This is the first in a series of posts about the Platonic solids. These posts will show how to find the corners of six Platonic solids: tetrahedron, cube (hexahedron), octahedron, dodecahedron, and icosahedron.
Deriving these values requires only algebra and a little trigonometry. Even so, they can be a bit involved so it’s not hard to make mistakes. After using the formulas shown to find the points, the posts include 3D WPF programs that draw the solids to verify the results. Those programs also calculate the lengths of the solids’ sides to verify that they all have the same length.
What are the Platonic Solids?
A regular polygon is a two-dimensional shape where each edge has the same length and the edges all make the same angles with respect to each other. Figure 1 shows two quadrilaterals. The square on the left is a regular polygon because all its sides are the same length and they all meet at 90 degree angles. The parallelogram on the right is not regular. While its sides have the same lengths, they do not all meet at the same angles.
A polytope is basically a “polygon” or “closed figure” in a given dimension. For example, cubes and pyramids are three-dimensional polytopes.
The Platonic solids were defined by the Greek mathematician and philosopher Plato (427-347 BC). They are all of the three-dimensional solids that you can define using faces that are identical regular polygons. These solids are also known as the three-dimensional regular polytopes or the regular solids.
The Platonic solids include the tetrahedron (4 triangular faces), cube or hexahedron (6 square faces), octahedron (8 triangular faces), dodecahedron (12 pentagonal faces), and the icosahedron (20 triangular faces).
No other regular polytopes are possible in three dimensions. You may have seen solids made up of more faces, each of which is identical. For example, some game stores sell 30-sided and even 100-sided dice. The faces of these solids are parallelograms not regular polygons, so they are not regular polytopes. Similarly you can make geodesic domes using identical triangles. The triangles are not equilateral (they have different side lengths), however, so the dome is not a regular polytope.
There are other regular polytopes in higher dimensions. For example, “cubes” are defined for all higher-dimensional spaces. There are four-dimensional cubes, five-dimensional cubes, an so forth. There are even some regular polytopes that do not correspond to any two- or three-dimensional polytopes.
Two-dimensional space has an infinite number of regular polytopes because you can make a regular polygon with any number of sides: triangle, square, pentagon, hexagon, and so forth. Four-dimensional space has the next most regular polytopes, although I can’t remember how many it has or how many faces they have. I think I remember it having four-dimensional counterparts to tetrahedrons, cubes, and octahedrons (I think all dimensions have those) but I don’t remember what else. If someone knows, post a comment.
The Platonic solids have some rather interesting dual relationships. To make the dual of a solid, you place a vertex in the center of each of the solid’s faces. Then connect each vertex to the vertices on the adjacent faces. For the Platonic solids, the result is another Platonic solid.
Figure 2 shows a cube and its dual: an octahedron.
Table 1 lists the Platonic solids and their duals.
Table 1: The duals of the Platonic solids.
The number of vertices, faces, and edges in duals have a reciprocal relationship. For example, a cube has 6 faces and 8 vertices while an octahedron has 8 faces and 6 vertices. Both have 12 edges. Table 2 lists the number of faces, vertices, and edges in the Platonic solids.
Table 2: The number of faces, vertices, and edges in the Platonic solids.
The number of faces, edges, and vertices are related to the shape and arrangement of the faces. To understand the relationship, define the following values:
|F||= Total faces in the solid|
|E||= Total edges in the solid|
|V||= Total vertices in the solid|
|EF||= Number of edges on each face|
|VF||= Number of vertices on each face|
|SE||= Number of faces that share each edge (always 2)|
|SV||= Number of faces that share each vertex|
|E||= F * EF / SE|
|V||= F * VF / SV|
For example, an icosahedron has 20 triangular faces with each vertex shared by 5 faces so F = 20, EF = 3, VF = 3, SE = 2, and SV = 5. Plugging these numbers into the previous equations gives:
|E||= 20 * 3 / 2||= 30|
|V||= 20 * 3 / 5||= 12|
That’s enough background to get started. In my next post, I’ll show how to find the vertices for a tetrahedron.