Dear Uncle Colin,

Why are there only five platonic solids?

- Pentagons Look Awful. Try Octagons!

Hi, PLATO, and thanks for your message!

A platonic solid is a three-dimensional shape with the following rules:

• Each face is the same regular polygon
• The same number of edges meet at every vertex

There are (in three dimensions), exactly five of them:

• the tetrahedron, made up of four equilateral triangles (three of which meet at each vertex)
• the octahedron, made up of eight equilateral triangles (four of which meet at each vertex)
• the icosahedron, made up of twenty equilateral triangles (five of which meet at each vertex)
• the cube, made up of six squares (three of which meet at each vertex)
• the dodecahedron, made up of twelve pentagons (three of which meet at each vertex)

### How come these are the only ones?

There’s a tiny clue in how I’ve listed them: it’s to do with fitting the shapes around each vertex. If you arrange your regular polygons around a point, you need to be able to do so without any overlap - but more to the point, you need some empty angle so you can fold the shape up! You also need to have at least three faces meeting at each vertex, or else there would be a gap, and we can’t have that.

So, with triangles, we can place three around a point with no problem; with four or five, we still have a gap; but when we put six next to each other, it all lies flat - we can’t make a regular polyhedron out of triangles with six or more meeting at each vertex.

With squares: three is fine, but four would also tile the plane.

With pentagons: we’d need to think. The interior angle, the one we care about, is $\frac{3}{5}\pi$… oh, fine, 108º. Three of those make $\frac{9}{5}\pi$, which is less than $2\pi$ (or, if you must, 324º, which is less than 360º); four of them make $\frac{12}{5}\pi$ (or 432º), which give you an overlap. Three pentagons would be fine; four would not.

With regular hexagons, having three meet at a vertex means they tile the plane again. With any larger number of sides, three shapes meeting at a vertex would overlap.

And that’s it: the only regular polygons that can meet at-least-three-to-a-vertex and leave a gap are triangles (3, 4 or 5), squares (3) and pentagons (3). Each of these gives a different platonic solid, making five altogether.

Hope that helps!

- Uncle Colin