CHAPTER 5
EUCLID AND HIS “ELEMENTS”
When one thinks of Greek geometry, one thinks of Euclid and of his masterwork, the Elements. In antiquity Euclid was often referred to simply as “the Geometer.” It is disappointing that so little is known about his life. We cannot identify the place of his birth or even a reasonably accurate birth or death year. Most books on the history of mathematics do not venture a guess at his exact dates, saying instead that he was alive during the year 300 BCE.
Euclid learned mathematics and discovered the great works of Theaetetus and the other Platonists at Plato’s Academy in Athens. Later in life he moved to Alexandria. This was during the time that the great library and museum were being constructed. Euclid founded a spectacularly successful and influential school of mathematics there.
Euclid wrote several books, but his eternal fame is due to one. In approximately 300 BCE he penned his magnum opus: the Elements. It was written as a textbook for elementary geometry, number theory, and geometric algebra. Euclid is not known for his new contributions to mathematics; much, if not most, of the material found in the Elements was first proved by others. Proclus wrote that Euclid “put together the elements, arranging in order many of Eudoxus’ theorems, perfecting many of Theaetetus’, and also bringing to irrefutable demonstration the things which had been only loosely proved by his predecessors.”2
The Elements lacks much in the way of presentation; it does not place the mathematics in historical context, motivation is absent, and applications are not presented. However, the exposition and the logical treatment of the material were superior to anything that had come before it. Euclid began with five seemingly “self-evident” assumptions, and based only on these simple postulates he developed the grand theories of geometry. Proclus praised the Elements as follows:
Figure 5.1. Artist’s rendering of Euclid.
[Euclid] included not everything which he could have said, but only such things as were suitable for the building up of the elements. He used all the various forms of deductive arguments, some getting their plausibility from first principles, some starting from demonstrations, but all irrefutable and accurate and in harmony with science . . . Further, we must make mention of the continuity of the proofs, the disposition and arrangement of the things which precede and those which follow, and the power with which he treats each detail.3
This logical treatment fulfilled the dreams of Pythagoras from several centuries before. The impact to future scientists was profound. Armed with self-evident, fundamental truths, one tried to deduce all the laws of science. This ideal approach to science proved to be too simplistic; there are few laws of science akin to Euclid’s five postulates. Nevertheless, the deductive, Euclidean approach to mathematics and science is still important today.
The Elements is the earliest major mathematical work created by the Greeks that has survived to this day. It was copied and recopied by hand numerous times until the first printed version appeared in Venice in 1482. Since then there have been an estimated one thousand printings.
Most of Book XIII, the final book of the Elements, is devoted to the Platonic solids. Some historians contend that the other twelve books were written only to prepare the reader for this final book. As we have said, the proofs found in Book XIII are most likely not due to Euclid but to Theaetetus. Some scholars contend that Euclid reprinted Theaetetus’s work without editing it.4
Figure 5.2. Splayed vertices from convex polyhedra (left and center), plus a non-convex polyhedron (right) for contrast.
The most important contribution of Book XIII is the proof that there are five and only five Platonic solids. First Euclid shows that there are at least five Platonic solids—that the tetrahedron, the octahedron, the icosahedron, the cube, and the dodecahedron are indeed regular. Then he proves that there are no more than these five. To accomplish the first of these tasks, Euclid gives explicit instructions on how to build each of the five Platonic solids—that is, he constructs the Platonic solids inside spheres. We will not repeat Euclid’s constructions here. We will, however, present his argument that there are no more than these five. Later we give a different proof of this theorem, one that uses Euler’s formula.
In his proof, Euclid uses a fact about plane angles. An angle in the face of a polyhedron is a plane angle (a cube has 24 plane angles measuring 90°). In Book XI Euclid proves that the plane angles that meet at any vertex of a convex polyhedron must sum to a value less than 360°. We omit the proof, but pictorially it is easy to see why the theorem is true. If we take the faces that meet at a vertex of a convex polyhedron and splay them out on a flat surface (to do so we must cut along one edge), the faces will not overlap one another and the two cut edges will not meet (see figure 5.2). This can only happen when the sum of the plane angles is strictly less than 360°.
Now consider a regular polyhedron. Each face is a regular polygon having n sides, and m edges of the polyhedron meet at each vertex. Because every face must have at least three sides, n ≥ 3, and because at least three edges meet at each vertex, m ≥ 3. Every angle of every face has the same measure; call this angle θ. At each vertex there are m faces, each contributing a plane angle with measure θ. From Euclid’s theorem, it follows that mθ must be less than 360°. For which m and n is this possible?
Figure 5.3. The five possible vertices for a Platonic solid, both splayed and unsplayed.
When n = 3, the faces are equilateral triangles, so θ = 60°. (The measure of an interior angle of a regular n-sided polygon is 180° (n − 2)/n.) Insisting that mθ < 360°, we have m(60°) < 360°, or m < 6. So m = 3, 4, or 5 are the only possibilities (see figure 5.3). These values of m yield the tetrahedron, the octahedron, and the icosahedron, respectively.
