Polygons and PolyhedraPlatonic Solids

At the beginning of this course we defined regular polygons as particularly “symmetric” polygons, where all sides and angles are the same. We can do something similar for polyhedra.

In a regular polyhedron all faces are all the same kind of regular polygon, and the same number of faces meet at every vertex. Polyhedra with these two properties are called Platonic solids, named after the Greek philosopher Plato.

So what do the Platonic solids look like – and how many of them are there? To make a three-dimensional shape, we need at least faces to meet at every vertex. Let’s start systematically with the smallest regular polygon: equilateral triangles:

And seven or more triangles at every vertex also don’t produce new polyhedra: there is not enough space around a vertex, to fit that many triangles.

This means we’ve found Platonic solids consisting of triangles. Let’s move on to the next regular polygon: squares.

Next, let’s try regular pentagons:

If pentagons meet at every vertex, we get the Dodecahedron. It has faces. (“Dodeca” means “twelve” in Greek.)

The next regular polygon to try are hexagons:

If three hexagons meet at every vertex, we immediately get a tessellationpolyhedronhexahedron. Since there is no space for more than three, it seems like there are no Platonic solids consisting of hexagons.

The same also happens for all regular polygons with more than six sides. They don’t tessellate, and we certainly don’t get any three-dimensional polygons.

This means that there are just Platonic solids! Let’s have a look at all of them together:

Notice how the number of faces and vertices are swapped aroundthe same for cube and octahedron, as well as dodecahedron and icosahedron, while the number of edges stays the sameare different. These pairs of Platonic solids are called dual solids.

We can turn a polyhedron into its dual, by “replacing” every face with a vertex, and every vertex with a face. These animations show how:

The tetrahedron is dual with itself. Since it has the same number of faces and vertices, swapping them wouldn’t change anything.

Plato believed that all matter in the Universe consists of four elements: Air, Earth, Water and Fire. He thought that every element correspond to one of the Platonic solids, while the fifth one would represent the universe as a whole. Today we know that there are more than 100 different elements which consist of spherical atoms, not polyhedra.

Images from Johannes Kepler’s book “Harmonices Mundi” (1619)

Archimedean Solids

Platonic solids are particularly important polyhedra, but there are countless others.

Archimedean solids, for example, still have to be made up of regular polygons, but you can use multiple different types. They are named after another Greek mathematician, Archimedes of Syracuse, and there are 13 of them:

Applications

Plato was wrong in believing that all elements consists of Platonic solids. But regular polyhedra have many special properties that make them appear elsewhere in nature – and we can copy these properties in science and engineering.