CHAPTER 3
THE FIVE PERFECT BODIES
Modern geometry, and indeed much of modern mathematics, can trace its roots back to the work of the Greeks. During the period from Thales (c. 624-547 BCE) to the death of Apollonius (c. 262-190 BCE), the Greeks produced an astonishing body of mathematical works, and the names of many of the scholars of this era are familiar to schoolchildren everywhere: Pythagoras, Plato, Euclid, Archimedes, Zeno, and so on.
Although the mathematics from Egypt, Mesopotamia, China, and India may have influenced the Greeks, they quickly made the discipline their own. As Plato wrote in Epinomis, “Whenever Greeks borrow anything from non-Greeks, they finally carry it to a higher perfection.”2 Unlike earlier civilizations for whom utility was the prime objective, the Greeks wanted to understand the ideas of mathematics and have rigorous proofs of the assertions. Gone were the formulas used for approximations. Exactness, logic, and truth were the aims of their investigation.
The Greeks were fascinated with geometry, and their achievements in this field are far too numerous to list. It is certainly no exaggeration to say that most of the geometry now taught in schools was discovered by the Greeks. We shall focus on a Greek theorem about regular polyhedra (we will define “regular” shortly). It is one of the most celebrated and beautiful theorems in all of mathematics (fourth-most beautiful, according to the survey mentioned in chapter 1).
There are exactly five regular polyhedra.
These five polyhedra are shown in figure 3.1. Three of the shapes are composed of equilateral triangular faces—the tetrahedron (the 4-sided pyramid), the octahedron (the double-pyramid with 8 faces), and the 20-sided icosahedron. The cube is composed of 6 squares and the dodecahedron is the 12-sided figure made of regular pentagons. (Appendix A contains templates for making the regular polyhedra out of paper.)
Figure 3.1. The five regular polyhedra: the tetrahedron, the octahedron, the icosahedron, the cube, and the dodecahedron.
Figure 3.2. Regular polygons with 3, 4, 5, 6, 7, and 8 sides.
The colorful history of these intriguing polyhedra begins with the Greeks and continues through the Renaissance to today. A proof that there are only five regular polyhedra is found in the final book of Euclid’s Elements. (In chapter 8 we will present another proof using Euler’s polyhedron formula.) Plato believed that the regular polyhedra were the building blocks of all matter. Because he incorporated them into an atomic theory, they are now called the Platonic solids. The astronomer Johannes Kepler (1571–1630) used the regular solids to create an early model of the solar system.
Beauty is often found in regularity, symmetry, and perfection. We are all familiar with the 2-dimensional regular polygons. A polygon is regular if every side has the same length and every interior angle has the same measure. The equilateral triangle is the only regular three-sided polygon, the square is the only regular four-sided polygon, and so forth (see figure 3.2). There are infinitely many regular polygons, one n-sided polygon for every integer n greater than 2.
The 3-dimensional analogue of a polygon is a polyhedron. The study of regularity in polyhedra is much more interesting than for polygons. While there are infinitely many regular polygons, the five figures shown in figure 3.1 are the only regular polyhedra.
Figure 3.3. Polyhedra that are not regular. Each fails one of the four conditions for regularity.
What, exactly, are the criteria for a polyhedron to be regular? As with our definition of polyhedron, we must be careful not to include or exclude anything unintentionally. A regular polyhedron or regular solid is a polyhedron satisfying the following conditions.
1. The polyhedron is convex.
2. Every face is a regular polygon.
3. All of the faces are congruent (identical).
4. Every vertex is surrounded by the same number of faces.
Each of these criteria is necessary to obtain the desired collection of polyhedra. In figure 3.3 we give examples of polyhedra that fail to meet exactly one of the criteria. The first polyhedron satisfies every criterion except that it is nonconvex. The second polyhedron, a distorted octahedron, would be regular if the faces were equilateral triangles. The soccer ball is not regular because the faces are regular pentagons and regular hexagons. The last polyhedron consists of equilateral triangles, but four faces meet at all of the equatorial vertices, while five meet at the north and south poles.
Regular polyhedra are found in nature. Crystals are the most obvious naturally occurring polyhedra, and some crystals are regular. For instance, sodium chloride can take the form of a cube, sodium sulphantimoniate can take the form of a tetrahedron, and chrome alum can take the form of an octahedron. Pyrite, commonly known as fool’s gold, can form a crystal with twelve pentagonal faces; however, the crystal is not a dodecahedron because the crystal structure creates pentagonal faces that are irregular.
In the 1880s, Ernst Haeckel of the HMS Challenger discovered and drew pictures of single-celled organisms called radiolaria. The skeletons of some of these organisms bear a striking resemblance to the regular polyhedra (figure 3.4).
Figure 3.4. Radiolaria resembling the regular polyhedra.
There are some early man-made examples of regular solids. The cube and the tetrahedron, being relatively simple and common, have appeared in many physical creations throughout history. A dodecahedron dating back to at least 500 BCE was unearthed in an excavation on Mt. Loffa near Padua, Italy. An ancient icosahedral die was found in Egypt, but its origin is unknown.
What about the octahedron? This was probably the last of the five solids to be created by man. It is not as simple as the cube or the tetrahedron, so no everyday object would take this shape. It is not as exotic as the icosahedron or the dodecahedron, being nothing more than two square-based pyramids fused along their bases, so if it was encountered, it was likely ignored. Mathematics historian William Waterhouse argued that until someone discovered and put forth the notion of regularity, the octahedron was nothing special. He wrote, “The octahedron became an object of special mathematical study only when someone discovered a role for it to play.”3
This discussion of the octahedron is enlightening. We see that there are three important stages in the development of the theory of regular polyhedra. The first is the construction of the objects themselves. Initially, their construction may be nothing more than making them out of clay, but eventually matters must become mathematical—they must be constructed geometrically. The second stage is the discovery of the abstract notion of regularity. This idea is obvious only in retrospect. Imagine handing the five regular solids to a passerby and asking, “what do these five objects have in common?” Chances are, he or she would not formulate the abstract notion of regularity. As Waterhouse said, “The discovery of this or that particular body was secondary; the crucial discovery was the very concept of a regular solid.”4 Finally, the third stage is proving that there are only five regular solids. It must be shown, using rigorous mathematics, that there are five, and no more than five, of these beautiful objects. The development of this theory, from discovery, to abstraction, to proof, is due to the Greeks.