Viktor Blåsjö The how and why of constructions in classical geometry NAW 5/17 nr. 4 december 2016 283 Viktor Blåsjö Mathematical Institute Utrecht University [email protected] History The how and why of constructions in classical geometry What can we learn today from the classical tradition of geometrical constructions? Viktor Urbe Condita, XXV.31) and as depicted in Blåsjö looks at construction methods through the ages, and points to lessons for teachers Figure 2. Figure 3 again shows geometry as well as philosophers. drawn in Greek sand. As the inscription ex- plains, it is based on a story about ship- Mathematics was for many thousands of only one is actually doing something with wrecked men finding hope in these signs years something you drew in the sand. It an instrument: Euclid, who is busy draw- of intelligent life on the beach where they was active, hands-on. We can see this for ing figures with his compass. Likewise, washed ashore. instance in Rafael’s fresco The School of Archimedes was killed by invading sol- This do-it-yourself element of ancient Athens (Figure 1). Among the many phi- diers while tracing figures in the sand, mathematics makes it a lot of fun and losophers and scientists shown, one and as ancient sources report (e.g., Livy, Ab also well-suited to modern pedagogical Figure 1 Rafael’s School of Athens. Early 16th century, the Vatican. 284 NAW 5/17 nr. 4 december 2016 The how and why of constructions in classical geometry Viktor Blåsjö in exact numerical form? Does it even exist at all? If we take the ruler and the compass as the basis of mathematical ontology this problem disappears at once. By the simple construction of Figure 5 the elusive 7 is readily caught on paper in the form of a line segment as humble as any other. Fur- thermore the construction is as precise as it is simple. I carried out this construction with my students the other day. We did it back-of-an-envelope style, with some ba- sic compasses I bought at a toy store. We made no special efforts to ensure accuracy. Yet when we measured our 7 segment and averaged our values we were off by Figure 2 The death of Archimedes in an 18th-century engraving. From Giovanni Maria Mazzuchelli, Notizie Isotoriche e Critiche Intorno alla Vita, alle Invenzioni, ed agli Scritti di Archimede Siracusano. less than 0.0001 from the true value. But there is much more at stake here inclinations. But at the time the purpose ing the very essence of mathematical than the problem of incommensurability. of this kind of mathematics was not ped- meaning and method, a fact that is easily Zeuthen famously advocated the thesis agogical, for engaging students. Rather, it obscured by how deceptively simple and of “the geometrical construction as ‘exis- was cutting-edge research pursued by the natural they are. Indeed, one could argue tence proof’ in ancient geometry”, i.e., that very best mathematicians of their time, that the ruler and the compass are as old “the construction ... served to ensure the and, as we shall see, these mathemati- as geometry itself. According to ancient existence of that which were to be con- cians had excellent and sophisticated rea- sources [15, p. 52], geometry began, as its structed” [19, p. 223]. Phenomena such as sons for insisting on drawing everything name suggests, as a form of ‘earth-mea- the fact that 7 “does not exist” in the in the sand. surement’. This was necessitated by the world of rational numbers could very well yearly overflowing of the Nile in Egypt: the have suggested that concerns of this type The significance of the ruler and compass flooding made the banks of the river fer- are not as paranoid as they may seem at The simplest embodiment of the tradition tile in an otherwise desert land, but it also first sight. And legitimate existence doubts of which I speak is the ruler and compass wiped away boundaries between plots, so are by no means confined to intuitively of Euclidean geometry. In classical times, a geometer, or ‘earth-measurer’, had to be obvious matters: the existence of the five these simple tools were seen as embody- called upon to redraw a fair division of the regular polyhedra, for example, is far from precious farmable land. In fact, Egyptian obvious by any standard, until one sees geometers were not called ‘earth-measur- constructions of them such as those with ers’ as in Greek but rather ‘rope-stretchers’. which Euclid crowned the Elements around The rope was their basic tool, as Figure 4 300 BC. illustrates. There are two basic ways of But the matter goes deeper still. Even generating a curve using a rope: if you pull existence questions aside, constructions both ends you get a straight line, and if are in the Greek tradition the very source you hold one end