind there is no inner difference, no polarity precise geometric notion of bilateral sym- between left and right, as there is for instance metry begins to dissolve into the vague notion in the contrast of male and female, or of the of Ausgewogenheit, balanced design with which anterior and posterior ends of an animal. It we started. "Symmetry," says Dagobert requires an arbitrary act of choice to deter- Frey in an article On the Problem of Symmetry in mine what is left and what is right. But 3 Art, "signifies rest and binding, asymmetry after it is made for one body it is determined motion and loosening, the one order and law, for every body. I must try to make this a the other arbitrariness and accident, the one little clearer. In space the distinction of left formal rigidity and constraint, the other life, and right concerns the orientation of a screw. play and freedom." Wherever God or Christ If you speak of turning left you mean that the are represented as symbols for everlasting sense in which you turn combined with the truth or justice they are given in the sym- upward direction from foot to head of your metric frontal view, not in profile. Probably body forms a left screw. The daily rotation for similar reasons public buildings and of the earth together with the direction of its houses of worship, whether they are Greek axis from South to North Pole is a left screw,

temples or Christian basilicas and cathedrals, it is a right screw if you give the axis the are bilaterally symmetric. It is, however, opposite direction. There are certain crystal- true that not infrequently the two towers of line substances called optically active which Gothic cathedrals are different, as for instance betray the inner asymmetry of their constitu- in Chartres. But in practically every case tion by turning the polarization plane of

this seems to be due to the history of the polarized light sent through them either to cathedral, namely to the fact that the towers the left or to the right; by this, of course, we were built in different periods. It is under- mean that the sense in which the plane rotates standable that a later time was no longer satis- while the light travels in a definite direction,

fied with the design of an earlier period; hence combined with that direction, forms a left one may speak here of historic asymmetry. screw (or a right one, as the case may be). Mirror images occur where there is a mirror, Hence when we said above and now repeat

be it a lake reflecting a landscape or a glass in a terminology due to Leibniz, that left mirror into which a woman looks. Nature and right are indiscernible, we want to express as well as painters make use of this motif. I that the inner structure of space does not

trust, examples will easily come to your mind. permit us, except by arbitrary choice, to dis- The one most familiar to me, because I look tinguish a left from a right screw.

at it in my study every day, is Hodlcr's Lake I wish to make this fundamental notion still of Silvaplana. more precise, for on it depends the entire While we are about to turn from art to theory of relativity, which is but another nature, let us tarry a few minutes and first aspect of symmetry. According to Euclid consider what one may call the mathematical one can describe the structure of space by a

philosophy of left and right. To the scientific number of basic relations between points,

3 Studium Generale, p. 276. such as ABC lie on a straight line, ABCD lie

16 17 the world is revealed na- in a plane, AB is congruent CD. Perhaps by the general laws of space best way of describing the structure of ture. They are formulated in terms of cer- the single tain basic quantities are functions in is the one Helmholtz adopted: by which notion of congruence of figures. A mapping S space and time. We would conclude that the point physical of space associates with every point p a structure of space "contains a of in s s''-p—*P'> screw," to use a suggestive figure of speech, if p''-p—*P'' A Pair maPP S $* the p> _> f which the one is the inverse of p y S' other, so that if S carries p into p' then

FIG. 13

carries p' back into p and vice versa, is spoken of as a pair of one-to-one mappings or transformations. A transformation which preserves the structure of space—and if we define this structure in the Helmholtz way, that would figures mean that it carries any two congruent these laws were not invariant throughout with into two congruent ones—is called an auto- respect to reflection. Ernst Mach tells of the morphism by the mathematicians. Leibniz intellectual shock he received when he learned

recognized that this is the idea underlying as a boy that a magnetic needle is deflected in the geometric concept of similarity. An a certain sense, to the left or to the right, if automorphism carries a figure into one that suspended parallel to a wire through which

in Leibniz' words is "indiscernible from it if an electric current is sent in a definite direc-

each of the two figures is considered by itself." tion (Fig. 14). Since the whole geometric What we mean then by stating that left and and physical configuration, including the right are of the same essence is the fact that electric current and the south and north poles

reflection in a plane is an automorphism. of the magnetic needle, to all appearances,

Space as such is studied by geometry. But are symmetric with respect to the plane E oc- space is also the medium of all physical laid through the wire and the needle, the currences. The structure of the physical needle should react like Buridan's ass between

18 19 equal bundles of hay and refuse to decide space and time considers motion a proof between left and right, just as scales of equal of the creation of the world out of God's arms with equal weights neither go down on arbitrary will, for otherwise it would be in- their left nor on their right side but stay explicable why matter moves in this rather horizontal. But appearances are sometimes than in any other direction. Leibniz is loath deceptive. Young Mach's dilemma was the to burden God with such decisions as lack result of a too hasty assumption concerning "sufficient reason." Says he, "Under the as- the effect of reflection in E on the electric sumption that space be something in itself it current and the positive and negative mag- is impossible to give a reason why God should poles of the needle: while we know a netic have put the bodies (without tampering with priori how geometric entities fare under their mutual distances and relative positions) reflection, we have to learn from nature how just at this particular place and not some- the physical quantities behave. And this is where else; for instance, why He should not plane we find : under reflection in the E what have arranged everything in the opposite the electric current preserves its direction, but order by turning East and West about. If, the magnetic south and north poles are inter- on the other hand, space is nothing more than changed. Of course this way out, which re- the spatial order and relation of things then establishes the equivalence of left and right, is the two states supposed above, the actual possible only because of the essential equality one and its transposition, are in no way dif- of positive and negative magnetism. All ferent from each other . . . and therefore it doubts were dispelled when one found that is a quite inadmissible question to ask why the magnetism of the needle has its origin in one state was preferred to the other." By molecular electric currents circulating around pondering the problem of left and right Kant the needle's direction; it is clear that under was first led to his conception of space and reflection in the plane E such currents time as forms of intuition. 5 Kant's opinion change the sense in which they flow. seems to have been this: If the first creative The net result is that in all physics nothing act of God had been the forming of a left has shown up indicating an intrinsic differ- hand then this hand, even at the time when ence of left and right. Just as all points and it could be compared to nothing else, had all directions in space are equivalent, so are the distinctive character of left, which can left and right. Position, direction, left and only intuitively but never conceptually be ap- right are relative concepts. In language tinged prehended. Leibniz contradicts: According with theology this issue of relativity was dis- to him it would have made no difference if cussed at great length in a famous controversy God had created a "right" hand first rather between Leibniz and Clarke, the latter a than a "left" one. One must follow the clergyman acting as the spokesman for world's creation a step further before a differ- Newton. 4 Newton with his belief in absolute ence can appear. Had God, rather than

4 See G. W. Leibniz, Pkilosophische Schriften, ed. 5 Besides his "Kritik der rcinen Vernunft" see es- Gerhardt (Berlin 1875 scq.), vu, pp. 352-440, in pecially §13 of the Prolegomena zu einer jeden kunftigen particular Leibniz' third letter, §5. Metaphysik. . .

20 21 making first a left and then a right hand, started with a right hand and then formed another right hand, He would have changed the plan of the universe not in the first but in the second act, by bringing forth a hand which was equally rather than oppositely oriented to the first-created specimen. Scientific thinking sides with Leibniz. Mythical thinking has always taken the con-

trary view as is evinced by its usage of right and left as symbols for such polar opposites as good and evil. You need only think of the double meaning of the word right itself. In this detail from Michelangelo's famous Crea-

tion of Adam from the Sistine Ceiling (Fig. 15) God's right hand, on the right, touches

life into Adam's left.

People shake right hands. Sinister is the FIG. 15 Latin word for left, and heraldry still speaks

of the left side of the shield as its sinister side. I remember a lecture Heinrich Wolfflin But sinistrum is at the same time that which once delivered in Zurich on "Right and left is evil, and in common English only this paintings"; together with an article on figurative meaning of the Latin word sur- in 6 problem of inversion (Umkehrung) in vives. Of the two malefactors who were "The Raphael's tapistry cartoons," you now find it crucified with Christ, the one who goes with printed in abbreviated form in his Gedanken Him to paradise is on His right. St. Matthew,

Kunstgeschichte, 1941 . By a number of ex- Chapter 25, describes the last judgment as zur amples, as Raphael's Sistine Madonna and follows: "And he shall set the sheep on his Rembrandt's etching Landscape with the three right hand but the goats on the left. Then Wolfflin tries to show that right in shall the King say unto them on his right trees, painting has another Stimmungswert than left. hand, Come ye, blessed of my Father, in- Practically all methods of reproduction inter- herit the Kingdom prepared for you from change left and right, and it seems that former the foundation of the world. . . . Then he times were much less sensitive than we are shall say also unto them on the left hand, such inversion. (Even Rembrandt Depart from me, ye cursed, into everlasting toward did not hesitate to bring his Descent from fire, prepared for the devil and his angels." the Cross as a converse etching upon the market.) Considering that we do a lot more 6 I am not unaware of the strange fact that as a reading than the people, say, of the sixteenth terminus technicus in the language of the Roman augurs century, this suggests the hypothesis that the sinistrum had just the opposite meaning of propitious.

22 23 1 to the inversion of time as they are with re- difference pointed out by Wolfflin is connected spect to the interchange of left and right. with our habit of reading from left to right. Leibniz made it clear that the temporal modi far as I remember, he himself rejected this As past and future refer to the causal structure of as well as a number of other psychological the world. Even if it is true that the exact forward in the discussion explanations put "wave laws" formulated by quantum physics after his lecture. The printed text concludes are not altered by letting time flow backward, the remark that the problem "obviously with the metaphysical idea of causation, and with has deep roots, roots which reach down to it the one way character of time, may enter the very foundations of our sensuous nature." physics through the statistical interpretation part disinclined to take the I for my am of those laws in terms of probability and matter that seriously. 7 particles. Our present physical knowledge equivalence of In science the belief in the leaves us even more uncertain about the upheld even in the left and right has been equivalence or non-equivalence of positive of certain biological facts presently to face and negative electricity. It seems difficult to which seem to suggest their be mentioned devise physical laws in which they are not strongly than does inequivalence even more intrinsically alike; but the negative counter- deviation of the magnetic needle which the part of the positively charged proton still re- same problem shocked young Mach. The mains to be discovered. and of equivalence arises with respect to past This half-philosophical excursion was interchanged by inverting future, which are needed as a background for the discussion of and with respect to the direction of time, the left-right symmetry in nature; we had electricity. In these cases, positive and negative to understand that the general organization it is perhaps clearer especially in the second, of nature possesses that symmetry. But one that a priori evi- than for the pair left-right will not expect that any special object of na- dence is not sufficient to settle the question; ture shows it to perfection. Even so, it is sur- empirical facts have to be consulted. the prising to what extent it prevails. There be sure, the role which past and future To must be a reason for this, and it is not far to play in our consciousness would indicate seek: a state of equilibrium is likely to be sym- the past knowable their intrinsic difference— metric. More precisely, under conditions future unknown and and unchangeable, the which determine a unique state of equilib- taken and still alterable by decisions now— rium the symmetry of the conditions must that this difference has its one would expect carry over to the state of equilibrium. There- of nature. But basis in the physical laws fore tennis balls and stars are spheres; the laws of which we can boast a reasonably those earth would be a sphere too if it did not rotate certain knowledge are invariant with respect around an axis. The rotation flattens it at 7 Cf. also A. Faistauer, "Links und rechts im the poles but the rotational or cylindrical sym- Bilde," Amicis, Jahrbuch der osterreichischen Galerie, metry around its axis is preserved. The fea- 1926, p. 77; Julius v. Schlosser, "Intorno alia lettura dei quadri," Critica 28, 1930, p. 72; Paul Oppe, ture that needs explanation is, therefore, "Right and left in Raphael's cartoons," Journal of not the rotational symmetry of its shape but the Warburg and Courtauld Institutes 7, 1944, p. 82. 25 24 the deviations from this symmetry as exhibited is definitely wrong and Clarke on the right by the irregular distribution of land and water track. But it would have been more sincere and by the minute crinkles of mountains on to deny the principle of sufficient reason al- its surface. It is for such reasons that in his together instead of making God responsible monograph the left-right problem on in for all that is unreason in the world. On the zoology Wilhelm Ludwig says hardly a word other hand Leibniz was right against Newton about the origin of bilateral the symmetry and Clarke with his insight into the principle prevailing in the animal the kingdom from of relativity. The truth as we see it today is echinoderms upward, but in great detail dis- this: The laws of nature do not determine cusses all sorts of secondary asymmetries uniquely the one world that actually exists, superimposed upon the symmetrical ground not even if one concedes that two worlds plan. 8 I quote: "The human body like that arising from each other by an automorphic of the other vertebrates is basically built transformation, i.e., by a transformation bilateral-symmetrically. All asymmetries oc- which preserves the universal laws of nature, curring are of secondary character, and the are to be considered the same world. more important ones affecting the inner If for a lump of matter the overall sym- organs are chiefly conditioned by the neces- metry inherent in the laws of nature is sity for the intestinal tube to increase its limited by nothing but the accident of its surface out of proportion to the growth of position P then it will assume the form of a the body, which lengthening led an asym- to sphere around the center P. Thus the lowest metric folding and rolling-up. And in the forms of animals, small creatures suspended course of phylogenetic evolution these first in water, are more or less spherical. For asymmetries concerning the intestinal system forms fixed to the bottom of the ocean the di- with its appendant organs brought about rection of gravity is an important factor, nar- asymmetries in other organ systems." It is rowing the set of symmetry operations from all well known that the heart of mammals is an rotations around the center P to all rotations asymmetric screw, as shown by the schematic about an axis. But for animals capable of drawing of Fig. 16. FIG. 16 self-motion in water, air, or on land both the If nature were all lawfulness then every postero-anterior direction in which their phenomenon would share the full symmetry body moves and the direction of gravity are of the universal laws of nature as formulated of decisive influence. After determination of by the theory of relativity. The mere fact the antero-posterior, the dorso-ventral, and that this is not so proves that contingency is an thereby of the left-right axes, only the dis- essential feature of the world. Clarke in his tinction between left and right remains controversy with Leibniz admitted the latter's arbitrary, and at this stage no higher sym- principle of sufficient reason but added that metry than the bilateral type can be expected. the sufficient reason often lies in the mere will Factors in the phylogenetic evolution that of God. I think, here Leibniz the rationalist tend to introduce inheritable differences be- 8 W. Ludwig, Rechts-links-Problem im Tierreick und tween left and right are likely to be held in beim Menschen, Berlin 1932. check by the advantage an animal derives

26 27 from the bilateral formation of its organs of in a laevo- and dextro-form, each form being motion, cilia or muscles and limbs: in case of a mirror image of the other, like left and their asymmetric development a screw-wise right hands. A substance which is optically instead of a straight-forward motion would active, i.e., turns the plane of polarized light naturally result. This may help to explain either left or right, can be expected to crystal- why our limbs obey the law of symmetry lize in such asymmetric forms. If the laevo- more strictly than our inner organs. Aristo- form exists in nature one would assume that phanes in Plato's Symposium tells a different the dextro-form exists likewise, and that in story of how the transition from spherical to the average both occur with equal fre- bilateral symmetry came about. Originally, quencies. In 1848 Pasteur made the dis- he says, man was round, his back and sides covery that when the sodium ammonium salt forming a circle. To humble their pride and of optically inactive racemic acid was recrys- might Zeus cut them into two and had tallized from an aqueous solution at a lower Apollo turn their faces and genitals around; temperature the deposit consisted of two and Zeus has threatened, "If they continue kinds of tiny crystals which were mirror insolent I will split them again and they images of each other. They were carefully shall hop around on a single leg." separated, and the acids set free from the The most striking examples of symmetry one and the other proved to have the same in the inorganic world are the crystals. The chemical composition as the racemic acid, gaseous and the crystalline are two clear-cut but one was optically laevo-active, the other states of matter which physics finds relatively dextro-active. The latter was found to be easy to explain; the states in between these identical with the tartaric acid present in two extremes, like the fluid and the plastic fermenting grapes, the other had never be- states, are somewhat less amenable to theory. fore been observed in nature. "Seldom," In the gaseous state molecules move freely says F. M. Jaeger in his lectures On the around in space with mutually independent principle of symmetry and its applications in natural random positions and velocities. In the science, "has a scientific discovery had such far- crystalline state atoms oscillate about posi- reaching consequences as this one had." tions of equilibrium as if they were tied to Quite obviously some accidents hard to them by elastic strings. These positions of control decide whether at a spot of the solu- equilibrium form a fixed regular configuration a laevo- or dextro-crystal comes into tion in space. What we mean by regular and being; and thus in agreement with the sym- how the visible symmetry of crystals derives metric and optically inactive character of from the regular atomic arrangement will be the solution as a whole and with the law of explained in a subsequent lecture. While chance the amounts of substance deposited most of the thirty-two geometrically possible in the one and the other form at any moment systems of crystal symmetry involve bilateral of the process of crystallization are equal or symmetry, not all of them do. Where it very nearly equal. On the other hand na- is not involved we have the possibility of ture, in giving us the wonderful gift of grapes so-called enantiomorph crystals which exist so much enjoyed by Noah, produced only one

28 29 of the forms, and it remained for Pasteur to of the failure of all attempts to "activate" by produce the other! This is strange indeed. mere chemical means optically inactive It is a fact 9 that most of the numerous car- material, it is understandable that Pasteur bonic compounds occur in nature in one, clung to the opinion that the production of either the laevo- or the dextro-form only. single optically active compounds was the The sense in wrote, which a snail's shell winds is an very prerogative of life. In 1860 he inheritable character founded in its genetic "This is perhaps the only well-marked line constitution, as is the "left heart" and the of demarkation that can at present be drawn winding of the intestinal duct in the species between the chemistry of dead and living Homo sapiens. This does not exclude that in- matter." Pasteur tried to explain his very

inversions occur, e.g. situs inversus of the intes- first experiment where racemic acid was tines of man occurs with a frequency of about transformed by recrystallization into a mix- 0.02 per cent; we shall come back to that ture of laevo- and dextro-tartaric acid by the later Also on his ! the deeper chemical constitution action of bacteria in the atmosphere of our human body shows that we have a neutral solution. It is quite certain today screw, a screw that is turning the same way that he was wrong; the sober physical ex- in every one of us. Thus our body contains planation lies in the fact that at lower temper- the dextro-rotatory form of glucose and laevo- ature a mixture of the two oppositely active

rotatory form offructose. A horrid manifesta- tartaric forms is more stable than the in- tion of difference this genotypical asymmetry is a meta- active racemic form. If there is a bolic disease called phenylketonuria, leading in principle between life and death it does not to insanity, that man contracts when a small lie in the chemistry of the material substra- quantity certain ever since of laevo-phenylalanine is added to tum; this has been fairly his food, while the dextro-form has no such Wohler in 1828 synthesized urea from purely disastrous effects. To the asymmetric chem- mineral material. But even as late as 1898 ical constitution of living organisms one must F. R. Japp in a famous lecture on "Stereo- attribute the success of Pasteur's method of chemistry and Vitalism" before the British isolating the laevo- and dextro-forms of sub- Association upheld Pasteur's view in the stances by means of the enzymatic action of modified form: "Only the living organisms, bacteria, conception moulds, yeasts, and the like. Thus or the living intelligence with its he found that an originally inactive solution of symmetry can produce this result (i.e. asym- of some racemate became gradually laevo- metric compounds)." Does he really mean rotatory if intelligence that, by devis- Penicillium glaucum was grown in it. that it is Pasteur's Clearly its great sur- the organism selected for its nutri- ing the experiment but to own ment that form of the tartaric acid molecule prise, creates the dual tartaric crystals? Japp asym- which best suited its own asymmetric chem- continues, "Only asymmetry can beget ical constitution. The image of lock and key 9 has There is known today one clear instance, the been used to illustrate this specificity of reaction of nitrocinnaminacid with bromine where the action of organisms. circular-polarized light generates an optically active In view of the facts mentioned and in view substance.

