Is this the best sequence ever?

platonic solids

I was idly browsing Quora, when I came upon the following sequence:

1, ∞, 5, 6, 3, 3, 3, ...

It is highly unusual to have infinity as a term in the middle of a sequence – whatever could it be?

Well, the nth term is the number of convex regular polytopes in n dimensions. Ah! That explains it?

It’s easiest to start with the second term of the sequence. It’s the number of regular polygons that you can make. A regular polygon is a plane figure with straight sides all of equal length. The most obvious is perhaps a square. But we additionally require the polygons to be convex: this means that the sides cannot turn in on themselves. The figure below on the left is a convex regular pentagon; the figure on the right is also a regular pentagon, but it is not convex.

not a convex polygon

It should be fairly obvious that you can construct regular polygons with any number of sides:

convex polygons

So the number of convex regular polygons is infinite. And that’s the second term of the sequence.

What about the others? Let’s look at the third term: 5. This is the number of regular polyhedra. That is, the number of three dimensional shapes, where each face is a regular polygon. (Again, we require them to be convex.) The five regular polyhedra are called the Platonic solids, and they are illustrated in the photograph at the top of this blog. From left to right: icosahedron, dodecahedron, cube, octahedron and tetrahedron. It is not possible to construct any other regular polyhedra: the angles won’t fit together to form a closed solid.

In informal language, then, the sequence is the number of regular shapes in one dimension, two dimensions, three dimensions, and so on. The word “polytope” is the generic term that covers polygons (two dimensions), polyhedra (three dimensions), and all the others in higher dimensions. Perhaps surprisingly, it is the two-dimensional world that offers the greatest variety.