Avenues for Polyhedral Research

by Magnus J. Wenninger

In the epilogue of my book Polyhedron Models I alluded Topologie Structurale #5, 1980 Structural Topology #5,1980 to an avenue for polyhedral research along which I myself have continued to journey, involving Ar- chimedean stellations and duals (Wenninger 1971). By way of introduction to what follows here it may be noted that the five regular convex polyhedra belong to a larger set known as uniform’ polyhedra. Their stellated forms are derived from the process of extending their R&urn6 facial planes while preserving symmetry. In particular the tetrahedron and the hexahedron have no stellated forms. The octahedron has one, the dodecahedron Cet article est le sommaire d’une etude et de three and the icosahedron a total of fifty eight (Coxeter recherches faites par I’auteur depuis une dizaine 1951). The octahedral stellation is a compound of two d’annees, impliquant les polyedres archimediens Abstract tetrahedra, classified as a uniform compound. The &oil& et les duals. La presentation en est faite a three dodecahedral stellations are non-convex regular partir d’un passe historique, et les travaux This article is a summary of investigations and solids and belong to the set of uniform polyhedra. Only d’auteurs recents sont brievement rappel& pour study done by the author over the past ten years, one icosahedral stellation is a non-convex regular la place importante qu’ils occupent dans I’etude involving Archimedean stellations and duals. The polyhedron and hence also a uniform polyhedron, but des formes polyedriques. L’auteur enonce une presentation is set into its historical background, several other icosahedral stellations are uniform com- regle &n&ale par laquelle on peut t,rouver un and the works of recent authors are briefly sket- pounds, namely the compound of five octahedra, five dual a partir de n’importe quel polyedre uniforme ched for their place in the continuing study of tetrahedra and ten tetrahedra. Their duals are also well non-convexe et il nous montre comment ap- polyhedral shapes. The author enunciates a known. pliquer la regle a I’aide de dessins. II nous general rule by which the dual of any non-convex presente egalement des photographies de uniform polyhedron can be discovered, and he modeles qu’il a fabriques lui-meme. C’est ainsi gives drawings to show how the rule applies. He The dual of any polyhedron is one which has the same que nous pouvons voir certains duals qu’ils a also presents photographs of models he has number of edges as the original from which it is derived, decouverts dernierement. II invite les techniciens made. These show some duals he has himself but there is an interchange in the number of faces and specialises en ordinateurs, les ingenieurs en recently discovered. He invites computer vertices. The kinds of faces and vertices are such, design et les artistes-decorateurs a considerer ces technicians, design engineers and decorator ar- however, that an n-sided polygon face in the original duals et les transformations possibles d’autres tists to consider these and other possible polyhedron yields an n-edged vertex in the dual. polyedres comme un champ de recherche a polyhedral transformations as an area for further developper et egalement comme elements research and for their usefulness as beautiful There are two so-called quasi-regular convex decoratifs aussi bien utiles que gracieux. decorative devices. polyhedra, the cuboctahedron and the 5 icosidodecahedron. Their stellated forms, especially Figure 1. The cuboctahedron and its complete stellation patterns.