30 31 metry." The truth of that statement I am the sun. But neither the earth's rotation nor willing to admit; but it is of little help since the combined magnetic fields of earth and there is no symmetry in the accidental past sun are of immediate help in this regard. and present set-up of the actual world which Another possibility would be to assume that begets the future. development actually started from an equal There is however a real difficulty: Why distribution of the enantiomorph forms, but should nature produce only one of the doub- that this is an unstable equilibrium which lets of so many enantiomorphic forms the under a slight chance disturbance tumbled origin of which most certainly lies in living over. organisms? Pascual Jordan points to this From the phylogenetic problems of left fact as a support for his opinion that the be- and right let us finally turn to their onto- ginnings of life are not due to chance events genesis. Two questions arise: Does the first which, once a certain stage of evolution is division of the fertilized egg of an animal into reached, are apt to occur continuously now two cells fix the median plane, so that one here now there, but rather to an event of of the cells contains the potencies for its left, quite singular and improbable character, the other for its right half? Secondly what occurring once by accident and then starting determines the plane of the first division? I an avalanche byautocatalytic multiplication. begin with the second question. The egg of Indeed had the asymmetric protein molecules any animal above the protozoa possesses from found in plants and animals an independent the beginning a polar axis connecting what origin in many places at many times, then develops into the animal and the vegetative their laevo- and dextro-varieties should show poles of the blastula. This axis together with nearly the same abundance. Thus it looks the point where the fertilizing spermatozoon as if there is some truth in the story of Adam enters the egg determines a plane, and it and Eve, if not for the origin of mankind would be quite natural to assume that this then for that of the primordial forms of life. is the plane of the first division. And indeed It was in reference to these biological facts there is evidence that it is so in many cases. when I said before that if taken at their face Present opinion seems to incline toward the value they suggest an intrinsic difference assumption that the primary polarity as well between left and right, at least as far as the as the subsequent bilateral symmetry come constitution of the organic world is concerned. about by external factors actualizing poten- But we may be sure the answer to our riddle tialities inherent in the genetic constitution. does not lie in any universal biological laws In many instances the direction of the polar but in the accidents of the genesis of the axis is obviously determined by the attach- organismic world . Pascual Jordan shows one ment of the oozyte to the wall of the ovary, way out; one would like to find a less radical and the point of entrance of the fertilizing one, for instance by reducing the asymmetry sperm is, as we said, at least one, and often of the inhabitants on earth to some inherent, the most decisive, of the determining factors though accidental, asymmetry of the earth for the median plane. But other agencies itself, or of the light received on earth from may also be responsible for the fixation of the 32 33 one and the other. In the sea-weed Fucus that a single blastomere isolated from its light or electric fields or chemical gradients partner in the two-cell stage develops into a determine the polar axis, and in some insects whole gastrula differing from the normal one

and cephalopods the median plane appears to only by its smaller size. Here are Driesch's be fixed by ovarian influences before fertiliza- famous pictures. It must be admitted that

• ° dis- tion . The underlying constitution on which this is not so for all species. Driesch's these agencies work is sought by some biolo- covery led to the distinction between the ac- gists in an intimate preformed structure, of tual and the potential destiny of the several which we do not yet have a clear picture. parts of an egg. Driesch himself speaks of Thus Conklin has spoken of a spongioplasmic prospective significance (prospektive Bedeu- framework, others of a cytoskeleton, and as tung), as against prospective potency (pros- there is now a strong tendency among bio- pektive Potenz); the latter is wider than the chemists to reduce structural properties to former, but shrinks in the course of develop- fibers, so much so that Joseph Needham in ment. Let me illustrate this basic point by

his Terry Lectures on Order and life (1936) another example taken from the determina-

dares the aphorism that biology is largely tion of limb-buds of amphibia. According the study of fibers, one may expect them to to experiments performed by R. G. Harrison, find that that intimate structure of the egg who transplanted discs of the outer wall of consists of a framework of elongated protein molecules or fluid crystals.

We know a little more about our first ques-

tion whether the first mitosis of the cell di-

vides it into left and right. Because of the fundamental character of bilateral symmetry

the hypothesis that this is so seems plausible enough. However, the answer cannot be an

unqualified affirmation. Even if the hypothesis should be true for the normal develop-

ment we know from experiments first performed by Hans Driesch on the sea urchin 10 Julian S. Huxley and G. R. de Beer in their classical Elements of embryology (Cambridge Uni- versity Press, 1934) give this formulation (Chapter xrv, Summary, p. 438): "In the earliest stages, the egg acquires a unitary organization of the gradient- FIG. 17 field type in which quantitative differentials of one or more kinds extend across the substance of the egg Experiments on pluripotence in Echinus. in one or more directions. The constitution of the ai and bi. Normal gastrula and normal pluteus. egg predetermines it to be able to produce a gradient- a2 and b 2 . Half-gastrula and half-pluteus, ex- field of a particular type; however, the localization pected by Driesch. of the gradients is not predetermined, but is brought a3 and b3. The small but whole gastrula and about by agencies external to the egg." pluteus, which he actually obtained.

34 35 the body representing the buds of future parts of whose nervous system are asym- limbs, the antero-posterior axis is determined metric. First the fertilized egg splits into a at different a time when transplantation may still cell / and a smaller P of obviously invert the dorso-ventral and the medio-lateral nature (Fig. 18). In the next stage they di- axes; thus at this stage the opposites of left and vide along two perpendicular planes into right still belong to the prospective potencies /' + I" and Pi + P2 respectively. There- of the discs, and it depends on the influence after the handle Pi + P2 turns about so that of the surrounding tissues in which way this P2 comes into contact with either P or/";

potency will be actualized. call the one it contacts B, the other A. We now have a sort of rhomboid and roughly

APi is the antero-posterior axis and PPi the dorsal-ventral one. Only the next division which along a plane perpendicular to the one separating A and B splits A as well as B into

symmetric halves A = a -f- oc, B = b + /3, is that which determines left and right. A further slight shift of the configuration destroys this bilateral symmetry. The question arises whether the direction of the two consecutive

shifts is a chance event which decides first FIG. 18 between anterior and posterior and then dorsal anieriqr....p between left and right, or whether the con-

stitution of the egg in its one-cell stage contains specific agents which determine the direction of these shifts. The hypothesis of the mosaic egg favoring the second hypothesis seems more likely for the species Ascaris. There are known a number of cases of poster/or genotypical inversion where the genetic con- d stitutions of two species are in the same relation as the atomic constitutions of two enantiomoi ph crystals. More frequent, how-

Driesch's violent encroachment on the ever, is phenotypical inversion. Left-hand-

normal development proves that the first edness in man is an example. I give another cell division may not fix left and right of the more interesting one. Several Crustacea of growing organism for good. But even in the lobster type have two morphologically

normal development the plane of the first and functionally different claws, a bigger A division may not be the median. The first and a smaller a. Assume that in normally stages of cell division have been closely developed individuals of our species, A is the studied for the worm Ascaris megalocephala, right claw. If in a young animal you cut off

36 37 the right claw, inversive regeneration takes place: TRANSLATORY, ROTATIONAL, AND the left claw develops into the bigger form A while at the place of the right claw a small SYMMETRIES one of type a is regenerated. One has RELATED to infer from such and similar experiences the bipotentiality of plasma, namely that all gen- erative tissues which contain the potency of an asymmetric character have the potency of bringing forth both forms, so however that in normal development always one form develops, the left or the right. Which one is genetically determined, but abnormal external circumstances may cause inversion. On the basis of the strange phenomenon of inversive regeneration Wilhelm Ludwig developed the hypothesis that the decisive factors in asymmetry may not be such specific potencies as, say, the development of a "right claw of type A," but two R and L (right and left) agents which are distributed in the organism with a certain gradient, the concentration of one falling off from right to left, the other in the opposite direction. The essential point is that there is not one but that there are two opposite gradient fields R and L. Which is produced in greater strength is determined by the genetic constitution. If, however, by some damage to the prevalent agent the other previously suppressed one becomes prevalent, then inversion takes place. Being a mathematician and not a biologist I report with the utmost caution on these matters, which seem to me of highly hypothetical nature. But it is clear that the contrast of left and right is connected with the deepest problems concerning the phylogenesis as well as the ontogenesis of organisms.

38 TRANSLATORS ROTATIONAL, AND RELATED SYMMETRIES

From bilateral we shall now turn to other kinds of geometric symmetry. Even in dis- cussing the bilateral type I could not help drawing in now and then such other symmetries as the cylindrical or the spherical ones. It seems best to fix the underlying general concept with some precision beforehand, and to that end a little mathematics is needed, for which I ask your patience. I have spoken of transformations. A mapping S of space associates with every space point p a point p' as its image. A special such mapping is the identity / carrying every point /; into itself. Given two mappings S, T, one can perform one after the other: if S carries p into p' and T carries p' into p" then the resulting mapping, which we denote by ST, carries p into p". A mapping may have an inverse S' such that SS' = / and S'S = I; in other words, if S carries the arbitrary point p into p' then S' carries p' back into p, and a similar condition prevails with S' performed in the first and 6" in the second place. For such a one-to-one mapping S the word transformation was used in the first lecture; let the inverse be denoted by $~~\ Of course, the identity / is a transformation, and / itself is its inverse. Reflection in a plane, the basic operation of bilateral symmetry, is such that its iteration SS results in the identity; in other words, it is its own inverse. In general composition of mappings is not commutative; ST need not be the same as TS. Take for instance a point o in a plane and let S be a

41 horizontal translation carrying o into 0\ and say that the automorphisms form a group. Any 90°. T a rotation around o by Then ST totality, any set T of transformations form a carries o into the point 02 (Fig- 19), but TS group provided the following conditions are

carries into 0* If S is a transformation with satisfied: (1) the identity / belongs to F; (2) l is transforma- -1 the inverse S~~\ then S~ also a if S belongs to T then its inverse S does; of tion and its inverse is S. The composite (3) if S and T belong to T then the composite two transformations ST is a transformation ST does. 1 -I -1 again, and (ST)' equals T S (in this One way of describing the structure of order!). With this rule, although perhaps space, preferred by both Newton and Helm- its expression, are not with mathematical you holtz, is through the notion of congruence. all familiar. When you dress, it is not imma- Congruent parts of space V, V are such as terial in which order you perform the opera- can be occupied by the same rigid body in in dressing start with the lions; and when you two of its positions. If you move the body shirt and end up with the coat, then in un- from the one into the other position the par- dressing you observe the opposite order; first ticle of the body covering a point p of V will take off the coat and the shirt comes last. afterwards cover a certain point p' of V, and

thus the result of the motion is a mapping p —* p' of V upon V. We can extend the rigid body either actually or in imagination so as to cover an arbitrarily given point p of FIG. 19 °1 space, and hence the congruent mapping — /; > p' can be extended to the entire space. I have further spoken of a special kind of Any such congruent transformation— I call transformations of space called similarity by it by that name because it evidently has an the geometers. But I preferred the name of inverse />'—>/>—is a similarity or an auto- automorphisms for them, defining them with morphism; you can easily convince yourselves Leibniz as those transformations which leave that this follows from the very concepts. It the structure of space unchanged. For the is evident moreover that the congruent trans- moment it is immaterial wherein that struc- formations form a group, a subgroup of the ture consists. From the very definition it is group of automorphisms. In more detail the clear that the identity / is an automorphism, situation is this. Among the similarities if l and S is, so is the inverse S~ . Moreover there are those which do not change the the composite ST of two automorphisms S, T dimensions of a body; we shall now call them is again an automorphism. This is only an- congruences. A congruence is either proper, other way of saying that (1) every figure is carrying a left screw into a left and a right similar to itself, if figure F' is similar to (2) one into a right, or it is improper or reflexive, then is similar to F', if is similar F F and (3) F changing a left screw into a right one and F' to F' and to F" then F is similar to F". vice versa. The proper congruences are The mathematicians have adopted the word those transformations which a moment ago group to describe this situation and therefore we called congruent transformations, con-

42 43 and length as AA', in other words the vector necting the positions of points of a rigid BB' = AA'. 1 The translations form a group; body before and after a motion. We shall indeed the succession of the two translations now call them simply motions (in a non- kinematic geometric sense) and call the AB, BC results in the translation AC. improper congruences reflections, after the What has all this to do with symmetry?

most important example: reflection in a It provides the adequate mathematical

plane, by which a body goes over into its language to define it. Given a spatial con- mirror of space image. Thus we have this step-wise figuration ft, those automorphisms arrangement: similarities —* congruences = which leave g unchanged form a group T, similarities without change of scale —* motions = proper congruences. The congruences form a subgroup of the similarities, the motions form a subgroup of the group of con-

gruences, of index 2. The latter addition

means that if B is any given improper con- and this group describes exactly the symmetry pos- gruence, we obtain all improper congruences sessed by %. Space itself has the full sym- in the form BS by composing B with all pos- metry corresponding to the group of all sible proper congruences S. Hence the automorphisms, of all similarities. The sym- FIG. 21 proper congruences form one half, and the metry of any figure in space is described by a improper ones another half, of the group of all subgroup of that group. Take for instance congruences. But only the first half is a the famous pentagram by which Dr. Faust group; for the composite AB of two improper banned Mephistopheles the devil. It is congruences A, B is a proper congruence. carried into itself by the five proper rotations A congruence leaving the point fixed around its center 0, the angles of which are may be called rotation around 0; thus there multiples of 360°/5 (including the identity), are proper and improper rotations. The and then by the five reflections in the lines joining with the five vertices. These ten operations form a group, and that group tells us what sort of symmetry the pentagram

1 While a segment has only length, a vector has FIG. 20 length and direction. A vector is really the same thing as a translation, although one uses different phrase- ologies for vectors and translations. Instead of speaking of the translation a which carries the point A rotations around a given center form a and in- group. The simplest type of congruences into A' one speaks of the vector a = AA'; phrase: the translation a carries A into are the translations. A translation may be stead of the a A' one says that A' is the end point of the vector laid off from B ends represented by a vector AA'; for if a transla- laid off from A. The same vector translation carrying A into A' carries B tion carries a point A into A' and the point in B' if the into B'. B into B' then BB' has the same direction 45 44 3 • For such reasons as these we shall possesses. Hence the natural generalization a , groups of con- which leads from bilateral symmetry to sym- almost exclusively consider have to do with actually metry in this wider geometric sense consists gruences—even if we such as in replacing reflection in a plane by any group or potentially infinite configurations of automorphisms. The circle in a plane band ornaments and the like. consid- with center and the sphere in space around After these general mathematical some special have the symmetry described by the group erations let us now take up are important in of all plane or spatial rotations respectively. groups of symmetry which operation which defines If a figure % does not extend to infinity then art or nature. The mirror reflection, is es- an automorphism leaving the figure invariant bilateral symmetry, operation. A must be scale-preserving and hence a con- sentially a one-dimensional reflected in any of its gruence, unless the figure consists of one straight line can be reflection carries a point P into point only. Here is the simple proof. Had points 0; this same distance from we an automorphism leaving g unchanged, that point P' that has the but lies on the other side. Such reflections changing the scale, then either this but improper congruences of the one- automorphism or its inverse would increase are the only dimensional line, whereas its only proper (and not decrease) all linear dimensions in a congruences are the translations. Reflection certain proportion a:\ where a is a number greater in followed by the translation OA yields than 1. Call that automorphism S, reflection in that point A\ which halves the and let a, /3 be two different points of our figure distance OA. A figure which is invariant g. They have a positive distance d. translation / shows what in the art Iterate the transformation S, under a of ornament is called "infinite rapport," i.e. S = S\ = 2 = SS . S . , SSS S\ . . repetition in a regular spatial rhythm. A

pattern invariant under the translation t is The H-times iterated transformation S" carries l 2 3 t its iterations t , t , , a and /3 into two also invariant under points

n • • • under the identity t° = I, whose distance , moreover is d a . With increasing _1 under the inverse / of t and its itera- exponent n this distance tends to infinity. and

2 3 • • • . If t shifts the line But if figure tions r\ r , r , our J is bounded, there is a by the amount a then I" shifts it by the amount number c such that no two points of % have a distance greater than c. Hence a contradic- na (n = 0, ±1, ±2, )• tion arises as soon as n becomes so large that

• characterize a translation / by d a" > c. The argument shows another Hence if we the iteration or thing: Any finite group of automorphisms the shift a it effects then consists t" is characterized by the multiple na. exclusively of congruences. For if power 6" carrying into itself a given it contains an that enlarges linear dimen- All translations sions at pattern of infinite rapport on a straight line the ratio a : 1, a > 1, then all the

l 2 z multiples na of one basic infinitely many • • • are in this sense iterations S ,S ,S , con- This rhythmic may be com- tained in the group would be different be- translation a. symmetry. If so the cause they enlarge at different scales 2 bined with reflexive a\ a , 46 47 centers of reflections follow each other at half the distance Y2 a. Only these two types of symmetry, as illustrated by Fig. 22, are possible for a one-dimensional pattern or "ornament." (The crosses X mark the centers of reflection.)

*- a

FIG. 22

Of course the real band ornaments are not strictly one-dimensional, but their sym-

metry as far as we have described it now

makes use of their longitudinal dimension FIG. 24 only. Here are some simple examples from Monreale mosaic of the Lord's Ascension Greek art. The first (Fig. 23) which shows (Fig. 10), but this time drawing your atten- a very frequent motif, the palmette, is of tion to the band ornaments framing it. The type i (translation reflection). The next + widest, carried out in a peculiar technique, (Fig. 24) are without reflections (type n). later taken up by the Cosmati, displays the This frieze of Persian bowmen from Darius' translatory symmetry only by repetition of palace in Susa (Fig. 25) is pure translation; the outer contour of the basic tree-like motif, but you should notice that the basic transla- FIG. 25 tion covers twice the distance from man to man because the costumes of the bowmen alternate. Once more I shall point out the FIG. 23

48 49 while each copy is filled by a different highly plaits of some sort, the design of which sug- symmetric two-dimensional mosaic. The gests that one strand crosses the other in palace of the doges in Venice (Fig. 26) may space (and thus makes part of it invisible). stand for translator)' symmetry in archi- If this interpretation is accepted, further tecture. Innumerable examples could be operations become possible; for example, re- added. flection in the plane of the ornament would As I said before, band ornaments really change a strand slightly above the plane into consist of a two-dimensional strip around a one below. All this can be thoroughly central line and thus have a second trans- analyzed in terms of group theory as is for instance done in a section of Andreas Spciser's book, Theorie der Gruppen von endlicher Ordnung, quoted in the Preface. In the organic world the translatory symmetry, which the zoologists call metamerism,

is seldom as regular as bilateral symmetry

frequently is. A maple shoot and a shoot of Angraecum distichum (Fig. 27) may serve 2 as examples. In the latter case translation

is accompanied by longitudinal slip reflection. Of course the pattern does not go on into infinity (nor does a band ornament),

but one may say that it is potentially infinite at least in one direction, as in the course of time ever new segments separated from each other by a bud come into being. Goethe said of the tails of vertebrates that they allude 28

as it were to the potential infinity of organic existence. The central part of the animal shown in this picture, a scolopendrid (Fig. 28), FIG. 26 possesses fairly regular translational, cem- versal dimension. As such they can have bincd with bilateral, symmetry, the basic further symmetries. The pattern may be operations of which are translation by one carried into itself by reflection in the central segment and longitudinal reflection. line /; let us distinguish this as longitudinal In one-dimensional time repetition at equal

reflection from the transversal reflection in a intervals is the musical principle of rhythm. line perpendicular to /. Or the pattern may As a shoot grows it translates, one might say, be carried into itself by longitudinal reflec- a slow temporal into a spatial rhythm. Re- tion combined with the translation by %a 2 This and the next picture are taken from Studium (longitudinal slip reflection). A frequent Generate, p. 249 and p. 241 (article by W. Troll, motif in band ornaments are cords, strings, or "Symmetriebetrachtung in der Biologic").

