So I was watching this experimental gameplay video for Miegakure, a 4d indie video game. It's like Fez, which was a 2d platformer in a 3d world that involved "rotating" things in 3d to solve puzzles, except lifted one level, so it's a 3d game in a 4d world.

Anyway, it mentioned the 24-cell, a weird 4d shape that I've always wanted to understand better, so I started on a Wikipedia binge. This immediately led me back to the platonic solids, when I realized something.

It always struck me as weird that there are an infinite number of regular 2d polygons, one for every number of sides - equilateral triangle, square, regular pentagon, regular hexagon, etc - but there were only five regular 3d polyhedrons - with dice names, the d4, d6, d8, d12, and d20. Going from infinite to finite is always interesting, and I didn't get it before, but I got to thinking about the patterns in the shapes, and suddenly it clicked - there just isn't enough space for any more regular polyhedrons!

See, the dice shapes can be split into two lines. The d4, d6, and d12 form the "increasing polygon side-count" line, while the d4, d8, and d20 form the "increasing number of triangles" line. (Yes, the d4 is the base of both.)

The "increasing polygon side-count" line goes from triangle (3) sides, to square (4) sides, to pentagon (5) sides. As you increase the number of sides, the angles get larger, so you have to lay the shapes out "flatter" to get them to fit together snugly, reducing the curvature and increasing the size of the die. But what happens when you fit hexagons together? They tile the plane. Put hexagons together snugly, and they perfectly fill up a flat 2d plane - 120° × 3 = 360°. You can't curve them at all, there's not enough space. So there's no way to make a polyhedron with a regular hexagon - it can't curve around to form a ball!

The same argument happens with "increasing numbers of triangles". It's easy to fit together three triangles - you get the d4. Spread them out a little and you make space for a fourth, getting the d8. A little more flattening makes room for a fifth, making the d20. You can still squeeze in one more triangle, putting six together snugly, but only by flattening them into a plane again. It's the same problem as the hexagon tiling - 60° × 6 = 360° again. The triangles don't curve around at all, so you once again can't form a ball!

7 (whether 7-sided shapes, or 7 triangles) is of course right out. You can't fit those together no matter how you try.

I presume this is why there are only three regular solids in 5+ dimension: the simplex (analogue of the d4), the measure complex (analogue of the d6), and the cross complex (analogue of the d8). In higher dimensions the volumes take up more "room", so you just can't fit as many together, and aren't able to go up to the "5" shapes (analogues of the five-sided or 5-triangled shapes).

This still doesn't answer what the fuck is up with four dimensions, tho. 4d has six regular solids - 5 analogues of the 3d solids, and 1 special. There's the three made out of increasing-size dice: the 5-cell, made out of d4s; the 8-cell, made out of d6s; and the 120-cell, made out of d12s. There's the three made out of tetrahedrons: the 5-cell, where 4 meet at each vertex; the 16-cell, where 5 meet at each vertex; and the 600-cell, where 6 meet at each vertex.

Then there's the 24-cell, made out of d8s. What the fuck. Geometry is weird.