Figure 2. The truncated cube and its complete stellation patterns. 6 those of the latter, are extremely numerous, the total solid has more and more faces, the related stellation tahedral symmetry group. Computer graphics could number being unknown. But the stellation pattern, patterns take on more and-more intersecting lines, and produce motion pictures of such a transformation called also the ((complete face)), has intrinsic interest, hence more and -more cells are generated leading to process in action. Verheyen sees this process at work because it is from this pattern that new stellated forms greater and greater complexity in these stellated forms. together with rotational transformations along axes of are discoverable. Complete faces for some Ar- Yet those who, like me, have spent a good deal of time symmetry in the generation of all uniform polyhedra, chimedean solids are shown in Figures 1,2,3. The final investigating these myriad shapes cannot help but be and he alludes to the possibility of making computer stellation is of special interest for any given solid overawed by the beauty that lies hidden here, just generated movies (Verheyen 1979). because this is unique (Wenninger 1971). waiting to be discovered and brought to the attention of artists and design engineers. The investigation of duals opens an equally interesting Apart from what has appeared in (Coxeter 1973, 1951) avenue for polyhedral research. The dual of the cuboc- and (Wenninger 1971) very little has been published An avenue of research as yet largely unexplored is that tahedron is the rhombic dodecahedron. The dual of the about stellated forms. So here is an avenue for of continuous transformations of polyhedral shapes. It icosidodecahedron is the rhombic triacontahedron. polyhedral research. Some closely related in- may be one thing to stellate the cuboctahedron, for just as the regular and semiregular convex solids can be vestigations have been done recently by Alexanderson, example, but a similar investigation of the truncated stellated by extending or producing their facial planes, Kerr and Wetzel; see (Alexanderson 1979, 1971), (Kerr cube and the truncated octahedron will show that these so these convex solids can be stellated by the same 1978) and (Wetzel 1978). Modern computer graphics are merely transformations of the cuboctahedron, process. A distinct advantage enters here because these can depict stellation patterns. In fact I have been which itself is a transformation of the cube and the oc- duals have each a single stellation pattern due to the greatly assisted in my work with such drawings supplied tahedron. This is already evident from the stellation pat- fact that dual forms are isohedral just as the originals are for my privately by J. Skilling, who has used computer terns shown in Figures 1, 2, 3. Such relationships for isogonal. analysis for his work; see (Skilling 1975, 1976). Com- the convex forms are beautifully shown in photographs puter graphics can also show individual polyhedral by (Holden 1971). In stellated forms the intersection of Stellations of the rhombic dodecahedron have been shapes, given data for vertices and their interconnec- facial planes for all of these is merely a parallel tran- studied by D. Luke; see (Cundy 1961). Interesting tedness; see the work of (Norman 1973) and (Smith slation into or away from a center of symmetry common varieties of interlocking puzzles, based on the rhombic 1973, 1974). In the stellation process as the original to all of them. In the example cited this is the oc- dodecahedron and rhombic triacontahedron, have

7 Figure 3. The truncated octahedron and its complete stellation patterns. been skilfully made in wood by (Coffin 1971). (Holden attempts to show how the Archimedean duals can be A third attempt to show the generation of convex duals 1971) depicts these shapes as rigid models. I have done geometrically derived by a process of erecting pyramids is that of (Fleurent 1979), who derived his inspiration some of these puzzle arrangemens as models in paper; bn the faces of a chosen basic polyhedron, these from Graziotti. This author seeks to introduce a suitable see (Wenninger 1963) and an unpublished typescript pyramids then determining the vertices of the related symbolism, and his work as yet unpublished could with diagrams in my files. dual. The heights of these pyramids are a crucial factor prove helpful to further understanding of the process of in this process. Unfortunately Graziotti has oversim- dualization as well as other relationships between con- Stellations of the rhombic triacontahedron have been plified the process to the extent that actual calculation vex solids. studied by (Ede 1958). An attempt at a systematic disqualifies seven of the thirteen cases he so beautifully enumeration of possible forms under some restrictive illustrates in his otherwise ingenious approach. rules came from the work of (Pawley 1973). Of special Thus, although the study of convex polyhedra and their interest is the fact that among the stellations of the Another more successful attempt is that of (Lalvani 1977), duals is still attracting attention today, myriads of non- rhombic triacontahedron is a uniform compound, the whose explosion-implosion process leads from the origi- convex forms and their duals remain largely unknown compound of five cubes, and the duals of two non- nal polyhedron through a stage he calls transpolyhedron and unexplored. M. Bruckner at the beginning of this convex uniform polyhedra (Coxeter 1973), (Cundy to the true dual form. In this work he alludes to the twentieth century was a pioneer explorer here, but it 1961) and (Holden 1971). The entire topic of uniform applicability of the process to uniform polyhedra and seems that the trail he blazed has not been taken up by compounds of uniform polyhedra has been thoroughly notes that this needs a more in-depth study. others. researched by (Skilling 1976). Skilling has also given the definitive enumeration of uniform polyhedra, bringing to a close the investigation of (Coxeter 1954) and (Skilling 1975).

So besides the two quasi-regular convex solids and their duals, avenues for polyhedral research are still open for nine other semiregular convex solids and two convex snub polyhedra along with all their duals. Little is known about their stellated forms at the present time. The one hundred twenty faces of the hexakisicosahedron pose a challenge almost too great for human endurance to penetrate, hence the use of computer graphics is undoubtedly a necessity here. But when this is seen in historical perspective such investigation is calling out for its completion.