50 flection, inversion in time, plays a far less But such schemes fall hardly under the important part in music than rhythm 3 does. heading of symmetry. A melody changes its character to a consid- We return to symmetry in space. Take a erable degree if played backward, I, and who band ornament where the individual section am a poor musician, find it hard to recog- repeated again and again is of length a and nize reflection when it is used in the construc- cir- sling it around a circular cylinder, the tion of a fugue; it certainly has no such spon- cumference of which is an integral multiple taneous effect as rhythm. All musicians of a, for instance 25a. You then obtain a agree that underlying the emotional element pattern which is carried over into itself of music is a strong formal element. It may through the rotation around the cylinder axis be that it is capable of some such mathe- by a = 360°/25 and its repetitions. The matical treatment as has proved successful 360°, twenty-fifth iteration is the rotation by for the art of ornaments. If so, we have or the identity. We thus get a finite group probably not yet discovered the appropriate of rotations of order 25, i.e. one consisting of mathematical tools. This would not be so 25 operations. The cylinder may be re- surprising. For after all, the Egyptians ex- placed by any surface of cylindrical sym- celled in the ornamental art four thousand Birkhoff 3 The reader should compare what G. D. years before the mathematicians discovered and music has to say on the mathematics of poetry in Lecture i, note 1. the group concept the proper mathematical in the two publications quoted in instrument for the treatment of ornaments and for the derivation of their possible symmetry classes. Andreas Speiser, who has taken a special interest in the group-theoretic aspect of ornaments, tried to apply combina- torial principles of a mathematical nature ; also to the formal problems of music. There is a chapter with this title in his book, "Die mathematische Denkweise," (Zurich, 1932). As an example, he analyzes Beethoven's pastoral sonata for piano, opus 28, and he also points to Alfred Lorenz's investigations on the formal structure of Richard Wagner's

chief works. Metrics in poetry is closely related, and here, so Speiser maintains, science has penetrated much deeper. A common principle in music and prosody

seems to be the configuration a a b which is

often called a bar: a theme a that is repeated

and then followed by the "envoy" b; strophe, antistrophe, and epode in Greek choric lyrics. FIG. 29 FIG. 30

52 53 metry, namely by one that is carried into itself by all rotations around a certain axis, for instance by a vase. Fig. 29 shows an attic vase of the geometric period which displays quite a number of simple ornaments of this

type. The principle of symmetry is the same, although the style is no longer "geometric," in this Rhodian pitcher (Fig. 30), Ionian school of the seventh century b.c. Other illustrations are such capitals as these from early Egypt (Fig. 31). Any finite group of proper rotations around a point in a plane, or around a given axis in space, contains a

primitive rotation t whose angle is an aliquot part 360% of the full rotation by 360°, and consists of 2 /»-', its iterations . . t\ t , . , n t = identity. The order n completely characterizes this .group. The result follows from the analogous fact that any group of trans-

lations of a line, provided it contains no operations arbitrarily near to the identity except the identity itself, consists of the itera-

FIG. 31

FIG. 32

tions va of a single translation a (v = 0, ±1,

±2, • • • )• The wooden dome in the Bardo of Tunis, once the palace of the Beys of Tunis (Fig. 32), may serve as an example from interior architecture. The next picture (Fig. 33) takes you to Pisa; the Baptisterium with the tiny- looking statue of John the Baptist on top is a central building in whose exterior you can distinguish six horizontal layers each of rotary symmetry of a different order n. One could make the picture still more impressive by adding the leaning tower with its six gal- leries of arcades all having rotary symmetry

54 55 "

the round arcs of the friezes, octagonal central

symmetry (n = 8, a low value compared to those embodied in the several layers of the Pisa Baplisterium) in the small rosette and the three towers, while bilateral symmetry rules the structure as a whole as well as almost every detail.

Cyclic symmetry appears in its simplest

form if the surface of fully cylindrical sym-

metry is a plane perpendicular to the axis. We then can limit ourselves to the two-

FIG. 34 * it n i^ as SV'SM^

FIG. 33

of the same high order and the dome itself, the exterior of whose nave displays in columns and friezes patterns of the lineal translatory

type of symmetry while the cupola is sur- rounded by a colonnade of high order rotary symmetry.

An entirely different spirit speaks to us from the view, seen from the rear of the choir, of the Romanesque cathedral in Mainz, Germany (Fig. 34). Yet again repetition in

56 57 '

FIG. 35

dimensional plane with a center 0. Mag- nificent examples of such central plane symmetry are provided by the rose windows of Gothic cathedrals with their brilliant-colored

glasswork. The richest I remember is the rosette of St. Pierre in Troyes, France, which

is based on the number 3 throughout. Flowers, nature's gentlest children, are also conspicuous for their colors and their cyclic

symmetry. Here (Fig. 35) is a picture of an iris with its triple pole. The symmetry of 5 is most frequent among flowers. A page like the following (Fig. 36) from Ernst

58 Haeckel's Kunstjormen der Natur seems to indicate that it also occurs not infrequently among the lower animals. But the biologists warn me that the outward appearance of

these echinoderms of the class of Ophiodea is to a certain degree deceptive; their larvae are organized according to the principle of bilateral symmetry. No such objection at- taches to the next picture from the same source (Fig. 37), a Discomedusa of octagonal symmetry. For the coelentera occupy a place in the phylogenetic evolution where cyclic has not yet given way to bilateral symmetry. Haeckel's extraordinary work, in which his interest in the concrete forms of organisms finds expression in countless draw-

ings executed in minutest detail, is a true nature's codex of symmetry. Equally re- vealing for Haeckel, the biologist, are the

thousands and thousands of figures in his Challenger Monograph, in which he describes for the first time 3,508 new species of radiolarians discovered by him on the Challenger Expedition, 1887. One should not forget these accomplishments over the often all-too-

speculative phylogenetic constructions in which this enthusiastic apostle of Darwinism indulged, and over his rather shallow mate- rialistic philosophy of monism, which made quite a splash in Germany around the turn of the century.

Speaking of Medusae I cannot resist the temptation of quoting a few lines from D'Arcy Thompson's classic work on Growth and Form, a masterpiece of English literature, which combines profound knowledge in geometry, physics, and biology with humanistic erudi- tion and scientific insight of unusual origi- nality. Thompson reports on physical experiments with hanging drops which serve

60 FIG. 37 ^

of to illustrate by analogy the formation medusae. "The living medusa," he says, "has geometrical symmetry so marked and regular as to suggest a physical or mechanical element in the little creatures' growth and construction. It has, to begin with, its vortex-like bell or umbrella, with its symmetrical handle or manubrium. The bell four or in is traversed by radial canals, ten- multiples of four; its edge is beset with regular tacles, smooth or often beaded, at intervals or of graded sizes; and certain sensory structures, including solid concre- tions or 'otoliths,' are also symmetrically interspersed. No sooner made, then it begins to pulsate; the bell begins to 'ring.' Buds, miniature replicas of the parent- organism, are very apt to appear on the tentacles, or on the manubrium or sometimes on the edge of the bell; we seem to see one vortex producing others before our eyes. The development of a medusoid deserves to be studied without prejudice from this point of view. Certain it is that the tiny medusoids of Obelia, for instance, are budded off with a rapidity and a complete perfection which suggests an automatic and all but instantane- ous act of conformation, rather than a gradual process of growth." While pentagonal symmetry is frequent in the organic world, one does not find it among the most perfectly symmetrical crea- tions of inorganic nature, among the crystals. There no other rotational symmetries are possible than those of order 2, 3, 4, and 6. Snow crystals provide the best known speci- mens of hexagonal symmetry. Fig. 38 shows some of these little marvels of frozen water. In my youth, when they came down from heaven around Christmastime blanketing the

63 —

landscape, they were the delight of old and tion from absolute symmetry in their colum- young. Now only the skiers like them, while nar structures." 5 they have become the abomination of motor- Up to now we have paid attention to ists. Those versed in English literature will proper rotations only. If improper rotations remember Sir Thomas Browne's quaint ac- are taken into consideration, we have the two count in his Garden of Cyrus (1658) of hexag- following possibilities for finite groups of onal and "quincuncial" symmetry which rotations around a center in plane geom-

"doth neatly declare how nature Geome- etry, which correspond to the two possibilities trizeth and observeth order in all things." we encountered for ornamental symmetry on One versed in German literature will re- a line: (1) the group consisting of the repeti- member how Thomas Mann in his Magic tions of a single proper rotation by an aliquot Mountain* describes the "hexagonale Un- part a = 360°/w of 360°; (2) the group of wesen" of the snow storm in which his hero, these rotations combined with the reflections

Hans Castorp, nearly perishes when he falls in n axes forming angles of J^a. The first asleep with exhaustion and leaning against group is called the cyclic group Cn and the a barn dreams his deep dream of death and second the dihedral group D n . Thus these love. An hour before when Hans sets out are the only possible central symmetries in on his unwarranted expedition on skis he two-dimensions: enjoys the play of the flakes "and among • • Cz, • • ;D D 2 ,D,, • . these (1) C, C2, h myriads of enchanting little stars," so he philosophizes, "in their hidden splendor, Ci means no symmetry at all, D\ bilateral too small for man's naked eye to see, there symmetry and nothing else. In architecture was not one like unto another; an endless the symmetry of 4 prevails. Towers often inventiveness governed the development and have hexagonal symmetry. Central build- unthinkable differentiation of one and the ings with the symmetry of 6 are much less same basic scheme, the equilateral, equi- frequent. The first pure central building angled hexagon. Yet each in itself—this was after antiquity, S. Maria degli Angeli in the uncanny, the antiorganic, the life-denying Florence (begun 1434), is an octagon. character of them all—each of them was ab- Pentagons are very rare. When once before solutely symmetrical, icily regular in form. I lectured on symmetry in Vienna in 1937 I

They were too regular, as substance adapted said I knew of only one example and that a to life never was to this degree—the living very inconspicuous one, forming the passage- principle shuddered at this perfect precision, way from San Michele di Murano in Venice found it deathly, the very marrow of death 8 Diirer considered his canon of the human figure Hans Castorp felt he understood now the more as a standard from which to deviate than as a toward which to strive. Vitruvius' tem- reason why the builders of antiquity pur- standard peralurae seem to have the same sense, and maybe posely and secretly introduced minute varia- the little word "almost" in the statement ascribed to 4 1 quote Helen Lowe- Porter's translation, Knopf Polykleitos and mentioned in Lecture i, note 1, New York, 1927 and 1939. points in the same direction.

64 65 —

to the hexagonal Capella Emiliana. Now, of course, we have the Pentagon building in

Washington. By its size and distinctive shape, it provides an attractive landmark for bombers. Leonardo da Vinci engaged in systematically determining the possible symmetries of a central building and how to attach chapels and niches without destroying the symmetry of the nucleus. In abstract

modern terminology, his result is essentially our above table of the possible finite groups of rotations (proper and improper) in two dimensions.

So far the rotational symmetry in a plane had always been accompanied by reflective symmetry; I have shown you quite a number FIG. 39 of examples for the dihedral group Dn and for none the simpler cyclic group Cn . But this is more or less accidental. Here (Fig.

39) are two flowers, a geranium (i) with the symmetry group Ds, while Vinca herbacea (n) has the more restricted group Cs owing to the asymmetry of its petals. Fig. 40 shows what is perhaps the simplest figure with rotational symmetry, the tripod (n = 3). When FIG. 41 FIG. 40 one wants to eliminate the attending reflective symmetry, one puts little flags unto the short time before Hitler's hordes occupied

arms and obtains the triquetrum, an old Austria, I added concerning the swastika: a magic symbol. The Greeks, for instance, "In our days it has become the symbol of used it with the Medusa's head in the center terror far more terrible than the snake-girdled as the symbol for the three-cornered Sicily. Medusa's head"—and a pandemonium of in (Mathematicians are familiar with it as the applause and booing broke loose the seal on the cover of the Rendiconti del Circolo audience. It seems that the origin of the Matematico di Palermo.) The modification magic power ascribed to these patterns lies

with four instead of three arms is the swastika, in their startling incomplete symmetry which need not be shown here—one of the rotations without reflections. Here (Fig. of the most primeval symbols of mankind, common 41) is the gracefully designed staircase possession of a number of apparently inde- pulpit of the Stephan's dome in Vienna; a pendent civilizations. In my lecture on triquetrum alternates with a swastika-like symmetry in Vienna in the fall of 1937, a wheel.

66 67 i.e. if during equal time intervals always the So much about rotational symmetry in two same motion is repeated. Then the fluid is dimensions. If dealing with potentially in- in "uniform motion." The simple group finite patterns like band ornaments or with law infinite groups, the operation under which S(h)S(h) = S(h + h) the pattern is invariant is not of necessity a congruence but could be a similarity. A expresses that the motions during two con- similarity in one dimension that is not a mere secutive time intervals l u h result in the translation has a motion fixed point and is a motion during the time t\ + t 2 . The dilatation s from in a certain ratio a:\ during 1 minute leads to a definite trans- where a 9* 1. It is no essential restriction formation s = S(\), and for all integers n the to assume a > 0. Indefinite iteration of motion S(n) performed during n minutes is this operation generates n 2 a group 2 consisting the iteration s : the discontinuous group of the dilatations consisting of the iterations of s is embedded in the continuous group with the parameter t (2) (n = Q, ±1, ±2, • • • ). consisting of the motions S(t). One could A good example of this type of symmetry say that the continuous motion consists of the is shown by the shell of Turrilella duplicate endless repetition of the same infinitesimal (Fig. 42). It is really quite remarkable motion in consecutive infinitely small time how exactly the widths of the consecutive intervals of equal length. whorls of this shell follow the law of geometric We could have applied this consideration progression. to the rotations of a plane disc as well as to The hands of some clocks perform a con- dilatations. We now envisage any proper tinuous uniform rotation, others jump from similarity s, i.e. one which does not inter- minute to minute. The rotations by an change left and right. If, as we assume, it is integral number of FIG. 42 minutes form a discon- not a mere translation, it has a fixed point tinuous subgroup within the continuous and consists of a rotation about combined group of all rotations, and it is natural to with a dilatation from the center 0. It can consider a rotation s and its iterations (2) as be obtained as the stage S{\) reached after contained in the continuous group. We can 1 minute by a continuous process S(t) of com- apply this viewpoint to any similarity in 1, bined uniform rotation and expansion. or 2, 3 dimensions, as a matter of fact to any This process carries a point t* along a transformation s. The continuous motion of so-called logarithmic or equiangular spiral. a space-filling substance, a "fluid," can This curve, therefore, shares with straight mathematically be described by giving the line and circle the important property of transformation U(t,l') which carries the going over into itself by a continuous group position Pt of any point of the fluid at the of similarities. The words by which James

moment / over into its . position Pt at the time Bernoulli had the spira mirabilis adorned on /'. These transformations form a one- his tombstone in the Munster at Basle, parameter group if U(l,l') depends on the "Eadem mutata resurgo," are a grandilo- time difference/' - /only, U(t/) = S(l' - /), 68 69 The most general rigid motion in three-

dimensional space is a screw motion s, combination of a rotation around an axis with a translation along that axis. Under the influence of the corresponding continuous uniform motion any point not on the axis describes a screw-line or helix which, of course, could say of itself with the same right as the logarithmic spiral: eadem resurgo. The stages P n which the moving point reaches at the equi-

FIG. 43

quent expression of this property. Straight line and circle are limiting cases of the logarithmic spiral, which arise when in the combination rotation-plus-dilatation one of the two components happens to be the identity. The stages reached by the process at the times

• t = n • • (3) = , -2, -1,0,1,2,

form the group consisting of the iterations (2). The well-known shell of Nautilus (Fig. 43) shows this sort of symmetry to an astonishing perfection. You see here not only the continuous logarithmic spiral, but the potentially infinite sequence of chambers has a symmetry described by the discontinuous group 2. For everybody looking at this picture (Fig. 44) of a giant sunflower, Hehanthus maximus, the florets will naturally

arrange themselves into logarithmic spirals, two sets of spirals of opposite sense of coiling.

70 distant moments are (3) equidistributed over stem of a plant with its leaves, a fir-cone with the helix like stairs on a winding staircase. its scales, and the discoidal inflorescence of

If the angle of rotation of the j- operation is a Helianthus with its florets. Where one can fraction p/v of the full 360° angle expressible check the numbers (4) best, namely for the

in terms of small integers /x, v then every i>th arrangement of scales on a fir-cone, the ac- point of the sequence lies Pn on the same curacy is not too good nor are considerable

vertical, and y. full turnings of the screw are deviations too rare. P. G. Tait, in the necessary to get from to Pn the point Pn+V Proceedings of the Royal Society of Edinburgh above it. The leaves around the shoot of a (1872), has tried to give a simple explanation, plant often show such a regular spiral ar- while A. H. Church in his voluminuous rangement. Goethe spoke of a spiral tend- treatise Relations of phyllotaxis to mechanical ency in nature, and under the name of laws (Oxford, 1901-1903) sees in the arith- phyllolaxis this phenomenon, since the days metics of phyllotaxis an organic mystery. I of Charles Bonnet (1754), has been the sub- am afraid modern botanists take this whole ject of much investigation and more specula- doctrine of phyllotaxis less seriously than tion botanists. 6 among One has found that their forefathers. the fractions n/v representing the screw-like Apart from reflection all symmetries so far arrangement of leaves quite often are mem- considered are described by a group consist- bers of the "Fibonacci sequence" ing of the iterations of one operation s. In the most (4) 13 one case, and that is undoubtedly H, y2 , H, H, H, Hs, Ai, %, important, the resulting group is finite, which results from the expansion into a con- namely if one takes for s a rotation by an tinued_ fraction of the irrational number angle a = 360°/n which is an aliquot part — 1). This number is no other M(V5 but of the full rotation 360°. For the two-di- the ratio known as the aurea sectio, which has mensional plane there are no other finite played such a role in attempts to reduce groups of proper rotations than these; witness beauty of proportion to a mathematical • • • Leonardo's the first line, Ch C2, C 3, of formula. The cylinder on which the screw table (1). The simplest figures which have is wound could be replaced by a cone; this the corresponding symmetry are the regular amounts to replacing the screw motion s by polygons: the regular triangle, the square, any proper similarity rotation — combined the regular pentagon, etc. The fact that

with dilatation. The arrangement of scales • there is for every number n = 3, 4, 5, • • on a fir-cone falls under this slightly more a regular polygon of n sides is closely related general form of symmetry in phyllotaxis. to the existence for every n of a rotational The transition from cylinder over cone to group of order n in plane geometry. Both disc is obvious, illustrated by the cylindrical facts are far from trivial. Indeed, the situa- 6 This phenomenon plays also a role in J. Ham- tion in three dimensions is altogether differ- bidge's constructions. His Dynamic symmetry contains ent: exist infinitely many regular on pp. 146-157 detailed notes by the mathematician there do not R. C. Archibald on the logarithmic spiral, golden polyhedra in 3-space, but not more than five, section, and the Fibonacci series. often called the Platonic solids because they

72 73 Greeks never used the word "symmetric" play an eminent role in Plato's natural phi- in our modern sense. In common usage losophy. They are the regular tetrahedron, (xvfj.fi€Tpos means proportionate, while in Euclid the cube, the octahedron, moreover the it is equivalent to our commensurable: side and pentagondodecahedron, the sides of which diagonal of a square are incommensurable are twelve regular pentagons, and the icosa- quantities, durvnnerpa neykdr). hedron bounded by twenty regular triangles. Here (Fig. 45) is a page from Haeckel's One might say that the existence of the first Challenger monograph showing the skeletons three is a fairly trivial geometric fact. But

the discovery of the last two is certainly one of the most beautiful singular discoveries and FIG. 45 made in the whole history of mathematics.

