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In real life, almost nothing is perfect. But in mathematics, things can be. Scholars consider circles and spheres perfect because of their high symmetry, which is one of the reasons why it was difficult to abandon the concept of circular planetary orbits and accept the elliptical orbits postulated by astronomer Johannes Kepler. If we focus on polyhedrons (three-dimensional polygons), five of them stand out as perfect, forming the so-called Platonic solids: the tetrahedron, cube, octahedron, dodecahedron and icosahedron.
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Admittedly, their names offer no special aesthetic appeal. But their forms are unique. Platonic solids are polyhedrons with the highest possible symmetry. Each of their faces is bounded by the same number of edges, and the same number of faces meet at each vertex.
Only five polygons meet those criteria, assuming we restrict ourselves to those without holes or indentations. The reason for this lies in one of my favorite quantities from mathematics: the Euler characteristic. This is a measure from topology, an abstract mathematical subfield focused on the fundamental properties of objects. Imagine molding dough into various shapes. If two shapes can be transformed into each other simply by kneading, without tearing or sticking, then they are the same shape from a topologist’s point of view. This means a baguette and a roll are the same—but a bagel and a pretzel are different.
Although the first topological ideas only emerged in the 18th century, the history of the Platonic solids stretches back much further. Scholars from ancient Greece—most notably Plato, for whom they are named—studied them extensively. Plato assumed that the four basic elements (fire, water, earth, air) were each composed of a Platonic solid: fire from the tetrahedron, a pointed shape that can cause pain upon contact; air from the octahedron, a shape so pliable that the particles are barely perceptible; water from the icosahedron, a fluid shape that could flow through one’s hands; and earth from the cube, a crumbly shape whose particles can completely fill three-dimensional space, making the earth particularly stable. The dodecahedron, on the other hand, was associated with the aether and the starry sky. Using geometric methods, Theaetetus, a mathematician and a contemporary of Plato, succeeded in proving that there are no more than five of these highly symmetrical bodies.
Nearly 2,000 years later, Kepler attempted to explain the known solar system using these five bodies. He failed. Nevertheless, his considerations ultimately led him to the now proven conclusion that planetary orbits are elliptical.
The fact that there are only five Platonic solids has been known since antiquity, but, thanks to an observation from the 18th-century Swiss mathematician Leonhard Euler, we can also prove it mathematically. Euler realized that if you add up all the vertices and faces of a polyhedron without holes and subtract the number of edges, you always get two—always!
Check it yourself. A cube, for example, has eight vertices, six faces, and 12 edges, which gives 8 + 6 − 12 = 2. A trigonal dodecahedron has eight vertices, 12 faces and 18 edges, which gives 8 + 12 − 18 = 2. If this relationship holds true for all polyhedrons without holes, it can be used to calculate how many Platonic solids there are.
But why should Euler’s polyhedron formula V − E + F = 2 always hold true for vertices, V, edges, E, and faces, F? As it turns out, this is a topological invariant: a quantity that is the same for all solids that can be deformed into one another without tearing or sticking together.
The Euler formula proves that there are only five Platonic solids. These symmetrical shapes each consist of faces that are identical polygons (n-gons), where each vertex is intersected by exactly m edges. To find out which polyhedrons can fulfill these properties, we look for all possible natural numbers n and m that satisfy Euler’s polyhedron formula.
To do this, one must first express the number of faces and vertices of a polyhedron in terms of the numbers n and m. If m edges intersect each vertex and each edge is bounded by two vertices, then m × V = 2E. At the same time, an edge always forms a boundary between two faces. Because each face consists of n edges, it follows that: n × F = 2E.
Substituting these findings into Euler’s polyhedron formula, one obtains (2E)/m − E + (2E)/n = 2. This expression can be simplified by finding a common denominator for the fractions, combining terms, and taking the reciprocal: 2n – mn + (2m)/(2mn) = 1/E.
A little further rearrangement then yields the expression 1/m + 1/n = 1/E+ 1/2. The term 1/E simply increases the value on the right-hand side because it is always positive, so we can instead consider the following inequality for n and m: 1/m + 1/n > 1/2.
Therefore, to generate a Platonic solid, the sum of the reciprocals of n and m must be greater than 1/2. Because n and m are natural numbers and must be greater than 3, given that otherwise no polyhedron is possible, this allows only five possibilities: (3,3), (3,4), (4,3), (3,5) and (5,3), which correspond to the five Platonic solids.
Thanks to Euler’s polyhedron formula, we have found conclusive proofthat there are exactly five Platonic solids. That equation represents a much more general concept that can be applied to all sorts of figures and spatial dimensions. For example, you can cover a surface with a net of triangles and then calculate the polyhedron formula V − E + F. The result will always be the same, regardless of whether the net consists of six or 600 triangles. For a sphere, as with polyhedrons, the result is two. But for other surfaces, such as a bagel, the value is zero.
The more general V − E + F = χ is also a topological invariant, and this characteristic distinguishes different objects from one another from a topological perspective. Therefore, if χ is different for two figures, then they are also topologically different. For ordinary two-dimensional surfaces, V − E + F = 2 – 2 × L, where L is the number of holes in the surface. Therefore, for a pretzel with three holes, χ = -4.
Thus, the formula proposed by Euler has become an indispensable mathematical tool. With its help, topologists come a step closer to their goal of finding suitable quantities to categorize different structures.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.