The five regular solids were known in ancient times and beyond doubt also the entire set of thirteen Ar- chimedean solids as well. The dual relationship does not seem to have been clearly recognized until the time 8 of J. Kepler, A.D. 1611. Other duals seem not to have “3 entered into historical perspective until the work of E. Catalan, a French mathematician, in 1862; for historical notes see (Coxeter 1973). The classic work of (Bruckner 1900) a German mathematician and great polyhedron model maker, is also rich in historical data. This work, surely should have been translated from the German original into other languages by now, but so far as I know this has not been done.

There have been recent attempts to present a systematic Figures 4, 5, and 6. The vertex figure of #77 with its circumcircle and tangents drawn to this circle, showing how the facial elaboration of the relationship between convex plane for the dual of #77 is derived. Then the vertex figure of #85 in relationship with the facial plane for the dual of #85. polyhedra and their duals. One such attempt is that The facial plane is a non-convex polygon. Finally, the vertex figure of MO3 in relationship with the facial plane for the dual of #103. (Graziotti 1962). Here this author who is also an artist The facial plane is a crossed polygon. 8 , v 4

v 8 86 v 8

V 8 69 & 86 V

v 8 69 V-! V 8 V s, 3 92 3

Figure 7. Partial stellation pattern for the duals of #69, X86, X92. The dual of #69 has facial planes which are non-convex polygons, while the dual of X86 9 has facial planes which are crossed polygons. The placement of hidden vertices is shown here. v 4 v 4 v 4

85 103 77

Figure 8. Partial stellation patterns for the duals of #77, X85, #103. 10 . \ l /

V 10 v ‘0, 5 3

72 & 74 94

Figure 9. Partial stellation patterns for the duals of #72, #74, #97. The dotted lines showing a semiregular pentagon may be compared with the regular pentagon in the central 11 portion of Figure 12. This shows the variation in the Archimedean form. The dual of #97 shows stippled area for hidden vertices. It is known that the faces of convex duals can be derived from the vertex figures of the original convex Table I solids. The faces of non-convex duals can similarly be derived from the vertex figures of the original non- convex solids. Some examples of this are shown in Figures 4, 5, 6. In the non-convex duals, just as in the original non-convex solids, the facial planes intersect each other, and these faces are sometimes crossed # Given solid Convex hull Dual of hull Dual* polygons. An added difficulty appears in the fact that sometimes vertices lie hidden below these intersecting ficial planes; see for example the vertex V{5/2} shown as Figure 6.4 C in (Coxeter 1973). The facial planes are discoverable from the respective stellation patterns, sin- 20 small stellated dodeca. icosa. dodeca. photo 1 ce they lie embedded in these patterns. This relation- 21 great dodeca. icosa. dodeca. photo 2 ship is shown in Figures 7,8,9,10. The cell structure of 22 great stellated dodeca. dodeca. icosa. photo 3 these non-convex duals can be determined in much the 41 great icosa. icosa. dodeca. photo 4 same way as in other stellated forms, and the hidden vertices can thus be clearly laid bare. 73 dodecadodeca. icosidodeca. rhombic triaconta. photo 5 94 great icosidodeca. icosidodeca. rhombic triaconta. photo 6 That the duals of non-convex uniform polyhedra remain largely unknown today is not surprising if one realizes 70 small ditrigonal icosidodeca. dodeca. icosa. photo 7 that the complete discovery of uniform polyhedra was 80 ditrigonal dodeca. dodeca. icosa. photo 8 not accomplished until the present century, (Coxeter 87 great ditrigonal icosidodeca. dodeca. icosa. photo 8 1954) and (Skilling 1975). 69 small cu bicu bocta. rhom bicu bocta. trap. icositetra. photo 9 In my own investigation I have found that a close 86 small rhombi hexa. rhombicubocta. trap. icositetra. photo 9 relationship exists between duality and the stellation 92 quasitruncated hexa. rhom bicu bocta. trap. icositetra. photo lo process. A general rule I have formulated is this: The dual of any given non-convex solid is a stellated form of 77 great cu bicu bocta. truncated hexa. triakisocta. photo 11 the dual of the convex hull of the given solid. In par- 85 quasirhombicubocta. truncated hexa. triakisocta. photo 12 ticular this is neatly exemplified by the four polyhedra 103 great rhombi hexa. truncated hexa. triakisocta. photo 13 known as the Kepler-Poinsot solids. If for example the given non-convex solid is taken to be the small stellated 72 small dodecicosidodeca. rhombicosidodeca. trap. hexeconta. photo 14 dodecahedron, the convex hull (which Coxeter calls 74 small rhombidodeca. rhombicosidodeca. trap. hexeconta. photo 14 the ({case,), namely the smallest convex solid that can 97 quasitruncated sm. st. dodeca. rhombicosidodeca. trap. hexeconta. photo 15 contain it, is the icosahedron whose dual is the dodecahedron among whose stellations is the great 81 great ditrig. dodecicosidodeca. truncated dodeca. triakisicosa. photo 16 dodecahedron, the dual of the given solid. just as the 88 great icosicosidodeca. truncated dodeca. triakisicosa. photo 17 five Platonic solids have duals within the same set of 101 great dodecicosa. truncated dodeca. triakisicosa. photo 18 five, so the four Kepler-Poinsot solids have their duals within the same set offour.