With a fair amount of certainty, it can be traced to the colonial Greeks in southern Italy. The suggestion has been made that they ab- stracted the regular dodecahedron from the crystals of pyrite, a sulphurous mineral abundant in Sicily. But as mentioned before, the symmetry of 5 so characteristic for the regular dodecahedron contradicts the laws of crystallography, and indeed one finds that the pentagons bounding the dodecahedra in which pyrite crystallizes have 4 edges of equal, but one of different, length. The

first exact construction of the regular penta-

gondodecahedron is probably due to Theaete-

tus. There is some evidence that dodecahedra were used as dice in Italy at a very early time and had some religious significance in Etruscan culture. Plato, in the dialogue Timaeus, associates the regular pyramid, octahedron, cube, icosahedron, with the four

elements of fire, air, earth, and water (in this order), while in the pentagondodecahedron he sees in some sense the image of the universe as a whole. A. Speiscr has advo- cated the view that the construction of the

five regular solids is the chief goal of the deductive system of geometry as erected by the Greeks and canonized in Euclid's Ele-

ments. May I mention, however, that the 75 74 .

of several Radiolarians. Nr. 2, 3, and 5 are about the three outer planets, Uranus, Nep- octahedron, icosahedron, and dodecahedron tune, and Pluto, which were discovered in in astonishingly regular form; 4 seems to have 1781, 1846, and 1930 respectively.) He tries a lower symmetry. to find the reasons why the Creator had chosen this order of the Platonic solids and draws parallels between the properties of the planets (astrological rather than astro- physical properties) and those of the corresponding regular bodies. A mighty hymn in which he proclaims his credo, "Credo spatioso numen in orbe," concludes his book.

We still share his belief in a mathematical harmony of the universe. It has withstood the test of ever widening experience. But we no longer seek this harmony in static forms like the regular solids, but in dynamic laws. As the regular polygons are connected with the finite groups of plane rotations, so must the regular polyhedra be intimately related to the finite groups of proper rotations around a center in space. From the study of plane rotations we at once obtain two types

FIG 46 of proper rotation groups in space. Indeed,

the group Cn of proper rotations in a horizontal plane around a center can be inter- Kepler, in his Myslerium cosmographicum, preted as consisting of rotations in space published in 1595, long before he discovered around the vertical axis through 0. Reflec- the three laws bearing his name today, made tion of the horizontal plane in a line / of the an attempt to reduce the distances in the plane can be brought about in space through planetary system to regular bodies which are a rotation around / by 180° (Umklappung) alternatingly inscribed and circumscribed to You may remember that we mentioned this spheres. Here (Fig. is 46) his construction, in connection with the analysis of a Sumerian by which he believed he had penetrated picture (Fig. 4). In this way the group deeply into the secrets of the Creator. The is into a D n in the horizontal plane changed six spheres correspond to the six planets, in space; it con- group D'n of proper rotations Saturn, Jupiter, Mars, Earth, Venus, Mer- tains the rotations around a vertical axis curius, separated in this order by cube, through by the multiples of 360°/n and tetrahedron, dodecahedron, octahedron, ico- the Umklappungen around n horizontal axes sahedron. (Of course, Kepler did not know through which form equal angles of

76 77 icosahedron) respectively. Their orders, i.e. 360°/2/z with each other. But it should be the number of operations in each of them, observed that the group D\ as well as C2 are 12, 24, 60 respectively. consists of the identity and the Umklappung simple It can be shown by a relatively around one line. These two groups are addition analysis (Appendix A) that with the therefore identical, and in a complete list of complete: of these three groups our table is the different groups of proper rotations in three

• • dimensions, D\ should be omitted if C2 is Cn (n = 1, 2, 3, •),

kept. Hence we start our list thus: • • •); (5) D'n (n = 2, 3, T, W, P. C2, • • Ci, Cs, C4, ;

£>' •• • the tabula- D' Z>' . equivalent to 2, 3 , 4 , This is the modern the Greeks. tion of the regular polyhedra by D'2 is the so-called four-group consisting of the These groups, in particular the last three, identity and the Umklappungen around three subject for are an immensely attractive mutually perpendicular axes. geometric investigation. For each one the of five regular bodies we improper What further possibilities arise if can construct the group of those proper rota- groups? rotations are also admitted to our tions which carry that body into itself. Does making use This question is best answered by this give rise to five new groups? No, only rotation, of one quite singular improper to three, and that for the following reason. point namely reflection in 0; it carries any Inscribe a sphere into a cube and an octa- respect to found into its antipode P' with hedron into the sphere such that P the corners the by joining P with and prolonging of the octahedron lie where the sides of the length: PO = straight line P0 by its own cube touch the sphere, namely in the centers with every OP'. This operation Z commutes of the six square sides. (Fig. 47 shows the let T be one of rotation S, regular tetrahedron, the cube (or octa- manner you get a group V'T which contains hedron), and the pentagondodecahedron (or 79 78 F while the other half of its operations are ORNAMENTAL SYMMETRY improper. For instance, T = C„ is a sub- of index group 2 of V = D'n ; the operations S' of Dn not contained in Cn are the Umklap- pungen around the n horizontal axes. The corresponding ^S' are the reflections in the vertical planes perpendicular to these axes.

Thus D„Cn consists of the rotations around the vertical axis, the angles of which are multiples of 360%?, and of the reflections in vertical planes through this axis forming angles of 360°/2tz with each other. You

might say that this is the group formerly

denoted by Dn - Another example, the

simplest of all: T = Cx is contained in V' = d. The one operation S' of C2 not contained in Ci is the rotation by 180° about the vertical

axis; <5" is reflection in the horizontal plane

through 0. Hence C>Ci is the group consisting of the identity and of the reflection in a given plane; in other words, the group to which bilateral symmetry refers. The two ways described are the only ones by which improper rotations may be included in our groups. (For the proof see Appendix

B.) Hence this is the complete table of all finite groups of (proper and improper) rotations:

c c,„ C/2nC/n (n = • n , 1, 2, 3,

D' n , D'n , D'na,

• (n = 2, 3, •

r, W, P T, W, P; WT.

The last group WT is made possible by the fact that the tetrahedral group T is a subgroup of index 2 of the octahedral group W. This list will be of importance to us when in the last lecture we shall consider the symmetry of crystals.

80 ORNAMENTAL SYMMETRY

This lecture will have a more systematic character than the preceding one, in as much special kind of as it will be dedicated to one geometric symmetry, the most complicated angle. but also the most interesting from every In two dimensions the art of surface ornaments deals with it, in three dimensions it characterizes the arrangement of atoms in a the orna- crystal. We shall therefore call it mental or crystallographic symmetry. Let us begin with an ornamental pattern occurs in two dimensions which probably more frequently than any other, both in art and nature: the hexagonal pattern so often used for tiled floors in bathrooms. You honeycomb as it see it here realized by the hivebees (Fig. 48). is built by our common the The bees' cells have prismatic shape, of these photograph is taken in the direction prisms. As a matter of fact, a honeycomb prisms consists of two layers of such cells, the the other of the one layer facing one way, ends of the opposite way. How the inner these two layers dovetail is a spatial problem which we shall presently take up. At the moment we are concerned with the simpler two-dimensional question. If you pile round- of shots or round beads in a heap they will themselves get arranged in the three-dimen- configura- sional analogue of the hexagonal pack tion. In two dimensions the task is to equal circles as compactly as possible. You circles that start with a horizontal row of touch each other. If you drop another will nest circle from above upon this row it between two adjacent circles of the row,

83 circumscribing the circle, and if you replace and the centers of the three circles will form circle by this hexagon you obtain the an equilateral triangle. From this upper each configuration of hexagons filling the circle there derives a second horizontal row of regular whole plane. circles nesting between those of the first row; According to the laws of capillarity a soap and so on (Fig. 49). The circles leave little spanned into a given contour made of lacunae between them. The tangents of a film thin wire assumes the shape of a minimal sur- circle at the points where it touches the six smaller area than any other surrounding circles form a regular hexagon face, i.e. it has

FIG. 48

FIG. 49

surface with the same contour. A soap bubble into which a quantum of air is blown assumes spherical form because the sphere encompasses the given volume with a minimum of surface. Thus is it not astonishing of that a froth of two-dimensional bubbles equal area will arrange itself in the hexagonal pattern because among all divisions of the plane into parts of equal area that is the one for which the net of contours has minimum length. We here suppose that the problem has been reduced to two dimensions by dealing with a horizontal layer of bubbles, say, between two horizontal glass plates. If the froth of vesicles has a boundary (an epidermal layer, as the biologist would say), we observe

84 85 FIG. 50

that it consists of circular arcs each forming

an angle of 120° with the adjacent cell wall FIG. 52 and the next arc, as is required by the law of minimal length. After this explanation surprised to find the hexagonal FIG. 51 one will not be pattern realized in such different structures of maize as for instance the parenchyma of our eyes, (Fig. 50), the retinal pigment which I the surface of many diatoms, of show here (Fig. 51) a beautiful specimen, the bees, and finally the honeycomb. As build their which are all of nearly equal size, cells will cells gyrating around in them, the form a densest packing of parallel circular cylinders which in cross section appear as our hexagonal pattern of circles. As long as the bees are at work the wax is in a semi-fluid proba- state, and thus the forces of capillarity from bly more than the pressures exerted within by the bees' bodies transform the (whose circles into circumscribed hexagons of corners however still show some remains the circular form). With the parenchym cel- of maize you may compare this artificial diffusion lular tissue (Fig. 52) formed by the in gelatin of drops of a solution of potassium

87 ferrocyanide. The regularity leaves something to be desired; there are even places where a pentagon is smuggled in instead of a hexagon. Here (Fig. 53 and 54) are two other artificial tissues of hexagonal pattern taken at random from a recent issue of Vogue (February 1951). The siliceous skeleton of

FIG. 53

FIG. 54

FIG. 55 one of Haeckel's Radiolarians which he called Aulonia hexagona (Fig. 55) seems to exhibit a

fairly regular hexagonal pattern spread out not in a plane but over a sphere. But a hexagonal net covering the sphere is impossible owing to a fundamental formula of topology. This formula refers to an arbitrary partition of the sphere into countries that border on each other along certain edges. It tells that the number A of coun-

tries, the number E of edges and the number C of corners (where at least three countries come together) satisfy the relation A + C — E = 2. Now for a hexagonal net we would

89 ;

- have E = 3A,C = 2A and hence A + C E in 1712 seems to be the first to have carried = see that some 0! And sure enough, we out fairly exact measurements, and he found of the meshes of the net of Aulonia are not that the three bottom rhombs of the cell have hexagons but pentagons. an obtuse angle a of about 110° and that the From the densest packing of circles in a angle /3 they form with the prism walls has plane let us now pass to the densest packing the same value. He asked himself the of equal spheres, of equal balls in space. We geometric question what the angle a of the start with one ball and a plane, the "hori- rhomb has to be so as to coincide exactly its center. densest zontal plane" through In with the latter angle 0. He finds a = /3 = packing this ball will touch on twelve others 109° 28' and thus assumes that the bees had ("like the seeds in a pomegranate," as Kepler solved this geometric problem. When prin- says), six in the horizontal plane, three below, ciples of minimum were introduced into the 1 and three above. If mutual penetration is study of curves and into mechanics, the idea arrangement, impossible the balls in this was not farfetched that the value of a is deter- under uniform expansion around their fixed mined by the most economical use of wax; centers, will change into rhombic dode- with every other angle more wax would be cahedra that fill the whole space. Mark that needed to form cells of the same volume. the individual dodecahedron is not a regular This conjecture of Reaumur was confirmed solid—whereas in the corresponding two- by the Swiss mathematician Samuel Koenig. dimensional problem a regular hexagon re- Somehow Koenig took Maraldi's theoretical sulted! The bees' cell consists of the lower value for the one he had actually measured, half of such a dodecahedron with the six and finding that his own theoretical value vertical sides so prolonged as to form a based on the minimum principle deviated hexagonal prism with an open end. Much from it by 2' (owing to an error of the tables has been written on this question of the he used in computing \/2) he concluded geometry of the honeycomb. The bee's that the bees commit an error of less than strange social habits and geometric talents 2' in solving this minimum problem, of which could not fail to attract the attention and he says that it lies beyond the reach of clas- admiration their excite the of human ob- sical geometry and requires the methods of servers and exploiters. "My house," says Newton and Leibniz. The ensuing discus- the bee in the Arabian Nights, "is constructed sion in the French Academy was summed up according to the laws of a most severe archi- by Fontenelle as Secretaire perpetuel in a tecture; and Euclid himself could learn from famous judgment in which he denied to the studying the geometry of my cells." Maraldi bees the geometric intelligence of a Newton

1 The arrangement is uniquely determined only if and Leibniz but concluded that in using the one requires the centers to form a lattice. For the highest mathematics they obeyed divine definition of a lattice see p. 96; for a fuller dis- guidance and command. In truth the cells cussion of the problem: D. Hilbert and S. Cohn- regular as Koenig assumed, it Vossen, Anschauliche Geometrie, Berlin, 1 932, pp. 40-41 are not as arid H. Minkowski, Diophantiscke Approximationen would be difficult to measure the angles even Leipzig, 1907, pp. 105-111. within a few degrees. But more than a

90 91 of surface hundred years later Darwin still spoke of the hedra gives even a better economy plane-faced bees' architecture as "the most wonderful of in relation to volume than the inclined to known instincts" and adds: "Beyond this rhombic dodecahedra. I am configuration gives stage of perfection in architecture natural believe that Lord Kelvin's know, selection (which now has replaced divine the absolute minimum; but so far as I guidance!) could not lead; for the comb of this has never been proved. the three-dimen- the hive-bee, as far as we can see, is abso- Let us now return from plane lutely perfect in economizing labor and wax." sional space to the two-dimensional

FIG. 57

and engage in a more systematic investigation of symmetry with double infinite rapport. First we have to make this notion precise. As was mentioned before, the translations, the parallel displacements of a plane form a group. A translation a can be completely described by fixing the point A' into which

it carries a given point A. The translation When one truncates the six corners of an translation octahedron in a suitable symmetric fashion or vector BB' is the same as the one obtains a polyhedron bounded by 6 AA' if BB' is parallel to AA' and of the same squares and 8 hexagons. This tetrakaideka- length. The composition of translations is hedron was known to Archimedes and redis- usually denoted by the sign +. Thus covered by the Russian crystallographer o + b is the translation resulting from first Fedorow. Copies of this solid obtained by carrying out a and then b. If a carries the suitable translations are capable of filling the point A into B and b carries B into C then whole space without overlappings and gaps, thus be c = a + b carries A into C and may just as the rhombic dodecahedron does (Fig. in the 56). In his Baltimore lectures Lord Kelvin indicated by the diagonal vector AC

showed how its faces have to be warped and parallelogram ABCD. Since here AD = BC edges curved to fulfill the condition of have minimal area. If this is done the partition = b and DC = AB = a (Fig. 57), we of space into equal and parallel tetrakaideka- the commutative law a + b = b + a for the

92 93 torture with composition of translations, or, as one also I am sorry that I had to you geometry. The says, for the addition of vectors. This addi- these elements of analytic invention of Descartes' is tion of vectors is nothing else but the law by purpose of this names to the points X in which two forces a, b unite to form a resultant nothing but to give plane by which we can distinguish and a -f- b = c according to the so-called paral- a be done in a lelogram of forces. We have the identity or recognize them. This has to are infinitely null vector which carries every point into systematic way because there it is the more necessary itself, and every translation a has its inverse many of them; and as points, unlike persons, are all completely — a such that a + ( — a) = 0. It is obvious alike, and hence we can distinguish them what 2a, 3a, 4a, . . . stand for, namely names attaching labels to them . The a-f-a, a + a-fa, a + a + a + a, etc. only by to be pairs of numbers The general rule by which the multiple na is we employ happen

defined for every integer n, positive, zero, or (x h Xi). commutative law, the addition negative, is expressed in the formulas Besides the of vectors—as a matter of fact, the composi- (n l)a = (na a and Oa = o. + )+ tion of any transformations—satisfies the as-

The vector b = ^£a is the unique solution of sociative law

the equation 3b = a. Hence it is clear what (a + b) + c = a + (b + c). Xa means if X is a fraction m/n with integral

. of vectors a, b, . . numerator m and denominator n, as for in- For the multiplication law . . . one has the stance % or —${3; and then also by con- by real numbers X, ix, tinuity what it means for any real number X, X(Ma) = (Xm)o whether rational or irrational. Two vectors distributive laws Ci, C2 are linearly independent if no linear and the two combination of them x&j + #262 is the null (X + m)o = (Xa) + (mo), vector unless the two real numbers x\ and X(a + b) = (Xa) + (Xb). *2 are zero. The plane is two-dimensional ask oneself how the coordinates because every vector 5 can be represented One must vector r change as uniquely as such a linear combination (xi, *») of an arbitrary from one vector basis (eg, e 2) to x\t\ + *2e2 in terms of two fixed linearly in- one passes expres- e' . The vectors ci, c' are dependent vectors d, C2. The coefficients another (ci, 2 ) 2 and vice versa: x\, X2 are called the coordinates of r with sible in terms of d, c 2 ,

respect to the basis (eg, C2). After fixing a a-ifa (1) ci = fluCi + 02^2, e'2 = 01261 + definite point as origin (and a vector basis and Ci, C2) we can ascribe to every point X two a' C a' e[ ^22*2 (V) Ci = a'n t'i + n t'2 , 2 = 12 + coordinates xh x 2 by OX = x\ti + ^262, and vice versa these coordinates xi, xz determine Represent the arbitrary vector J in terms of the position of X relative to the "coordinate the one and the other basis:

system" (0, eg, C2). ? = xid + -V2C2 = *iei + x'2 t'2 .