fixed and move the other of meaning in mathematics. It is a war- while holding the rope fully stretched you rant guaranteeing that every mathematical get a circle. In this way the ruler and the proposition, no matter how subtle, has a compass can be seen as arising immedi- definite ‘cash value’, as it were, i.e., that ately from the most basic motivation for it has theory-independent, jargon-free, em- geometry in practical necessity. pirical content. Constructions, then, serve But in Greek times, around the fifth cen- the purpose of grounding geometry in a tury BC, these simple tools took on a much concrete, pre-theoretical reality that is ac- more sophisticated theoretical importance cessible and indisputable even to outsid- (see for example the historical introduction ers. in [8]). Consider for instance 7. Today we This means in particular that any prop- think of 7 as a number, but this is highly osition is in principle checkable without problematic since it cannot be written as any understanding of its proof, since it a fraction of integers or a finite or repeat- can ultimately be boiled down to a con- ing decimal expansion. What, then, is this struction recipe and an empirically check- able assertion about the resulting figure. Illustration: Linda Hall Library of Science, Engineering & Technology Engineering Science, of Library Hall Linda Illustration: mysterious number called 7, which no That is to say, theorems in the Euclidean Figure 3 Frontispiece from Apollonii Pergaei Conicorum, one has ever seen actually written down Oxford, 1710. tradition are of the form “if you perform Viktor Blåsjö The how and why of constructions in classical geometry NAW 5/17 nr. 4 december 2016 285 Illustration: www.metmuseum.org Illustration: Figure 4 Egyptian geometers, or ‘rope-stretchers’, delineating a field by means of a stretched rope. From the tomb of Menna, Egypt, c. 14th century BC. such-and-such operations, this will result”, translate into a recipe for concrete mea- or that the earth was in the center of the e.g., if you draw a triangle and add up its surement, no matter how convincing a solar system, or that planetary motion con- angles they will make two right angles. By story one might be able to spin in such sists of combinations of circular motions, thus speaking about measurements and ‘metaphysical’ terms. Greek geometry lives despite these theories having been consid- relations in figures whose constructions by this principle too. It speaks of noth- ered virtually the pinnacle of human un- have been specified, theorems in the Eu- ing it cannot exhibit in the most tangible, derstanding for thousands of years. Geom- clidean tradition imply a recipe for check- concrete form right before our eyes. The etry, however, fared differently; it passed ing them empirically in as many instances definition of 7 as a number such that the test of time with flying colours. Not as desired. This has many potential uses, 77# = 7 could certainly be accused a single theorem of ancient mathematics from convincing sceptic outsiders to aiding of being ‘metaphysical’ and therefore an- needed to be revised. explorative research. It also makes it pos- ti-scientific in the positivist sense. But It was only natural, therefore, to seek sible to display expertise without reveal- once it has been concretely exhibited by a the distinguishing characteristic that set ing one’s methods — a common practice ruler-and-compass construction there is no geometry apart from the other sciences. in mathematics as late as the seventeenth longer any room for such a critique. And thinkers such as Descartes, Leibniz, century, where constructions published This foundational role of constructions and Hobbes found the answer in the con- without proofs are commonplace. These was arguably the key characteristic that structive character of geometry. It is in this kinds of advantages of construction-based separated geometry from other scientif- light that we must understand for instance mathematics are quite incompatible with ic and philosophical theories in Greek Hobbes’s otherwise peculiar-sounding the emphasis in modern mathematics on times. For instance, the Greeks (following claim (in 1656) that political philosophy, grand ‘systemic’ theorems such as Rolle’s a program laid out by Plato in the fourth rather than physics or astronomy, is the Theorem in analysis, Cayley’s Theorem in century BC) attempted to account for plan- field of knowledge most susceptible to group theory, and so on. These modern etary motions using combinations of cir- mathematical rigour: kinds of theorems are not of the construc- cles. This science was virtually a branch “Of arts, some are demonstrable, others tive, Euclidean type, whose very formula- of mathematics.