When n = 4, the faces are square, so θ = 90°. This implies that m(90°) < 360°, or m < 4. So we can have only m = 3, and we obtain the cube.
When n = 5, the faces are regular pentagons and θ = 108°. Thus m(108°) < 360°, or m < 10/3. So we can have only m = 3, and we obtain the dodecahedron.
When n = 6, the faces are regular hexagons and θ = 120°. But m(120°) < 360° implies m < 3, which is impossible. So there is no regular polyhedron with hexagonal faces. We encounter the same problem when n > 6. Thus there are no other Platonic solids.
Examining the proof, we see that Euclid overlooked some subtle details. In particular, he did not eliminate the possibility that there could exist two different polyhedra, both of which are made of regular n-gons and both of which have m faces meeting at each vertex. For instance, perhaps there is another polyhedron besides the icosahedron that is formed from equilateral triangles, five of which meet at each vertex. Euclid has the unstated assumption that this cannot happen. It turns out that Euclid is correct, so long as we are assuming convexity; but this fact needs to be proved. If we do not assume convexity, however, Euclid is wrong. In figure 5.4 we see a nonconvex polyhedron with the same properties as the icosahedron—it is composed of twenty equilateral triangles, five of which meet at each vertex. The only difference is that one of the vertices is pushed inward, making it nonconvex.
Figure 5.4. A nonconvex Platonic solid?
Pairs of polyhedra such as the icosahedron and the nonconvex icosahedron shown in figure 5.4 are called stereoisomers (borrowing a term from chemistry). They are constructed from identical collections of faces, and the faces are joined together along the same edges.
We must also consider the possibility that the polyhedra are flexible. Imagine making a polyhedron out of unbendable metal faces using hinges as edges. A conjecture that dates back at least to Euler is that such a polyhedron is not flexible even though all of the edges are hinged. Its shape cannot be changed by pulling, pushing, or squeezing. In 1766 Euler wrote that “[solid figures] can undergo change only to the extent that they are not undamaged or closed on all sides.”5 Proving this conjecture is important, because if one of the regular polyhedra is flexible, then we would have a whole family of stereoisomers, and thus an infinite number of subtly different regular polyhedra. This fact would destroy Euclid’s proof.
It turns out that Euclid was correct, but the rigorous justification came two thousand years later from the prolific French mathematician Augustin-Louis Cauchy (1789–1857). In 1811, Cauchy proved that any two convex stereoisomers must be identical.6 In other words, if we know the faces of a convex polyhedron and know which faces are neighbors, then we know the geometry of the polyhedron exactly. One consequence of this celebrated theorem is that the five Platonic solids are indeed unique. Another is that every convex hinged polyhedron is inflexible. This latter fact became known as the rigidity theorem for convex polyhedra. Remarkably, the rigidity conjecture does not hold for nonconvex hinged polyhedra, and this fact was not discovered until 1977. The American mathematician Robert Connelly constructed the first flexible nonconvex polyhedron.7
The Greeks’ final major contribution to the theory of regular solids is due to Archimedes of Syracuse. Archimedes introduced the notion of semiregular solids. Like a regular solid, a semiregular solid is a convex polyhedron whose faces are regular polygons, but we now allow more than one type of regular polygon as a face. In addition, we insist that all faces with the same number of sides be congruent and that all of the vertices are identical (that is, each vertex has the same ordering of polygons surrounding it, and any vertex can be rotated to form any other vertex with the rest of the polyhedron lining up perfectly). Three semiregular polyhedra are shown in figure 5.5. Archimedes’ work is lost, but according to the following passage by Pappus (c. 290-350 CE), we know that Archimedes found thirteen semiregular polyhedra:
Figure 5.5. Three of the thirteen Archimedean solids.
Although many solid figures having all kinds of faces can be conceived, those which appear to be regularly formed are most deserving of attention. These include not only the five figures found in the godlike Plato . . . but also the solids, thirteen in number, which were discovered by Archimedes and are contained by equilateral and equiangular, but not similar, polygons.8
The complete set of thirteen polyhedra was reconstructed in 1619 by Kepler, who was unaware of Archimedes’ work. Just as Theaetetus proved that the five Platonic solids are the only regular polyhedra, so did Kepler prove that there are only thirteen semiregular polyhedra. We should mention that there is an infinite collection of polyhedra called prisms and antiprisms that satisfy the semiregularity criteria, but historically these have not been called semiregular solids. Today the semiregular polyhedra are known as Archimedean solids.
Following the decline of Greek civilization, the center of mathematical activity moved to Persia (modern Iraq). Under royal patronage, Arabic mathematicians translated many of the Greek mathematical classics, including works of Euclid, Archimedes, Apollonius, Diophantus, Pappus, and Ptolemy. They were more than caretakers of the Greek texts, however. They are responsible for creating the field of algebra and for making substantial contributions to number theory, number systems, and trigonometry. The Arabic period of mathematical dominance lasted until approximately the fifteenth century.
Arabic mathematicians advanced the state of geometry, but they did not substantially add to the theory of polyhedra. For a renewed interest in polyhedra, mathematics had to wait for Europe to emerge from the medieval period.