94 95 except for*i = x 2 = 0. There substituting for t[, t' or (1') variables *i, x 2 By (1) 2 for e t , e 2 exist special coordinate systems, the Cartesian one finds that the coordinates xi, x 2 with form assumes the simple respect to the first basis are connected with ones, in which this expression x\ x\; they consist of two mu- the coordinates x[, * 2 in the second system + of equal by the two mutually inverse "homogeneous tually perpendicular vectors d, l-i Cartesian linear transformations" length 1. In metric geometry all coordinate systems are equally admissible. xi = a x = a (2) nx'i + ai2* 2, 2 2 i*i + 022*2; Transition from one to the other is affected f r,/\ I 1,1 I 1,1 by an orthogonal transformation, i.e. by a (2 ) x x = a nxi + a 12*2, * 2 = a 2l xi + a nx 2 . homogeneous linear transformation (2), (2 ) The coordinates x vary with the vector but y; which leaves the form x\ + x\ unaltered,

the coefficients 1 1 2 /2 1 '2 X\ + X\ = *! + X 2 . But with a slight modification such a trans- /in, oi 2\ a 'n, <*[ 2\ ( formation may also be interpreted as the Y*Ms 022/ \0 2 i, a 22/ algebraic expression of a rotation. If by a are constants. It is easy to see under what rotation around the origin the Cartesian linear like circumstances a transformation Cartesian basis e basis Ci, c 2 passes into the lf (2) has an inverse, namely, if and only if its vector r = X1C1 * 2 e 2 g°es into c 2 then the + so-called modul 011^22 — 012021 is different if you write this as r' = x x ci + x2 c 2, and from 0. basis e xiei + x'itz, using the original (d, 2) As long as we use no other concepts than see as frame of reference throughout, you those introduced so far, namely (1) addition that the vector with the coordinates *i, *2 of vectors a b, (2) multiplication of a vec- + goes into that with the coordinates x'u x 2 tor a by a number X, the operation by (3) where which two points A,B determine the vector

x[t x e 2 , x t t[ + x 2 e 2 = x + 2 AB, and concepts logically defined in terms of them, we do affine geometry. In affine and hence

geometry any vector basis d, e 2 is as good 022*2 (4) x[ = OllXl + 012*2, *J = ^21*1 + as any other. The notion of the length |r| [formulas (2) with the pairs (*i, x 2), (*i, *2) of a vector J transcends affine geometry and is basic for metric geometry. The square of interchanged]. If vectors are replaced by points the homo- the length of an arbitrary vector J is a replaced quadratic form geneous linear transformations are throughout by non-homogeneous ones. Let (3) £11*1 + 2guX 1X 2 + £22*2 *' the coordinates of the (*i, *2), (*i, 2) be same arbitrary point X in two coordinate of its coordinates x h x 2 with constant coeffi- we systems c 2 {0'\ e(, c 2 ). Then cients . This is the essential (0; d, ), gi U g i2, g 22 con- tent of Pythagoras' theorem. The metric have

ground form (3) is positive-definite, namely

its value is positive for any values of the OX = *id + *2d, O'X = *'xCi + 44

96 97 in any two such coordinate systems linked by and since OX = 00' + O'X: an orthogonal transformation are, as we shall xi = (5) anx'i + ai2x 2 + bi (i = 1, 2) say, orthogonally equivalent. Hence what Leonardo had done can now be formulated where we have set 00' = Mi + b 2 t 2 . The in algebraic language as follows: He made

non-homogeneous differ from homogeneous up a list of groups of orthogonal transforma-

transformations by the additional terms bi. tions such that (1) any two of the groups in

The mapping his list are orthogonally inequivalent, and

(2) any finite group of orthogonal trans- (6) x'i = anxi + ai2x2 + bi (i = 1, 2) formations is orthogonally equivalent to a carrying the point (xh x 2) into the point group occurring in his list. We say briefly: (x[, x in- 2) expresses a congruence provided He made up a complete list of orthogonally the homogeneous part of the transformation equivalent finite groups of orthogonal transformations. This seems an unnecessarily in- (4) x'i = a (1xi + ^2*2 (* » 1, 2), volved way of stating a simple situation; but

giving the corresponding mapping of the its advantages will presently become evident. vectors, is orthogonal. (Here of course the The symmetry of ornaments is concerned with coordinates refer to the same fixed coordinate discontinuous groups of congruent mappings system.) Under these circumstances we of the plane. If such a group A contains call also the non-homogeneous transforma- translations it would be absurd to postulate tion orthogonal. In particular, a transla- finiteness, for iteration of a translation a tion rise by the vector (b h b 2) is expressed by the (different from the identity 0) gives to

transformation infinitely many translations na (n = 0, ±1, finiteness • • Therefore we replace ** ±2 •)• *1 *l + ha *2 = x-i + b 2 . is no by discontinuity: it requires that there We now return to Leonardo's table of operation in the group arbitrarily close to finite rotation groups in the plane, the identity, except the identity itself. In other words, there is a positive number e Ci, C2, Cz, (7) such that any transformation (6) in our group D D 2 , Z) h 3 , for which the numbers The algebraic expression of the operations an — 1, 012, b of one of the groups C„ does not depend on an, a 22 — 1, b ;)2 the choice of the Cartesian vector basis. ( This is not so for the groups identity (for D n \ here we lie between -e and +e is the normalize the algebraic expression by intro- which all these numbers are zero). The ducing as the first basic vector Ci one that translations contained in our group form a lies in one of the reflection axes. A group discontinuous group A of translations. For of rotations is expressed in terms of the such a group there are three possibilities: Cartesian coordinate identity, system as a group of Either it consists of nothing but the orthogonal transformations. Its expressions the null vector 0; or all the translations in

98 99 of the the group are iterations xt of one basic trans- integral coefficients that has an inverse unimodular by the mathe- lation e 9* a (# « 0, ±1, ±2, • • •); or these same type is called translations (vectors) form a two-dimensional maticians; one easily sees that a linear trans- coefficients is lattice, namely consist of the linear combina- formation with integral tions if and only if its modul X\t\ + # 2 e 2 by integral coefficients Xj, x 2 unimodular equals or — 1. of two linearly independent vectors Ci, C2. fliit»22 — ai2«2i +1 all possible discon- The third case is that of double infinite In order to determine double rapport in which we are interested. Here tinuous groups of congruences with proceed as follows. the vectors Ci,C2 form what we call a lattice infinite rapport we now represent basis. Choose a point as origin; those We choose a point as origin and the lattice points into which goes by all the transla- the translations in our group A by point O. tions of the lattice form a parallelogramatic I of points into which they carry the be consid- lattice of points (Fig. 58). Any operation of our group may ered as a rotation around followed by a translation. The first, the rotary part, then carries the lattice into itself. Moreover these rotary parts form a discontinuous, and {A}. In hence finite group of rotations T = the terminology of the crystallographers it is

this group which determines the symmetry the class of the ornament. T must be one of groups in Leonardo's table (7), FIG. 58

= 1, 2, 3, • • •). (8) Cn , D n (n

operations carry the lattice L To what extent, we will ask at once, is the but one whose between the choice of the lattice basis for a given lattice into itself. This relationship lattice L imposes arbitrary? If e is rotation group T and the 1 ,C 2 another such basis we must have certain restrictions on both of them. from As far as T is concerned, it excludes t (1) x = fluCi -f- fl2iC 2 , e' = and + 02262 2 the table all values of n except n = 1, 2, 3, is among the ex- where the atj are integers. But also the coeffi- 4, 6. Notice that n = 5 the lattice permits the cients in the inverse transformation (1') must cluded values I Since be rotation by 180°, the smallest rotation leaving integers, otherwise el, c'2 would not consti- aliquot part of 180°, tute a lattice basis. For the coordinates one it invariant must be an gets two mutually inverse linear transforma- or of the form

tions (2), (2') with integral coefficients : 360° divided by 2 or 4 or 6 or 8 or .

on are (2") (*»' *A and fa j*\ We must show that the numbers from 8 \a 2 i, 022/ \a a 21 , 22/ impossible. Take the case of n = 8 and let one that A homogeneous linear transformation with A among all lattice points 9*0 be

100 101 A

lattice is may serve as lattice basis. The is nearest to (Fig. 59). Then the whole points are the corners O and the centers • octagon A = Ay, A2, A3, • • • which arises the rectangles. (It was an arrangement from A by rotating the plane around of of trees in such a lozenge lattice which through % of the full angle again and again Thomas Browne called quincuncial, thinking consists of lattice points. Since OAi, OA* the quincunx as its elementary are lattice vectors their difference, the vector of

although the lattice in fact has noth- A1A2, must also belong to our lattice, or the figure, ing to do with the number 5.) Shape and point B determined by OB = A A should X 2 size of the fundamental rectangle or rhomb be a lattice point. this leads to However a are arbitrary. contradiction, since B is nearer to than After having found the 10 possible groups V A = A\, indeed the side A\Ai of the regular of rotations and the lattices L left invariant octagon is smaller than its radius QA\. by each of them, one has to paste together a Hence for the group T we have only these 10 T with a corresponding L so as to obtain possibilities: the full group of congruent mappings. FIG. 59 Closer investigation has shown that, while (9) d, C2 , C3 , C4, C6 ; D h D 2, D 3, D 4 , D,. there are 10 possibilities for T, there are It is easy to see that for each of these groups exactly 17 essentially different possibilities there actually exist lattices left invariant by for the full group of congruences A. Thus the operations of the group. ^here are 17 essentially different kinds of Clearly, for d and & any lattice will do, symmetry possible for a two-dimensional for any lattice is invariant under the identity ornament with double infinite rapport. the rotation 180°. let and by But us con- Examples for all 17 groups of symmetry are sider D\, which consists of the identity and found among the decorative patterns of

the reflection / in an axis through 0. There antiquity, in particular among the Egyptian are two types of lattices left invariant by ornaments. One can hardly overestimate this group: the rectangular and the diamond the depth of geometric imagination and lattices (Fig. 60). The rectangular lattice inventiveness reflected in these patterns. is obtained by dividing the plane into equal Their construction is far from being mathe- rectangles along lines parallel II and perpen- matically trivial. The art of ornament con-

dicular to /. corners of the rectangles The tains in implicit form the oldest piece of are the lattice points. A natural basis for higher mathematics known to us. To be the lattice consists the two sides of Ci, C2 sure, the conceptual means for a complete ab- issuing from of the fundamental rectangle stract formulation of the underlying prob- left-lower corner is whose 0. The diamond lem, namely the mathematical notion of a lattice consists of equal rhombs into which FIG. 60 group of transformations, was not provided the plane is divided by the diagonal lines of before the nineteenth century, and only on the rectangular lattice. The two sides of the this basis is one able to prove that the 17 fundamental rhomb the left corner of which symmetries already implicitly known to the

102 103 Egyptian craftsmen exhaust all possibilities. procedure one takes first the metric structure Strangely enough the proof was carried out into account and thus introduces the Car- only as late as 1924 by George Polya, now tesian coordinate systems in terms of which teaching at Stanford. 2 The Arabs fumbled the metric ground form has a unique normal- around much with the number 5, but they ized expression x\ -f- x\, while in the algebraic were of course never able honestly to insert representation of the continuous manifold a central symmetry of 5 in their ornamental of invariant lattices there remains a variable designs of double infinite rapport. They element. But instead of adapting our co- tried various deceptive compromises, how- ordinates to the metric by admitting Car- ever. One might say that they proved ex- tesian coordinates only, we could put the perimentally the impossibility of a pentagon lattice structure first and adapt the coordi- in an ornament. nates to the lattice by choosing Ci, e 2 as a Whereas it was clear what we meant by lattice basis, with the effect that the lattice saying that there are no other groups of rota- is now normalized in a uniquely determined tions associable with an invariant lattice than manner when expressed in terms of the cor- the 10 groups (9), our assertion of the exist- responding coordinates xj, *2. Indeed the ence of not more than 17 different ornamental lattice vectors are now those whose coordi- groups calls for some explanation. For in- nates are integers. In general we cannot do stance, if T = C\ then the group A is one con- both at the same time: have a coordinate sisting of translations only; but any lattice system in which the metric ground form ap- is possible here, the fundamental parallelo- pears in the normalized form x\ -f- x 2 > and gram spanned by the two basic vectors of the the lattice consists of all vectors with integral lattice may be of any shape and size, we have coordinates *i, xi. We now follow the the choice in a continuously infinite manifold second procedure, which turns out to be of possibilities. In arriving at the number mathematically more advantageous. I con- 17 we count all these as one case only; but sider this analysis as one which is of basic with what right? Here I need my analytic importance for all morphology. geometry. If we look at our plane in the As an example consider once more D%. light of affine geometry it bears two struc- If the invariant lattice is rectangular and the tures: (i) the metric structure by which every lattice basis is chosen in the natural manner vector has a length the square of which y described above, then D\ consists of the is expressed in terms of coordinates by a identity and the operation positive definite quadratic form (3), the

metric ground form; (ii) a lattice structure, x i — x u x 2 = —X2. owing to the fact that the ornament endows the plane with a vector lattice. In the usual The metric ground form can be any positive

2 -+- Cf. his paper "Ueber die Analogie der Kristall- form of the special type a xx\ a-ix\. If the symmetrie in der Ebenc," ^eitschr. j. Kristallographie invariant lattice is a diamond lattice and the 60, pp. 278-282. sides of the fundamental diamond are chosen

104 105 as the lattice basis, then Di contains the any finite group of linear transformations identity and the further operation with real coefficients one may construct positive quadratic forms left invariant under these *i = *2, x 2 = *i- transformations. 3 How many different, i.e. The metric ground form may be any positive unimodularly inequivalent, finite groups of

form of the special type a(x\ + *l) + 2bx\x 2 . linear transformations with integral coeffi- But instead of Di we now obtain two groups cients in two variables, exist then? Ten, D", D\ of linear transformations with integral namely our old friends (9)? No, there are coefficients, which, though orthogonally, are more, since we have seen that D\, for instance, no longer unimodularly equivalent, the one breaks up into two inequivalent cases D\, D v consisting of the two operations with the The same happens to D 2 and D 3, with the

coefficient matrices result that there are exactly 1 3 unimodularly groups of linear operations O inequivalent finite with integral coefficients. From a mathe- n (!4) matical standpoint this is the really interest- the other of ing result rather than the table (9) of the (::> 10 groups of rotations with invariant lattices. In a last step one has to introduce the Two groups of homogeneous linear trans- translational parts of the operations, and one formations are, of course, called unimodularly obtains 17 unimodularly inequivalent dis- equivalent if they both represent the same continuous groups of non-homogeneous lin- group of operations, in of one, the one terms ear transformations which contain all the terms the other in of another lattice basis, translations i.e. if they change into each other by a uni- x[ = b x x b modular transformation of coordinates. xi + u 2 = 2 + 2 In the lattice-adapted coordinate system with integral b\,b 2 and no other translations. the operations of T now appear as homo- This last step offers little difficulty, and the geneous linear transformations with in- (4) remarks that remain to be made are better tegral coefficients aq\ for as each carries the based on the 1 3 finite groups T of homogene- lattice into itself, x[, x assume integral values 2 ous transformations which result from can- whenever integral values are assigned to *i celling the translatory parts. and x^ The arbitrariness in the choice of So far only the lattice structure of the plane the lattice basis finds its expression the by has been taken into account. Of course, agreement to consider unimodularly equiva- the metric of the plane cannot be ignored lent groups of linear transformations as the forever. And it is here that the continuous same thing. Besides having integral coeffi-

3 cients the transformations of V will leave a This is a fundamental theorem due to H. Maschke. The proof is simple enough: Take any positive quad- certain positive definite quadratic form (3) ratic form, e.g. x\ + x\, perform on it each of the invariant. But this is really no additional transformations 5 of our group and add the forms restriction; indeed it be for can shown that thus obtained: the result is an invariant positive form.

106 107 is fully revealed only if one passes from the aspect of the problem comes in. For each truncated homogeneous group T = [A] to of the 13 groups Y there exist invariant the full ornamental group A. The splitting positive quadratic forms into something discrete and something con- = G(x) £n*i + 2gi2*i*2 + gaa*f- tinuous seems to me a basic issue in all morphology, and the morphology of orna- Such a form is characterized by its coefficients ments and crystals establishes a paragon by (gn, £12, £22)- The form G(x) is not uniquely the clearcut way in which this distinction is determined by T; for instance G(x) may be carried out. replaced by any multiple c • G(x) with a real After all these somewhat abstract mathe- positive constant factor c. All positive matical generalities I am now going to show quadratic forms G(x) left invariant by the you a few pictures of surface ornaments with operations of T form a continuous convex double infinite rapport. You find them on "cone" of simple nature and of 1, 2, or 3 wall papers, carpets, tiled floors, parquets, dimensions. For instance in the cases D" all sorts of dress material, especially prints, and D\ we had the two-dimensional manifolds and so forth. Once one's eyes are opened, of all positive forms of the types a\x\ + 02*2 one will be surprised by the numerous sym- and a{x\ -+- x\) + 2bx&% respectively. The metric patterns which surround us in our metric ground form is always one in the daily lives. The greatest masters of the manifold of invariant forms. geometric art of ornament were the Arabs. In the full description of the ornamental The wealth of stucco ornaments decorating groups A we have now clearly divided those the walls of such buildings of Arabic origin features which are discrete, and those capable as the Alhambra in Granada is simply of varying over a continuous manifold. overwhelming. The discrete feature is exhibited by represent- For the purpose of description it is good to ing the group in terms of lattice-adapted know what a congruent mapping in two coordinates and turns out to be one of 17 dimensions looks like. A proper motion may- definite distinct groups. To each of them be either a translation or a rotation around a there corresponds a continuum of possibilities point 0. If such a rotation occurs in our for the metric ground form G(x), from which symmetry group and all the rotations around the one actual metric ground form must be occurring in it are multiples of the rotation picked. The advantage of adapting the co- by 360°/n, we call a pole of multiplicity n ordinate system to the lattice rather than to or simply rc-pole. We know that no other the metric . becomes visible in the fact that values except n = 2, 3, 4, 6 are possible. now the variable element G(x) varies over An improper congruence is either a reflection one simple convex continuous manifold, in a line /, or such a reflection combined with while in terms of the metric-adapted coordi- a translation a along /. If it occurs in our nates the lattice L, which then appears as group, / is called an axis or gliding axis re- the variable element, ranges over a con- spectively. In the latter case iteration of the tinuum that may consist of several parts, as congruence leads to the translation by the the example of Di showed. The advantage

108 109 the symmetry axes. Designs 3', 3a, and 3b reduce the multiplicity of those poles to 3; among them 3' is without symmetry axes, in 3a axes pass through every 3-pole, in 3b only through those (one-third of the whole number) which had been six-fold before. C&, D%, The homogeneous groups are D 6 , Cz, D\ respectively, where Z>£, D\ are the two unimodularly inequivalent forms assumed by coordinate system. D 3 in a lattice-adapted There now follow some actual ornaments of Moorish, Egyptian, and Chinese origin. This window of a mosque in Cairo, of the fourteenth century (Fig. 62), is of the hexa-

FIG. 62

FIG. 61

vector 2a; hence the gliding vector a must be one-half of a lattice vector of our group.

The first picture (Fig. 61) is a drawing of the hexagonal lattice, the discussion of which

started today's lecture. It has a very rich symmetry. There are poles of multiplicities

2, 3, and 6, indicated in the diagram by dots, small triangles, and hexagons respectively.

The vectors joining two 6-poles are the lattice vectors. The lines in the figure are the axes. There are also gliding axes, which have not been shown in the drawing; they run mid-way between and parallel to the axes. The possible symmetry groups of the hexagonal type are five in number and are obtained by putting one of the simple figures 6 or 6' or 3' or 3a or 3b into each of the six-poles. Designs 6 and 6' preserve the multiplicity 6 of these poles, but 6' destroys no 111 in Granada. The group is 3' or gonal class Z) 6 . The elementary figure is a Alhambra or not one takes trefoil knot the various units of which are 6', according to whether is one of the interlaced with superb artistry. Almost un- account of the colors. This art that the interrupted tracks cross the design in the finer tricks of the ornamental pattern as ex- three directions arising from the horizontal symmetry of the geometric by by rotations through 0°, 60°, 120°; the mid- pressed by a certain group A is reduced expressed lines of these tracks are gliding axes. You the coloring to a lower symmetry of the can easily discover lines that are ordinary by a subgroup of A. A symmetry exhibited by this (Fig. 64) axes. Such axes are absent from this square class D 4 is pavements; Azulejos ornament (Fig. 63) decorating the well-known design for brick that no ordi- back of the alcove in the Sala de Camas of the the amusing thing about it is

FIG. 63

FIG. 65 FIG. 64

nary axes, only gliding axes, pass through the 4-poles (one of which is marked). Of the same symmetry is the Egyptian ornament shown next (Fig. 65), as well as the two following Moorish ornaments (Fig. 66). A monumental work on our subject is Owen which Jones' Grammar of ornaments, from some of these illustrations are taken. Of a more special character is the Grammar of which Chinese lattice by Daniel Sheets Dye, deals with the lattice work the Chinese use for repro- the support of their paper windows. I duce here (Figs. 67 and 68) two character- the istic designs from that volume, one of

hexagonal and one of the Z) 4-type.