Table I is a summary of my investigation of non-convex uniform polyhedral duals to date. The numbers at the Notes: # = Wenninger number. * ((dual)) = dual of given solid. #80 and #87, #69 and #86, extreme left of the table are used to identify each given #72 and #74 are pairs which each have the same photo for their duals. Their facial planes polyhedron. These numbers are the same as those used however are distinct. They differ in the placement of hidden vertices. in my book Polyhedron models (Wenninger 1971). 12 Perhaps the one drawback of the general rule for duality 89 sm. icosihemidodeca. Also an examination of the complete list of 57 shows actual search through 91 sm.dodecahemidodeca. that only those listed in the table have a convex hull rect one is discovered. 100 sm.dodecahemicosa. which is a regular or semiregular convex solid. too difficult. For the However it may well be conjectured that the general a tedious task. Hence 102 gr.dodecahemicasa. rule still holds even when the convex hull has faces that a more analytical approach is needed, one which uses 106 gr.icosi hemidodeca. are not regular polygons. My investigation has led me the mathematical tools of coordinate geometry and vec- 107 gr. dodecahemidodeca. to see that convex hulls with faces that are not regular tor analysis as well as computer graphics. polygons are merely symmetrical transformations of and 119 dirhombicosidodecahedron. convex hulls with regular faces. So here also is a rich An examination of the complete list of 57 non-convex avenue for polyhedral research, one along which I uniform polyhedra shows that the following can have myself am presently proceeding, but one which un- no duals in finite space: All of these bear the designation ((-hemi-)+ except 119 doubtedly calls for computer technology to bring it to which could also bear it, since they all have faces which completion. 67 tetrahemihexa. pass through the center of symmetry of the solid. Hen- 68 octahemiocta. ce for all of these the corresponding vertices recede to My own work has led me to see there is no reason to 78 cubohemiocta. infinity. limit oneself to strictly Archimedean forms. In fact some variations lead to far more aesthetically pleasing results. It is generally agreed that convex Archimedean duals are not particularly attractive. The same may be said for some of the non-convex duals as well, as the reader may judge from the photographs of the models.

I have done a golden ratio truncation of the icosahedron which when stellated yields compounds of the icosahedral and dodecahedral stellations interpenetrating in a fashion far more beautiful than the same compounds derived from stellating the icosidodecahedron as shown in (Wenninger 1971). I found the Archimedean stellations of the truncated dodecahedron and the truncated icosahedron particularly disappointing. On the other hand I have found that stellated forms of Archimedean duals become far more attractive than the convex forms from which they are derived. It seems that the basic symmetry of these duals becomes far more pronounced and visibly explicit in the more complex shapes.