113 FIG. 68

of these orna- I wish I could analyze some prerequisite for such ments in detail. But a alge- an investigation would be an explicit ornamental braic description of the 17 at was groups. What this lecture aimed mathe- more a clarification of the general morphology matical principles underlying the than a group- of ornaments (and crystals) ornaments. theoretic analysis of individual from Shortness of time has prevented me abstract and doing justice to both sides, the explain the basic the concrete. I tried to some mathematical ideas, and I showed you I indi- pictures: the bridge between them over it step cated, but I could not lead you by step.

FIG. 67

114 115 CRYSTALS THE GENERAL MATHEMATICAL IDEA OF SYMMETRY CRYSTALS THE GENERAL MATHEMATICAL IDEA OF SYMMETRY

In the last lecture we considered for two dimensions the problem of making up a complete list (i) of all orthogonally inequivalent finite groups of homogeneous orthogonal transformations, (ii) of all such groups as have invariant lattices, (iii) of all unimodularly inequivalent finite groups of homo- genous transformations with integral coefficients, (iv) of all unimodularly inequivalent discontinuous groups of non-homogeneous linear transformations which contain the translations with integral coordinates but no other translations.

Problem (i) was answered by Leonardo's list

• C„, Dn (n = I, 2, 3, • •)> values (ii) by limiting the index n in it to the k n = 1, 2, 3, 4, 6. The numbers hi, n , hm, out to hj V of groups in these four lists turned be oo, 10, 13, 17 respectively. The most important problem

is doubtless (iii). One could have posed these same four questions for the one-dimensional line rather than the two-dimensional plane. There the answer would have been very simple, and one would have found all

h , equal to 2. In- the numbers hu kn , hux , JV deed in each of the cases (i), (ii), (iii) the group consists either of the identity x' = x alone, or of the identity and the reflection x' = -x.

119 throughout the nineteenth century. Dirich- But what we are intending now is not to let, Hermite, and more recently Minkowski descend from 2 to 1, but ascend from 2 to and Siegel have contributed to this line of 3 dimensions. All finite groups of rotations research. The investigation of ornamental in 3 dimensions were listed at the end of the symmetry in m dimensions is based on the second lecture; I repeat them here: results won by these authors and on the List A: algebraic and the more refined arithmetic C,„ C„, C2n Cn (n = 1, 2, 3, • •) theory of so-called algebras or hypercomplex

D' , D' last generation n n , D'2n D'n , D'nCn number systems, on which the (n = 2, 3, • •) of algebraists, in this country L. Dickson

T, W, P; T, W, P; WT. above all, have spent much effort. We decorate surfaces with flat ornaments; If one requires that the operations of the art has never gone in for solid ornaments. group leave a lattice invariant, only axes of But they are found in nature. The arrange- rotation of multiplicity 2, 3, 4, 6 are admis- ments of atoms in a crystal are such patterns. sible. And by this restriction our table re- The geometric forms of crystals with their duces to the following plane surfaces are an intriguing phenomenon List B: of nature. However the real physical sym-

C\, C?, Cz, C*, Ct; Ci, C2 , C3, C , Ce; 4 metry of a crystal is not so much shown by D' 2 , Z/ , D\, D' ; £>;, D' , D' , D[; 3 6 3 4 its outward appearance as by the inner C2C1, C4C2, CzCa; physical structure of the crystalline substance.

D' D' , D' D' t 2 6 3 ; Let us suppose that this substance fills the

D' C2 , D' C , D\C 2 3 3 h Dfa; entire space. Its macroscopic symmetry will T, W, T, W, WT. find its expression in a group Y of rotations. of the crystal in space It contains 32 members. One easily con- Only such orientations are carried vinces oneself that each of these 32 groups are physically indistinguishable as rotation of this group. possesses invariant lattices. In three di- into each other by a general propa- the For example, light, which in mensions numbers hu ku , km, hiv have di- the values gates with different velocities in different * , 32, 70, 230. rections in the crystalline medium, will propagate with the same speed in any two In its algebraic formulation our problem directions that arise from each other by a may be posed for any number m of variables, rotation of the group T. So for all other • • • *i, X2, , xm instead of just 2 or 3, and physical properties. For an isotropic me- the corresponding theorems of finiteness the group V consists of all rotations, have been proved. The methods are of the dium a crystal it is made up of a finite greatest mathematical interest. The com- but for number of rotations, sometimes even of bination "metric plus lattice" lies at the but the identity. Early in the his- bottom of the arithmetical theory of quadratic nothing crystallography the law of rational forms which, inaugurated by Gauss, played tory of derived from the arrangement of a central part in the theory of numbers indices was

121 120 '

the plane surfaces of crystals. It led to the While r = {A} describes the manifest macro- hypothesis of the lattice-like atomic structure scopic spatial and physical symmetry, A de- of crystals. This hypothesis, which explains fines the microscopic atomic symmetry hid-

the law of rational indices, has now been den behind it. You probably all know on definitely confirmed by the Laue interference what the success of von Laue's photography patterns, which are essentially X-ray photo- of crystals depends. The image of an object graphs of crystals. traced by light of a certain wave length will More exactly the hypothesis states that the be fairly accurate only with respect to details discontinuous group A of congruences which of considerably greater dimensions than that carry the arrangement of atoms in our wave length, whereas details of smaller di- crystal into itself contains the maximum mensions are leveled down. Now the wave

number 3 of linearly independent transla- length of ordinary light is about a thousand tions. Incidentally this hypothesis can be times as big as the atomic distances. How- reduced to much simpler requirements. ever X-rays are light rays the wave length 8 Atoms which go over into each other by an of which is exactly of the desirable order 10~ operation of A may be called equivalent. centimeters. In this way von Laue killed The equivalent atoms form a regular point- two birds with one stone: he confirmed the set in the sense that the set is carried into lattice structure of crystals, and proved what itself by every operation of A and that for had been merely a tentative hypothesis at the any two points in the set there is an operation time of his discovery (1912), that X-rays of A which carries the one into the other. consist of shortwave light. Even so, the Speaking of the arrangements of atoms I portraits of the atomic pattern which his dia- refer to their positions in equilibrium; in grams show are not likenesses in the literal

fact the atoms oscillate around these posi- sense. By observing a slit whose width is tions. Perhaps one should take a lead from only a few wave lengths, you obtain a some- quantum mechanics and substitute for the what contorted image of the slit made up by atoms' exact positions their average distribu- interference fringes. In the same sense these tion density: this density function in space Laue diagrams are interference patterns of

is invariant with respect to the operations of the atomic lattice. Yet one is able to com- A. The group r = {Aj of the rotational, pute from such photographs the actual ar- parts of those congruences that are members rangement of atoms, the scale being set by of A leaves the lattice L of points invariant the wave length of the illuminating X-rays. which arise from the origin by the transla- Here are two Laue diagrams (Figs. 69 and tions contained in A. The resulting 32 pos- 70), both of zinc-blende from Laue's original sibilities for T enumerated in List B corre- paper (1912); the pictures are taken in spond to the 32 existing symmetry classes of such directions as to exhibit the symmetry crystals. For the group A itself we have 230 around an axis of order 4 and 3 respectively. distinct possibilities, as was mentioned above. Whereas in the oral lecture I could show

1 See for instance P. Niggli, Geometrische Kristal- various three-dimensional (enlarged) models lographie des Diskontinuums, Berlin, 1920. of the actual arrangement of atoms, a photo-

122 123 FIG. 71 FIG. 69

graph of one such model (Fig. 71) must suffice

for this printed text: it represents a small part of an Anatas crystal of the chemical constitution TiO%\ the light balls are the Ti-, the dark ones the 0-atoms. In spite of all contortion which mars our X-ray likenesses, the symmetry of the crystal

is faithfully portrayed. This is a special case of the following general principle: If conditions which uniquely determine their effect possess certain symmetries, then the effect will exhibit the same symmetry. Thus Archimedes concluded a priori that equal weights balance in scales of equal arms. Indeed the whole configuration is symmetric with respect to the midplane of the scales,

and therefore it is impossible that one mounts reason FIG. 70 while the other sinks. For the same

124 125 we may be sure that in casting dice which vaporous depends on temperature. Tem-

are perfect cubes, each side has the same perature is the environmental factor kat' chance, %. Sometimes we are thus enabled exochen. The examples of crystallography, to make predictions a priori on account of chemistry, and genetics cause one to suspect symmetry for special cases, while the general that this duality, described by the biologists case, as for instance the law of equilibrium as that of genotype and phenotype or of for scales with arms of different lengths, can nature and nurture, is in some way bound up only be settled by experience or by physical with the distinction between discrete and principles ultimately based on experience. continuous; and we have seen how such a As far as I see, all a priori statements in splitting into the discrete and the continuous physics have their origin in symmetry. can be carried out for the characteristic To this epistemological remark about features of crystals in a most convincing way.

symmetry I add a second. The morpho- But I will not deny that the general prob-

logical laws of crystals are today understood lem is in need of further epistemological in terms of atomic dynamics: if equal atoms clarification.

exert forces upon each other that make pos- It is high time for me now to close the dis- sible a definite state of equilibrium for the cussion of geometric symmetries dwelling in atomic ensemble, then the atoms in equi- ornaments and crystals. The chief aim of

librium will of necessity arrange themselves this last lecture is to show the principle of in a regular system of points. The nature symmetry at work in questions of physics of the atoms composing the crystal determines and mathematics of a far more fundamental under given external conditions their metric nature, and to rise from these and its previous disposition, for which the purely morpho- applications to a final general statement of

logical investigation summed up in the 230 the principle itself. groups of symmetry A had still left a con- What the theory of relativity has to do with tinuous range of possibilities. The dynamics symmetry was briefly explained in the first of the crystal lattice is also responsible for lecture: before one studies geometric forms the crystal's physical behavior, in particular in space with regard to their symmetry one for the manner of its growth, and this in turn must examine the structure of space itself determines the peculiar shape it assumes under the same aspect. Empty space has a under the influence of environmental factors. very high grade of symmetry: every point is No wonder then that crystals actually occur- like any other, and at a point there is no ring in nature display the possible types of intrinsic difference between the several

symmetry in that abundance of different directions. I told you that Leibniz had forms at which Hans Castorp on his Magic given the geometric notion of similarity this Mountain marvelled. The visible charac- philosophical twist: Similar, he said, are teristics of physical objects usually are the two things which are indiscernible when each

results of constitution and environment. is considered by itself. Thus two squares in Whether water, whose molecule has a definite the same plane may show many differences chemical constitution, is solid, liquid, or when one regards their relation to each other;

126 127 for instance, the sides of the one may be in- similar; the dilatations are included among clined by 34° against the sides of the other. the automorphisms. But physics has re- length is But if each is taken by itself, any objective vealed that an absolute standard statement made about one will hold for the built into the constitution of the atom, or other; in this sense they are indiscernible and rather into that of the elementary particles, definite hence similar. What requirements an ob- in particular the electron with its standard jective statement has to meet I shall illustrate charge and mass. This atomic by the meaning of the word "vertical." length becomes available for practical meas- of the Contrary to Epicurus we moderns do not urements through the wave lengths by atoms. consider the statement about a line that it is spectral lines of the light emitted from vertical to be an objective one, because we The absolute standard thus derived the conven- see in it an abbreviation of the more complete nature itself is much better than statement that the line has the direction of tional standard of the platinum-iridium Comite gravity at a certain point P. Thus the meter bar kept in the vaults of the in Paris. gravitational field enters into the proposition International des Poids et Mesures described as a contingent factor, and moreover there I think the real situation has to be system enters into it an individually exhibited point as follows. Relative to a complete P on which we lay the finger by a demon- of reference not only the points in space but fixed by strative act as is expressed in words like I, also all physical quantities can be reference are here, now, this. Hence Epicurus' belief is numbers. Two systems of of them all shattered as soon as it is realized that the equally admissible if in both physical laws of direction of gravity is different here at the universal geometric and expression. place where I live and at the place where nature have the same algebraic between such Stalin lives, and that it can also be changed The transformations mediating form by a redistribution of matter. equally admissible systems of reference laws Let these brief remarks suffice here instead the group of physical automorphisms; the the of a more thorough analysis of objectivity. of nature are invariant with respect to It is a fact In concreto, as far as geometry is concerned, transformations of this group. we have followed Helmholtz in adopting as that a transformation of this group is uniquely the the one basic objective relation in space that determined by that part of it that concerns of congruence. At the beginning of the coordinates of space points. Thus we can second lecture we spoke of the group of con- speak of the physical automorphisms of space. the dilatations, gruent transformations, which is contained Their group does not include absolute as a subgroup in the group of all similarities. because the atomic laws fix an reflections because Before going on I wish to clarify a little further length, but it contains the the relationship of these two groups. For no law of nature indicates an intrinsic dif- right. Hence the there is the disquieting question of the rela- ference between left and the group tivity of length. group of physical automorphisms is map- In ordinary geometry length is relative: a of all proper and improper congruent in space building and a small-scale model of it are pings. If we call two configurations

128 129 deduce, as he said, the absolute congruent provided they are carried over question, to bodies from their differences, the into each other by a transformation of this motion of motions, and from the group, then bodies which are mirror images observable relative forces acting upon the bodies. But although of each other are congruent. I think it is believed in absolute space, i.e. in necessary to substitute this definition of con- he firmly of the statement that two gruence for that depending on the motion the objectivity in the same place, he did not of rigid bodies, for reasons similar to those events occur in objectively distinguishing rest of which induce the physicist to substitute the succeed from all other possible motions, thermodynamical definition of temperature a mass point in a straight line with uni- for an ordinary thermometer. Once the but only motion uniform transla- group of physical automorphisms = con- form velocity, the so-called other motions. Again, has gruent mappings has been established, one tion, from all that two events occur at the may define geometry as the science dealing the statement different places, say here with the relation of congruence between same time (but at Sirius) objective meaning? Until spatial figures, and then the geometric auto- and on The basis of this morphisms would be those transformations of Einstein, people said yes. people's habit of con- space which carry any two congruent figures conviction is obviously an event as happening at the moment into congruent figures—and one need not be sidering observe it. But the foundation surprised, as Kant was, that this group of when they of this belief was long ago shattered by Olaf geometric automorphisms is wider than that that light propagates not of physical automorphisms and includes the Roemer's discovery finite velocity. dilatations. instantaneously but with realize that in the four- All these considerations are deficient in one Thus one came to space-time only respect: they ignore that physical occurrences dimensional continuum of of two world points, "here- happen not only in space but in space and time; the coincidence or their immediate vicinity has a the world is spread out not as a three- but now," But whether a as a four-dimensional continuum. The sym- directiy verifiable meaning. this four-dimensional con- metry, relativity, or homogeneity of this four- stratification of in three-dimensional layers of simul- dimensional medium was first correctly de- tinuum one-dimen- scribed by Einstein. Has the statement, we taneity and a cross-fibration of world-lines of points resting ask, that two events occur at the same place sional fibers, the objective features of the an objective significance? We are inclined in space, describe world's structure became doubtful. What to say yes; but it is clear, if we do so, we this: without bias he col- understand position as position relative to Einstein did was all the physical evidence we have about the earth on which we spend our life. But lected structure of the four-dimensional is it sure that the earth rests? Even our the real continuum and thus derived its youngsters are now told in school that it space-time of automorphisms. It is called rotates and that it moves about in space. true group the Dutch physicist Newton wrote his treatise Philosophiae na- the Lorentz group after Einstein's the turalis principia mathematica to answer this H. A. Lorentz who, as John

130 131 —

by posing the general problem, Baptist, prepared the way for the gospel of eventualities given group of transformations to relativity. It turned out that according to how for a (invariant relations, in- this group there are neither invariant layers find its invariants quantities, etc.), and by solving it of simultaneity nor invariant fibers of rest. variant important special groups The light cone, the locus of all world-points for the more groups are known or are not in which a light signal given at a definite whether these to be the groups of automorphisms for world-point 0, "here-now," is received, known suggested by nature. This is divides the world into future and past, into certain fields Klein called "a geometry" in that part of the world which can still be in- what Felix abstract sense. A geometry, Klein said, fluenced by what I do at and the part the of transformations, and which cannot. This means that no effect is defined by a group everything that is invariant under travels faster than light, and that the world investigates given group. Of has an objective causal structure described the transformations of this with respect to a sub- by these light cones issuing from every world symmetry one speaks of the total group. Finite sub- point 0. Here is not the place to write group 7 special attention. A figure, down the Lorentz transformations and to groups deserve point-set, has the peculiar kind of sketch how special relativity theory with its i.e. any by the subgroup y if it fixed causal and inertial structure gave way symmetry defined itself by the transformations to general relativity where these structures goes over into have become flexible by their interaction of 7- 2 great events in twentieth century- with matter. I only want to point out that The two rise of relativity theory and of it is the inherent symmetry of the four- physics are the there also some con- dimensional continuum of space and time quantum mechanics. Is mechanics and that relativity deals with. nection between quantum Symmetry plays a We found that objectivity means invariancc symmetry? Yes indeed. ordering the atomic and molecu- with respect to the group of automorphisms. great role in understanding of which Reality may not always give a clear answer lar spectra, for the physics provide to the question what the actual group of the principles of quantum An enormous amount of empirical automorphisms is, and for the purpose of the key. concerning the spectral lines, their some investigations it may be quite useful to material lengths, and the regularities in their replace it by a wider group. For instance wave been collected before in plane geometry we may be interested only arrangement had scored its first success; in such relations as are invariant under quantum physics consisted in deriving the law parallel or central projections; this is the this success Balmer series in the spectrum origin of affine and projective geometry. of the so-called atom and in showing how The mathematician will prepare for all such of the hydrogen the characteristic constant entering into that 2 One may compare my recent lecture at the of the elec- law is related to charge and mass Munich meeting of the Gesellschaft deutscher Planck's famous constant of action Naturforscher: "50 Jahre Relativitatstheorie," Die tron and of the Naturwissenschqften 38 (1951), pp. 73-83. h. Since then the interpretation

132 133 corresponding set of spectra has accompanied the development of or if you will, of the

• • • Thus, for example, quantum physics; and the decisive new fea- points (Pi, , Pn). = 5 electrons the laws must be tures, the electronic spin and Pauli's strange in case of n the points P3, P\, P& are exclusion principle, were discovered in this unaffected if Ph P2, P2, Pi, P* [permutation way. It turned out that, once these founda- replaced by P3 , P6, 2-+ 3-»2, 4->l, 5 -> 4]. The tions had been laid, symmetry could be of 1-+3, 5, a group of order n\ = great help in elucidating the general char- permutations form

1 • • • • the second kind of sym- acter of the spectra. 2 n, and is expressed by this group of permuta- Approximately, an atom is a cloud of elec- metry mechanics represents the trons, say of n electrons, moving around a tions. Quantum state of a physical system by a vector in a fixed nucleus in 0. I say approximately, since space of actually of infinitely many, the assumption that the nucleus is fixed is many, dimensions. states that arise from each not exactly true and is even less justified than Two a virtual rotation of the treatment of the sun as the fixed center of our other, either by one of their permu- planetary system. For the mass of the sun system of electrons or by tations, are connected by a linear transforma- is 300,000 times as big as that of an individual that rotation or that planet like Earth, whereas the proton, the tion associated with permutation. Hence the profoundest and nucleus of the hydrogen atom, is less than group theory, the 2000 times as heavy as the electron. Even most systematic part of theory of representations of a group by linear so it is a good approximation! We dis- into play here, I tinguish the n electrons by attaching the transformations, comes must refrain from giving you a more precise • • • labels 1,2, , n to them; they enter into subject. But here the laws ruling their motion by the coordi- account of this difficult more has proved the clue to a • symmetry once nates of their positions Ph , Pn with importance. respect to a Cartesian coordinate system with field of great variety and art, from biology, from crystal- origin 0. The symmetry prevailing is two- From physics I finally turn to mathe- fold. First we must have invariance with lography and all the more be- respect to transition from one Cartesian co- matics, which I must include especially that ordinate system to another; this symmetry cause the essential concepts, from their comes from the rotational symmetry of space of a group, were first developed in mathematics, more particu- and is expressed by the group of geometric applications algebraic equations. rotations about O. Secondly, all electrons larly in the theory of deals in numbers, are alike; the distinction by their labels An algebraist is a man who the only operations he is able to perform 1,2, • • • ,n is one not by essence, but by name but *-. , The only: two constellations of electrons that arise are the four species +, — X, from each other by an arbitrary permutation numbers which arise by the four species numbers. of the electrons are indiscernible. A permu- from and 1 are the rational is closed with tation consists of a re-arrangement of the The field F of these numbers species, i.e. sum, differ- labels; it is really a one-to-one mapping of respect to the four rational numbers • • ence, and product of two the set of labels (1, 2, , n) into itself,

134 135 its coefficients a v , has n solutions, or "roots" are rational numbers, and so is their quotient ' ' ' (as one is wont to say) i?i, #2, > #n, so provided the divisor is different from zero. that the polynomial f(x) itself decomposes Thus the algebraist would have had no rea- into the n factors son to step outside this domain F, had not the

demands of geometry and physics forced /(*) = (x- *!)(* -*)•••(*- *»)• the mathematicians to engage in the dire Here a: is a variable or indeterminate, and the business of analysing continuity ;nd to embed equation is to be interpreted as stating that the rational numbers in the continuum of all the two polynomials on either side coincide real numbers. This necessity first appeared coefficient for coefficient. when the Greeks discovered that the diagonal Such relations between two indetermi- and side of a square are incommensurable. nate numbers x, y as the algebraist is able Not long afterwards Eudoxus formulated the to construct with his operations of addi- general principles on which to base the con- tion and multiplication can always be struction of a system of real numbers suitable brought into the form R(x, y) = where the for all measurements. Then during the function R(x, y) of the two variables x, y is a Renaissance the problem of solving algebraic polynomial, i.e. a finite sum of monomials equations led to the introduction of complex of the type numbers a -f- hi with real components (a,b).

that first them and • • •) The mystery shrouded a»W (Mj » = 0, 1, 2, their imaginary unit i — y/ — 1 was com- rela- with rational coefficients aM ,. These pletely dissolved when one recognized that tions are the "objective relations" accessible they are nothing but pairs (a, b) of ordinary to him. Given two complex numbers a, real numbers, pairs for which addition and he will therefore ask what polynomials multiplication are so defined as to preserve R(x, y) with rational coefficients exist which all the familiar laws of arithmetics. This get annulled by substituting the value a for can indeed be done in such a way that any the indeterminate x and ior y. From two real number a may be identified with the one may pass to any number of given com- complex number (a, 0) and that the square - * - The algebraist 2 plex numbers #1, , 0» i i = z' of i = (0, 1) equals —1, more ex- of this set 2 2 will ask for the automorphisms plicitly ( — 1,0). Thus the equation x +l =0 of numbers, namely for those permutations not solvable by any real number x be- of #!,••, #n which destroy none of came solvable. At the beginning of the ! the algebraic relations /?(#i, ' ,# n) = nineteenth century it was proved that the • • existing between them. Here R(x u • , x n) introduction of the complex numbers had is any polynomial with rational coeffi- made solvable not only this but all algebraic • cients of the n indeterminates xi, , xn equations: the equation which is annulled by substituting the values

• • • • • for , x n . The auto- (1) /(*)- &h , #„ *i, n 1 n~2 x -f fli*"- +- aox -f- + a n -ix + a n = morphisms form a group which is called the Galois group, after the French mathematician for the unknown .v, whatever its degree n and

136 137 x 2 - 2 = (x - Evariste Galois (1811-1832). As this de- V2)(* + V2).

is nothing else scription shows, Galois' theory As I mentioned a moment ago, they are ir- but the relativity theory for the set 2, a set rational. The deep impression which this which, by its discrete and finite character, is discovery, ascribed to the school of Pythag- conceptually so much simpler than the infi- oras, made on the thinkers of antiquity is nite set of points in space or space-time dealt evidenced by a number of passages in Plato's stay with by ordinary relativity theory. We dialogues. It was this insight which forced entirely within the confines of algebra when the Greeks to couch the general doctrine of we assume in particular that the members quantities in geometric rather than algebraic

• • • of the set 2 are defined as the t? l5 , #n terms. Let R(x h xz) be a polynomial of of algebraic equation (1 ),/(*) = 0, n roots an x h xi with rational coefficients vanishing

coefficients a . of nth. degree, with rational v (i.e. assuming the value zero) tor x\ = #i, the One then speaks of the Galois group of is X2 = # 2 . The question whether /?(# 2, #i) = 0. It may be difficult equation j{x) is also zero. If we can show that the the group, requiring enough to determine answer is affirmative for every R then the polynomials as it does a survey of all transposition

• conditions. R(x h ,x n ) satisfying certain (3) d 0! # 2 ->th But once it has been ascertained one can the structure of this group a lot learn from is an automorphism as well as the identity about the natural procedures by which to r?x —> t?!, t? 2 —* #2- The proof runs as follows. solve the equation. Galois' ideas, which for The polynomial R{x, —x) of one indeter- book with seven several decades remained a minate x vanishes for x = t?i. Its division exerted a more and more 2 seals but later by x - 2, profound influence upon the whole develop- R(x, -x) = (X* - 2) • d(x) (ax + b) ment of mathematics, are contained in a + friend on the eve farewell letter written to a leaves a remainder ax + b of degree 1 with in silly duel at of his death, which he met a rational coefficients a, b. Substitute #i for letter, if judged the age of twenty-one. This x: the resulting equation a&i +- b = con- ideas it con- by the novelty and profundity of tradicts the irrational nature of t?i = \/2 the substantial piece tains, is perhaps most unless a = 0, b = 0. Hence of writing in the whole literature of mankind. 2 R(x, -x) = (x - 2) • £(*), I give two examples of Galois' theory. antiquity. The The first is taken from and consequently R(&2, #i) = R(&2, —di) = and side of a ratio V2 between diagonal 0. Thus the fact that the group of auto- quadratic equa- square is determined by the morphisms contains the transposition (3) tion rational coefficients with besides the identity is equivalent to the irra-

1 tionality of y/l. (2) x - 2 = 0. My other example is Gauss' construction — = Its two roots are #i = \/2 and # 2 = &i of the regular 17-gon with ruler and compass, - V2,

138 139 which he found as a young lad of nineteen. arrangement: the dialing of this diagram,

Up to then he had vacillated between clas- i.e. iteration of its rotation by 3^6 of the whole sical philology and mathematics; this success periphery, gives the 16 automorphisms as was instrumental in bringing about his final permutations among the 16 roots. This decision in favor of mathematics. In a plane group Cie evidently has a subgroup Cs of

we represent any complex number z = index 2; it is obtained by turning the dial

x + yi by the point with the real Cartesian through 3^> %, %> ' " ' of the full angle. coordinates (x,y). The algebraic equation By repeating this process of skipping alternate points we find a chain of consecutive sub- tf - 1 =* groups O means "contains") has p roots which form the vertices of a regu- C 16 DCsDCtDCzDd lar p-gon. z = 1 is one vertex; and since which starts with the full group Ci6 and ends (z» - 1) - - with the group C\ consisting of the identity p 1 2 (z- \)'(z + s"- + ' ' ' +* + l). only, a chain in which each group is contained the others are the roots of the equation in the preceding one as a subgroup of index 2. Due to this circumstance one can determine 1" 1 (4) z + • • • + Z + 1 = 0. the roots of the equation (4) by a chain of 4

If p is a prime number, as we shall now as- consecutive equations of degree 2. Equa- sume, they are algebraically indiscernible tions of degree 2, quadratic equations, are solved (as the Sumerians already knew) by extraction of square roots. Hence the solution of our problem requires, besides the rational operations of addition, subtraction, multiplication, and division, four consecutive extractions of square roots. However, the four species and extraction of a square root are exactly those algebraic operations which may geometrically be carried out by

ruler and compass. This is the reason why the regular triangle, pentagon and 17-gon,

p = 3, 5, and 17, may be constructed by ruler and compass; for in each of these cases FIG. 72 the group of automorphisms is a cyclic group — and the group of automorphisms for the p 1 whose order p — 1 is a power of 2, roots is a cyclic group of order p — 1. I 1 2 4 3 = 2 + 1, 5 = 2 + l, 17 = 2 + 1. describe the situation for the case p = 17.

The left 17-point dial (see Fig. 72) shows the It is amusing to observe that, whereas the labeling of the vertices, the right 16-point (obvious) geometric symmetry of the 17-gon

dial the 1 6 roots of (4) in a mysterious cyclic is described by a cyclic group of order 17,

140 141 rational. its (hidden) algebraic symmetry, which deter- Obviously addition, subtraction, multiplication of numbers of the mines its constructibility, is described by one and two field rise to a of the field. of order 16. It is certain that the regular give number Nor does the operation of division lead beyond heptagon is not constructible nor are the field. let = b a regular polygons with 1 1 and 1 3 sides. the For a a -f- \/2 be number of the field different from zero with the Only if p is a prime number such that n a, b let = — is of — 1 = 2 then, rational components and a' P 1 a power 2, p , according to Gauss's analysis, the regular a — b y/2 be its "conjugate." Because 2 is not the \>f a rational number, the so- p-gon is constructible by ruler and compass. square called norm of a, the rational number act = However, p = 2" + 1 cannot be a prime 2 2 a is different zero, therefore number unless the exponent n is a power of 2. — 2b , from and 2* 2 For assume that is the exact power of one obtains the reciprocal - of a as a number by which n is divisible, so that n = 2" • m 2 " in the field as follows: where m is an odd number. Put 2 = a; n m then 2 + 1 = a + 1 • But for odd m the 1 ct a - bV2 m -\- 2 2 ' number a 1 is divisible by a + 1, a aa! ' a - 2b

Thus the field { \/2\ is closed with respect to 1 " 2 - • • • a (a + IK*"'-' a" + - + 1), the operations of addition, subtraction, multi- and hence a composite number with the plication, and division, with the self-understood exclusion of division by zero. We may factor a + 1 , unless /w = 1 . Therefore the ask for automorphisms of such a next number of the form 2" + 1 after 3, 5, now the and 17 which has a chance to be a prime field. An automorphism would be a one-to- 8 — number is 2 -f 1 = 257. As this is actually one mapping a > a* of the numbers of the

a prime number, the regular 257-gon is con- field such that a + /3 and a • /3 go into structible by ruler and compass. a* -\- 0* and a* ' /3* respectively, for any Galois' theory may be put in a slightly dif- numbers a, /3 in the field. It follows at

ferent form, as I shall illustrate by the equa- once that an automorphism changes every

tion (2). Let us consider all numbers of the rational number into itself and \/2 into a 2 form a = a + b y/2 with rational compo- number # satisfying the equation # — 2 = 0, — nents a, b; we call them the numbers of the thus either into y/2 or \/2. Hence field {\/2}- Because of the irrationality of there are only two possible automorphisms, the the \/2 such a number is zero only if a = 0, one which carries every number a of field into itself, and the other carrying b = 0. Consequently the rational compo- { \/2}

nents a, b are uniquely determined by a, for any number a = a + b y/2 into its conjugate a' = a — b is evident that this a + b \/2 = ai + fa y/l yields y/2. It second operation is an automorphism, and (a - oi) (b - bi) \/2 = 0; + one has thus determined the group of all a — a\ = 0, b — b\ = automorphisms for the field { \/2\.

A field is perhaps the simplest algebraic or a = a i, b = bi, provided a,b and a u bi are

142 143 group is structure we can invent. Its elements are of their composition is. If the tabulate the composition of numbers. Characteristic for its structure are finite one could abstract the operations of addition and multiplication. elements. The group scheme or is itself a structural These operations satisfy certain axioms, group thus obtained the law among them those that guarantee a unique entity, its structure represented by elements, inversion of addition, called subtraction, and or table of composition for its a unique inversion of multiplication (pro- st = u. Here the dog bites into its own tail, is clear enough warning for vided the multiplier is different from zero), and maybe that a ask with re- called division. Space is another example of us to stop. Indeed one may is the an entity endowed with a structure. Here spect to a given abstract group: What what are the the elements are points, and the structure is group of its automorphisms, the into established in terms of certain basic relations one-to-one mappings s —* s' of group into s't' while between points such as: A, B, C lie on a itself which make st go over /' s, t go over into /, straight line, AB is congruent CD, and the the arbitrary elements like. What we learn from our whole discus- respectively? sion and what has indeed become a guiding Symmetry is a vast subject, significant in principle in modern mathematics is this art and nature. Mathematics lies at its root, lesson: Whenever you have to do with a structure- and it would be hard to find a better one on endowed entity 2 try to determine its group of which to demonstrate the working of the automorphisms, the group of those element-wise mathematical intellect. I hope I have not transformations which leave all structural re- completely failed in giving you an indication lations undisturbed. You can expect to gain of its many ramifications, and in leading you a deep insight into the constitution of 2 in up the ladder from intuitive concepts to this way. After that you may start to investi- abstract ideas. gate symmetric configurations of elements, i.e. configurations which are invariant under a certain subgroup of the group of all auto-

morphisms; and it may be advisable, before looking for such configurations, to study the subgroups themselves, e.g. the subgroup of those automorphisms which leave one element fixed, or leave two distinct elements fixed, and investigate what discontinuous or finite subgroups there exist, and so forth. In the study of groups of transformations one does well to stress the mere structure of

such a group. This is accomplished by at-

taching arbitrary labels to its elements and then expressing in terms of these labels for any

two group elements s, t what the result u = st

144 145 APPENDICES .

APPENDIX A

DETERMINATION OF ALL FINITE GROUPS OF PROPER

ROTATIONS IN 3-SPACE (cf. p. 77).

A simple proof for the completeness of the list (5) in Lecture II is based on the fact first established by Leonhard Euler in the eighteenth century that every proper* rotation in 3-space which is not the identity / is rotation around an axis, i.e. it leaves fixed not only the origin but every point on a certain straight line through 0, the axis /. It is sufficient to consider the two-dimensional sphere 2 of unit radius around instead of the three-dimensional space; for every rotation carries 2 into itself and thus is a one-to- one mapping of 2 into itself. Every proper rotation 9* / has two fixed points on 2 which are antipodes of each other, namely the points where the axis / pierces the sphere. Given a finite group T of proper rotations of order N, we consider the fixed points of the

TV — 1 operations of T which are different from /. We call them poles. Each pole p has a definite multiplicity v (= 2 or 3 or 4

• • 5" or -) : The operations of our group which leave p invariant consist of the iterations of the rotation around the corresponding axis by 360°/"i and hence there are exactly v such operations S. They form a cyclic sub-

v. these operations group T p of order One of

is the identity, hence the number of operations 9* I leaving p fixed amounts to v — 1 For any point p on the sphere we may consider the finite set C of those points q into

which p is carried by the operations of the group; we call them points equivalent to p.

149 I, • • Sy are the v ele- Because T is a group this equivalence is of Thus if Si = S2, , the nature of an equality, i.e. the point p is ments leaving p fixed then

itself; if is equivalent to equivalent to q p X X Ti = L~ S XL, T2 = L~ S 2L, then is equivalent to q; and if both q\ and p T. = L- XS„L qz are equivalent to p then q\ and q 2 are equivalent among each other. We speak of are the v different operations leaving q our set as a class of equivalent points; any fixed. Moreover, the v different operations

• into q. Vice versa, point of the class may serve as its representa- S\L, , S„L carry p carrying into then tive p inasmuch as the class contains with p if U is an operation of V p q X of the all the points equivalent to p and no others. UL~ carries p into p and thus is one While the points of a sphere are indiscernible operations S leaving p fixed; therefore under the group of all proper rotations, the U = SL where 5 is one of the v operations

• • • let • • - be the points of a class remain even indiscernible S h , S,. Now ffc , qn after this group has been limited to the n different points of the class C = Cp and in carrying finite subgroup V. let Li be one of the operations T

• • Then all the Of how many points does the class Cp of p into §i (« 1, , »). the points equivalent to p consist? The n - v operations of the table answer: of JV points, that naturally sug- S\L\, , S,Li, gests itself, is correct provided / is the only SiL 2 , , SfLz, operation of the group which leaves p fixed.

For then any two different operations ShS2 SiL* , OyLn of T carry p into two different points q\ = pS\, are different from each other. Indeed each qi = pS2 since their coincidence q\ = q% ~ X line consists of different operations. would imply that the operation S\S2 carries individual ~ X all the operations of, say, the second p into itself, and would thus lead to S\S2 = /, And line must be different from those in the fifth S\ = S2 . But suppose now that p is a pole of multiplicity v so that v operations of the group line since the former carry p into q 2 and the

point =^ . Moreover carry p into itself. Then, I maintain, the latter into the q h q 2 operation of the group T is contained number of points q of which the class Cp con- every carries sists equals N/v. in the table because any one of them

• • • into of the points n , say Proof: Since the points of the class are in- p one qi, , q zth discernible even under the given group T, into qt, and must therefore figure in the

each must be of the same multiplicity v. line of our table. relation jV = nv and thus Let us first demonstrate this explicitly. If This proves the the operation L of V carries p into q then the fact that the multiplicity v is a divisor of

X use the notation v — v for the L~ SL carries q into q provided 5 carries p JV. We v pole know that it is the into p. Vice versa, if T is any operation of multiplicity of a p; we

X given class C, T carrying q into itself then S = LTL~ same for every pole p in a it also be denoted in an carries into and hence T is of the form and can therefore f> p X unambiguous manner by vc- The multi- L~ SL where S is an element of the group T v .

150 151 : .

plicity vc and the number n c of poles in the that JV is at least 2 and hence the left side v JV. class C are connected by the relation nc c = of our equation is at least 1, but less than 2. con- After these preparations let us now The first fact makes it impossible for the sider all pairs (S, p) consisting of an opera- sum to the right to consist of one term only. tion S t* I of the group T and a point p left Hence there are at least two classes C. But pole fixed by S—or, what is the same, of any certainly not more than 3. For as each vc leav- p and any operation S 9* I oi the group is at least 2, the sum to the right would at indi- ing p fixed. This double description least be 2 if it consisted of 4 or more terms. cates a double enumeration of those pairs. Consequently we have either two or three On the one hand there are JV — 1 operations classes of equivalent poles (Cases II and III S in the group that are different from /, and respectively) each has two antipodic fixed points; hence II. In this case our equation gives the number of the pairs equals 2(JV — 1). 2 On the other hand, for each pole p there are = i + i or 2 = ^+*- JV v\ vi "1 "2 — 5^ in the group leaving v p 1 operations / p fixed, and hence the number of the pairs But two positive integers n\ = JV/vi, equals the sum n.2 = N/v 2 can have the sum 2 only if each

equals 1 (», - i) I v% = v% = JV; n% = «2 = 1.

Hence each of the two classes of equivalent all poles collect the extending over p. We poles consists of one pole of multiplicity JV. equivalent poles and poles into classes C of What we find here is the cyclic group of rota- basic equation: thus obtain the following tions around a (vertical) axis of order JV. HI. In this case we have 2(JV- 1) = ^nc(vc - 1) c i + i + I = 1+ 2. "1 "2 v 3 JV where the sum to the right extends over all classes C of poles. On taking the equation Arrange the multiplicities v in ascending division by JV yields v order, v . nc vc = N into account, x < 2 < v 3 Not all three num- v the relation bers h v 2 , Vi can be greater than 2; for then

the left side would give a result that is 2 " fs % + M "KM * 1> contrary to the value ~NJV 2/ V *o/ of the right side. Hence pi = 2,

equation. What follows is a discussion of this i + ^^ ^+ 1 The most trivial case is the one in which the v 2 v z 2 JV group T consists of the identity only; then Not both numbers v2 , v s can be > 4, for JV = and there are no poles. 1, then the left sum would be < ^. Therefore Leaving aside this trivial case we can say C2 = 2 or 3.