Some variations of Archimedean duals also turn up some extremely unexpected results. For example, compounds of two dodecahedra and their interpenetrating stellations are found among the stellated forms of a variation of the tetrakishexahedron; see Figure 11. Compounds of five dodecahedra and their interpenetrating stellations are found among the stellated forms of a variation of the trapezoidal hexecontahedron; see Figure 12. For a commentary on these beautiful solids and for a further note on the five dodecahedra see (Norman 1973) and (Cundy 1976). Note however a printing error in the first formula on p. 218 in (Cundy 1976): r fl should read T/G. , I was ut- 13 Figure 10. Partial stellation pattern for the duals of #81, #88, #lOl . terly amazed to find after actual calculation that Figure 11 l Partial stellation pattern for compounds of two dodecahedra. 14 15 Figure 12. Partial stellation pattern for compounds of five dodecahedra. Graziotti’s trapezoidal hexecontahedron is this variation and not the true Archimedean dual he sought to show; see (Graziotti 1962), Plate X.

Another treatment of regular convex solids and their in- terpenetration can be found in the work of (Harman 1974). His study of what he calls symmetric compounds of regular convex solids could be extended to the study of semiregular compounds as well. Inter- penetrating stellated forms also become very attractive. So there are literally endless avenues for polyhedral research still unexplored.

Although my work has been done entirely in models using paper only, other more durable materials could be utilized, especially colorful translucent plastics for lighting fixtures and lampshades.- I suggested this at the end of my book on Spherical models (Wenninger 1979).

In conclusion I might add that the process of faceting has not been taken into account in this article. Faceting is the dual process to stellation. I have avoided it in my own work simply because the stellation process is easier to manipulate for purposes of model making. Prisms and antiprisms have also been omitted, perhaps because I so not find these to be attractive forms either. All reference to the dihedral symmetry group has also been omitted. A complete theory would have to take all of these into account.

18 1

19 ..,. . . - . Photoi3. - _ Photo 14. Photo 15. Photo 16. \

Photo 17.

21 Biographical note His published works include:

Rev. Magnus J. Wenninger, O.S.B., was born on Oc- Stellated Rhombic Dodecahedron Puzzle, The tober 31, 1919, Park Falls, Wisconsin, U.S.A. He Mathematics Teacher, March 1963. professed as a monk in the Order of St. Benedict, St. John’s Abbey, Collegeville, Minnesota, U.S.A. 1940, The World of Polyhedra. The Mathematics Teacher, was awarded a Bachelor of Arts in Philosophy at St. March 1965. John’s University, Collegeville, Minnesota, U.S.A. in 1942, and was ordained as a priest in the Roman Polyhedron Models for the Classroom. N.C.T.M. Catholic Church in 1945. Subsequent degrees included Publication, 1966. Second edition 1975. Spanish a Master of Arts in Philosophy at the University of Ot- language translation, Olsina, Spain, 1975. tawa, Ottowa, Ontario, Canada, in 1946, and a Master of Arts in Mathematics Education from Columbia Some Interesting Octahedral Compounds. The University Teachers College, New York, New York, Mathematical Gazette, February 1968. U.S.A in 1962. Rev. Wenninger was a teacher of mathematics and head of the mathematics department A New look for the Old Platonic Solids. Summation, at St. Augustine’s College, Nassau, Bahamas from 1946 Journal of the Association of Teachers of Mathematics, to 1971, and is presently accountant and comptroller at New York. Winter 1971. the Business Office of St. Augustine’s Monastery and College, Nassau, Bahamas. Polyhedron Models. Cam bridge University Press, Lon- don and New York, First printing 1971. Paperback 1974. Russian language translation, Mir, Moscow, 1974. Japanese language translation, Dainippon, Tokyo, 1979.

Spherical Models. Cam bridge University Press, London and New York, 1979.

Geodesic Domes by Euclidean Construction. The Mathematics Teacher, October 1978.

A Polyhedron Construction Kit, marketed by Creative P hotocpphs by Toog0od’s Photograph v, Nassau Visuals, Big Spring Texas. Sixty sheets of machine cut figures redrafted bv Nabil Macarios, Montr&al paper for making twenty four different models.

22 Bibliography

The code in the first block of each bibliographic item consists of The middle letter(s) indicates whether the piece was intended The key words or other annotations in the third column are inten- three parts, separated by dashes. The first letter indicates whether primarily for an audience of ded to show the relevance of the work to research in structural the item is a topology, and do not necessarily reflect its overall contents, or the M athematicians, intent of the author. B ook, A rchitects, or A rticle, E ngineers. P reprint, or C ourse notes. The final letter(s) indicates if the piece touches on one or more of the principal themes of structural topology:

G eometry, in general, P olyhedra, J uxtaposition, or R igidity.

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