152 153 First alternative Illtf v x = v 2 = 2,

. B N = 2v z APPENDIX

alternative III : v = 2, v 2 = 3; Second 2 x INCLUSION OF IMPROPER ROTATIONS (cf. p. 78). 1 = 1+2.

If the finite group F* of rotations in 3- = in IIIi. We have two Set c 3 n Case space contains improper rotations let A be one classes of poles of multiplicity 2 each con- of them and Sh ,Sn be the proper opera- consisting of sisting of n poles, and one class tions in P*. The latter form a subgroup F, easily seen two poles of multiplicity n. It is and T* contains one line of proper opera- fulfilled by the that these conditions are tions and another line of improper operations dihedral group D'n and by this group only. Si> " * * Sn For the second alternative III 2 we have, 0) j ,

three (2) ASh , AS„. in view of v 3 > v 2 = 3, the following possibilities: It contains no other operations. For if T is

X an improper operation in V* then A~ T is V3 = 3, N= 12; v 3 = 4, X = 24; proper and hence identical with one of the v 3 = 5, N = 60, operations in the first line, say Si, and there- respectively. which we denote by T, W, P fore T = AS,. Consequently the order of two classes of 4 three-poles T: There are T* is 2n, half of its operations are proper the poles of one class each. It is clear that forming the group T, the other half are must form a regular tetrahedron and those of improper. therefore the other are their antipodes. We We now distinguish two cases according to obtain the tetrahedral group. The 6 equiva- whether the improper operation ^ is or is are the projections from lent two-poles not contained in F*. In the first case we of the 6 edges. onto the sphere of the centers choose Z f°r ^ and thus get Y* = V. four-poles, forming the W: One class of 6 In the second case we may also write the the corners of a regular octahedron; hence second line in the form octahedral group. One class of 8 three-

• (2') • ' <^7" poles (corresponding to the centers of the Z^l, , n two-poles (correspond- sides); one class of 12 where the Ti are proper rotations. But in of the edges). ing to the centers this case all the Tt are different from all the class of 12 five-poles which Case P: One Si. Indeed were Ti = Sk then the group the corners of a regular icosa- must form T* would contain with %Ti = %Sk and 5* to s hedron. The 20 three-poles correspond also the element G3S»)j£T = Z-> contrary to 30 two-poles to the centers of the 20 sides, the the hypothesis. Under these circumstances the centers of the 30 edges of the polyhedron. the operations

5 ijl (3)

154 155 form a group T' of proper rotations of order In in which T is contained as a subgroup of ACKNOWLEDGMENTS index 2. Indeed the statement that the two

lines (3) form a group is, as one easily verifies, equivalent to the other that the lines (1) and I wish to acknowledge especially my debt to (2') constitute a group (namely the group Miss Helen Harris in the Marquand Library r*). Thus T* is what was denoted in the of Princeton University who helped me to main text by TT, and we have thus proved find suitable photographs of many of the art that the two methods mentioned there are objects pictured in this book. I am also the only ones by which finite groups contain- grateful to the publishers who generously ing improper rotations may be constructed. allowed me to reproduce illustrations from their publications. These publications are

listed below.

Figs. 10, 11, 26. Alinari photographs. Fig. 15. Anderson photograph. Figs. 67, 68. Dye, Daniel Sheets, A grammar of Chinese lattice, Figs. C9b, SI 2a. Harvard-Yenching Institute Monograph V. Cambridge, 1937.

Figs. 69, 70, 71. Ewald, P. P., Kristalle und Rbntgenstrahlen, Figs. 44, 45, 125. Springer, Berlin, 1923. Figs. 36, 37. Haeckel, Ernst, Kunstformen der Natur, Pis. 10, 28. Leipzig und Wien, 1899. Fig. 45. Haeckel, Ernst, Challenger monograph. Report on the scientific results of the voyage of H.M.S. Challenger, Vol. XVIII, PI. 117. H.M.S.O., 1887. Fig. 54. Hudnut Sales Co., Inc., advertise- ment in Vogue, February 1951. Figs. 23, 24, 31. Jones, Owen, The grammar of ornament, Pis. XVI, XVII, VI. Bernard Quaritch, London, 1868. Fig. 46. Kepler, Johannes, Mysterium Cosmo- graphicum. Tubingen, 1596.

157 Figs. 22, 58, 59, 60, 61, 64. Weyl, Hermann, Fig. 48. Photograph by I. Kitrosser. Reali- "Symmetry," Journal the Washington tes. ler no., Paris, 1950. of Academy Sciences, Vol. No. Fig. ' 32. Kiihnel, Ernst, Maurische Kunst, of 28, 6, 15, 1938. Figs. 2, 5, 6, 7, 8, 9. PI. 104. Bruno Cassirer Verlag, Berlin, June Wulff, 1924. Figs. 8, 12. O., Altchristliche und byzantinische Kunst; II, Die byzantinische Figs. 16, 18. Ludwig, W., Rechts-links- 514. Problem im Tierreich und beim Menschen, Kunst, Figs. 523, Akademische Verlagsgesellschaft Athenaion, Berlin, Figs. 81, 120a. Springer, Berlin, 1932. 1914. Fig. 17. Needham, Joseph, Order and life,

Fig. 5. Yale University Press, New Haven, 1936. Fig. 35. New York Botanical Garden, photograph of Iris rosiflora. Fig. 29. Pfuhl, Ernst, Malerei und geichnung der Griechen; III. Band, Verzeichnisse und Abbildungen, PL I (Fig. 10). F. Bruckmann, Munich, 1923. Figs. 62, 65. Speiser, A., Theorie der Gruppen von endlicher Ordnung, 3. Aufl., Figs. 40, 39. Springer, Berlin, 1924.

Figs. 3, 4, 6, 7, 9, 25, 30. Swindler, Mary H.,

Ancient painting, Figs. 91, (p. 45), 127, 192, 408, 125, 253. Yale University Press, New Haven, 1929. Figs. 42, 43, 44, 50, 51, 52, 55, 56. Thomp- son, D'Arcy W., On growth and form, Figs. 368, 418, 448, 156, 189, 181, 322, 213. New edition, Cambridge Uni- versity Press, Cambridge and New York, 1948. Fig. 53. Reprinted from Vogue Pattern Book, Conde Nast Publications, 1951. Figs. 27, 28, 39. Troll, Wilhelm, "Sym- metriebetrachtung in der Biologie," Studium Generate, 2. Jahrgang, Heft 4/5,

Figs. (1 9 & 20) , 1 , 1 5 . Berlin-Gottingen- Heidelberg, Juli, 1949. Fig. 38. U.S. Weather Bureau photograph by W. A. Bentley.

158 159 9 ,,

INDEX

absolute vs. relative space and time, 21 abstract group, 145 active (optically active) substances, 17, 29 Adam and Eve, 32 addition of vectors, 93 afEne geometry, 96, 132 algebra, 121, 135 Alhambra (Granada), 109, 112 analytic geometry, 94 Angraecum distichum, 51 a priori statements and symmetry, 126 Arabian Nights, 90 Archibald, R. C, 72 Archimedes, 92 Aristotle, 3 arithmetical theory of quadratic forms, 120 arrangement of atoms in a crystal, 122 Ascaris megalocephala, 36 associative law, 95 asymmetry and symmetry in art, 13, 16; historic — 16; — of crystals, 17, 29; — in the animal kingdom,

26; — the prerogative of life? 31 ; — and the origin of life, 32 aurea sectio, 72 automorphism, 18, 42; geometric and physical — 129-130; in algebra, 137; for any structure- endowed entity, 144 axis, 109, 149

Balmer series, 133 band ornaments, 48-51 bar, 52

beauty and symmetry, 3, 72 De Beer, G. R., 34 bees, their geometric intelligence, 91 Beethoven (pastoral sonata), 52 bilateral symmetry, Chap. I; definition, 4; — of animals, 26; aesthetic and vital value, 6, 28; of the human body, 8 bipotentiality of plasma, 38 Birkhoff, G. D., 3, 53 Bonnet, Charles, 72 Browne, Sir Thomas, 64, 103

Buridan's ass, 1

Capella Emiliana (Venice), 65 Cartesian coordinates, 97 causal structure, 25, 132

161 /I central symmetry, in two dimensions, 65; in three echinoderms, 60 dimensions, 79-80 Einstein, Albert, 130, 132 Ghartres, 16 enantiomorph crystals, 28 Chinese window lattice, 113 Entemena, King — , 8 Church, A. H., 73 Epicurus, 128 Clarke, 20 equiangular spiral, 69 class of crystals, 122; — of equivalent points, 150 equivalence of left and right, 19-20, 129; — of past Cohn-Vossen, S., 90 and future, 24; — of positive and negative elec- commensurable, 75; — and incommensurable, 135 tricity, 25 commutative law (in general not valid for composi- equivalent points, 150 vectors, tion of transformations), 41 ; for addition of Euclid, 17, 74 93 Eudoxus, 136 complex numbers, 136 Euler, L., 149; his topological formula ("polyhedron composition of mappings, 41, of elements in an formula"), 89 abstract group, 145 exclusion principle, 134 congruence and congruent transformations, 18, 43, 130 Faistauer, A., 24 CONKLIN, 34 Faust, 45 constitution and environment, 126 Fedorow, 92 construction of 17-gon by ruler and compass, 139, 141 Fibonacci series, 72 contingency (and lawfulness), 26 field of numbers, 143 flowers, continuity, analysis of — , 1 36 their symmetry, 58 continuous vs. discontinuous features, 108-109, 127; fontenelle, 91 — group and discontinuous subgroup, 68-69 Frey, Dagobert, 16 coordinates and coordinate system, 94 Fugus, 34 Corneto, 13 future, see: past Cosmati, 49 crystals, 28; enantiomorph —, 28; crystallographic Galen, 4 groups, 120, for any number of dimensions, 121 Galois, E., Galois group and Galois theory, 137, 142 cube, 74 Gauss, 120, 139, 142 cyclic group, 65 general relativity theory, 132 genetic constitution, 37 Darwin, Charles, 92 genotype and phenotype, 126 densest packing of circles, 83, of spheres, 90 gcnotypical and phenotypical inversion, 37 Descartes, 95 geometry, what is a — ?, 133 determination of left and right in animals, 33-38 geometric automorphism, 130 dextro- and laevo-forms of crystals, 29 geranium, 66 diamond lattice, 102 gliding axis, 109 diatoms, 87 Goethe, 51, 72 Dickson, L., 121 group of automorphisms, 42, 144; — of dilatations, dihedral group, 65 68; — of permutations, 135; — of transformations, dilatation and 'dilatory' symmetry, 68 43; abstract group, 145; finite group of proper rota- Diodore, 3 tions in 2 dimensions, 54, of proper and DlRICHLET, LEJEUNE, 121 improper rotations, 65; such groups having an discontinuous group, 68, 99; of translations, invariant lattice, 120; the same for 3 dimensions, 99-100; see also under: continuous 79, 149-154; 80, 120, 155-156; 120; finite group of distributive law, 95 unimodular transformations in 2 dimensions, 107, dodecahedron, 74 in 3 dimensions, 120. double infinite rapport, 100 Driesch, Hans, 35 Haeckel, Ernst, 60, 75, 88

Durer, 3, 65 Hambidce, J., 72 Dye, Daniel Sheets, 113 Harrison, R. G., 35 dynamic symmetry, 72 Helianthus maximus, 70, 73

162 163 longitudinal reflection, 50 helix, 71 Lorentz, H. A., 131; Lorentz group, 131 helladic, 12 Lorenz, Alfred, 52 Helmholtz, H., 18, 43, 128 Ludwig, W., 26, 38 heraldic symmetry, 8 Hermite, 121 Mach, Ernst, 19 Herzfeld, Ernst, 10 macroscopic and microscopic symmetry of crystals, hexagonal symmetry, 63; — lattice or pattern, 83, 110 123 Hilbert, D., 90 magnetism, positive and negative?, 20 historic asymmetry, 16 Mainz cathedral, 56 Hodler, F., 16 Mann, Thomas, 64 homogeneous linear transformation, 96 mapping, 18, 41 Homo sapiens, 30 Maraldi, 90, 91 honey-comb, 83, 90-91 Maschke, H. (and Maschke's theorem), 107 Huxley, Julian S., 34 Medusa, medusae, 63 hypercomplex number system, 121 66; metric ground form, 96; — structure and lattice structure, 104 icosahedron, 74 Michelangelo, 22 identity, 41 Minkowski, H., 90, 121 indiscernible, 17, 127 mitosis, 34 infinite rapport, 47 modul, 96 invariant quadratic form, 106, 108; — quantities, Monreale, 14 relations, etc., 133 morphology, 109, 126 inverse mapping, 41 motion (in the geometric sense), 44 inversion of left and right, 23; — of time, 24, 52; inversus, multiplication of a vector by a number, 94 genotypical and phenotypical — , 37; situs music, formal elements of —, 52 30 mythical thinking, 22 irrationality of square root of two, 129

Nautilus, 70 Jaeger, F. M., 29 Needham, Joseph, 34 Japp, F. R., 31 negative, see under: positive Jones, Owen, 113 Newton, 43 Jordan, Pascual, 32 20, 27, Nicomachean Ethics, 4 Nigc li, Paul, 122 Kant, I., 21, 130 Kelvin, Lord, 93 objective statements, 128 Kepler, 76, 90 octahedron, 74 Klein, Felix, 133 ontogenesis of bilateral symmetry (and asymmetry), Koenig, Samuel, 91 33 Oppe, Paul, 24 laevo-, see under: dextro- optically active, 17 lattice, 100; — basis, 100; — structure and metric order of a group, 79 structure, 104; Chinese window — , 113; rectangu- ornamental symmetry in one dimension, 48; in lar and diamond lattices, 102 two dimensions, 99-115 Laue, Max von, 122-124; Laue interference pattern orthogonal transformation, 97 or diagram, 124 orthogonally equivalent, 99 law of rational indices, 121 left and right: polar opposites or not?, 17, 22; concern parenchyma of maize, 87 a screw, 17; in paintings, 23 particle (and wave), 25 Leibniz, 17, 18, 20, 21, 27, 127 past and future, 24, 132 Leonardo da Vinci, 66, 99 Pasteur, 29-31 linear independence of vectors, 94; linear transforma- Pauli, W., 134 tion, homogeneous, 96, and non-homogeneous, 97 Penicillium glaucum, 30 logarithmic spiral, 69

164 165 pentagonal symmetry in organic nature, 63 rotational symmetry, 5; — — of animals, 27; as pentagram, 45 defined by a finite group of rotations, 53 permutation, 134 phenotypical inversion, 37 San Apollinare (Ravenna), 13 phenylketonuria, 30 San Marco (Venice), 14 phyllotaxis, 72 S. Maria degli Angeli (Florence), 65 phylogenesis of asymmetry, 33 St. Matthew, 22 physical automorphism, 129 San Michele di Murano (Venice), 65 Pisa Baptisterium, 55 St. Pierre (Troyes, France), 58 Planck, Max, 133 seal stones, 10 planetary orbits and the regular polyhedra, 76 sectio aurea, 72 L., Plato, 8, 28, 73; his Timaeus, 74; Platonic solids, 73 Siegel, C. 121 polar axis, 33; — polyhedra, 78 similarity, 18, 42 polarity, see under: equivalence sinister, 22, 127 pole, 109, 149; animal and vegetative poles of blastula, Sistine Chapel, 22 33; magnetic poles, 19 situs inversus, 30 Polya, G., 104 slip reflection, 50 polyhedral groups, 78, 154 space, 17; absolute or relative?, 21 131 POLYKLEITOS, 3, 65 space-time, positive and negative electricity, 24; mag- special relativity theory, 132 netism, 20 spectra, 133 potentially infinite, 51 Speiser, Andreas, 50, 52, 74 probability, 25 spherical symmetry, 25, 27 projective geometry, 132 spin, 134 proper and improper congruences, 43 'spiral' symmetry, 70 prospective potency and significance, 35 spira mirabilis, 69 Pythagoras' theorem, 96 Stephan's dome (Vienna), 67 structure of space, 17-18, 130; — of space-time, 131;

quadratic forms, 96, 108; their arithmetical theory, structure-endowed entity, 144; causal — , 25, 132 120 subgroup of index 2, 44 quantum physics, 25, 133 Sumerians, 8, 9 quincunx, 64, 103 swastika, 66 symmetry = harmony of proportions, 3, 16; — racemic acid, 29 defined by a group of automorphisms, 45, 133, 134; priori statements, radiolarians, 76, 88 — and equilibrium, 25; — and a Raphael (Sistine Madonna, tapestry cartoons), 23 126; the three stages of spherical, rotational and Reaumur, 91 bilateral symmetry in the animal kingdom, 27; classes rectangular lattice, 102 symmetry of crystals, 121-123; symmetry of dilatory, reflection, 4, — and rotation by 180°, 9, 77; — at crystals, 122; — of flowers, 58; bilateral, etc. symmetry, see under: bilateral, dilatory, etc. origin, 79; one-dimensional — , 47 regular point set, 122; — polygon, 73; — polyhedra, 73 Tait, P. G., 73 relativity of direction, of left and right, of position, 20, tartaric acid, 29 129; — of length, 128; relativity and symmetry, 17, tetrahedron, 74 45, 127, 133, 144; theory of—, 127; special and tetrakaidekahedron, 92 general theory of relativity, 132 theory of relativity, 1 27 Rembrandt, 23 Thompson, D'Arcy, 93 rhythm and rhythmic symmetry, 47, especially in Thomson, William (Lord Kelvin), 93 music, 51 time, absolute or relative?, 21; as the fourth dimen- right, double meaning of the word, 22; see also under: sion of the world, 1 30 left Tiryns, 12 roots of an algebraic equation, 137 Tomb of the Triclinium, 13 rotation, 44, 97, 149 Torcello, 12

166 167 transformation, 18, 41 translation, 44, 98 translatory symmetry, 47 transversal reflection, 50 tripod, 66 triquetrum, 66 Troll, W., 51 Turritella duplicata, 68

Umklappung, 77 unimodular transformation, 100 unimodularly equivalent, 106

vector, 44; vector calculus, 93-94 vertical, 128 Vinca herbacea, 66 Vitruvius, 3, 65

Wickham, Anna, 5 WOHLER, 31 Wolfflin, H., 23, 24 world = space-time, 131

X-rays, 122

168 Philosophy of Mathematics and Natural Science

By HERMANN WEYL

Scientists and philosophers will welcome this revised and augmented translation of Professor Weyl's "Philosophic der Mathc- matik und Naturwissenschaft," which originally appeared in Oldenbourg's Handbuch der Philosophie in 1927. Mr. Weyl has added six new appendices on such subjects as Foundations of Mathematics, Ars Com- binatoria, Quantum Mechanics, Physics and Biology, and Evolution.

Part 1, on mathematics, starts with some principles of mathematical logic, describes the axiomatic method, and then digs deeper into the problem of the infinite. Part 11 dis- cusses space, time, and the relative merits of the realistic and idealistic attitudes toward the external world; it also gives a general methodology of science and closes with an account of the development and of the main features of our modern world picture, under such titles as Matter, Cau- sality, Law, Chance, Freedom. "A rich and fascinating work by one of the best minds of our day."—Scientific American.

"A masterpiece of synthesis. ... It will serve as an introduction to this unfamiliar but magnificently complex and intricate world in which we live."— Thought. "A sensitive and informed appreciation of the history of ideas."—Journal of Philos- ophy. 376 pages. $6.00