Ciro Santilli

Classification of regular polytopes | 🗖 nosplit | ↑ parent "Regular polytope" | 205

The 3D regular convex polyhedrons are super famous, have the name: platonic solid, and have been known since antiquity. In particular, there are only 5 of them.
The counts are:
Table 1. Number of regular polytopes per dimension.
Dimension Count
2 Infinite
3 5
4 6
>4 3
The cool thing is that the 3 that exist in 5+ dimensions are all of one of the three families:
  • simplex: triangle, tetrahedron.
    Non-regular description: take convex hull take D + 1 vertices that are not on a single D-plan.
  • hypercube: square, cube. 4D case known as tesseract.
    Convex hull of all (Cartesian product power) D-tuples.
    Two are linked iff they differ by a single number. So each vertex has D neighbors.
  • cross polytope: square, octahedron.
    All permutations of
    Each edge E is linked to every other edge, except it's opposite -E.
Then, the 2 3D missing ones have 4D analogues and the sixth one in 4D does not have a 3D analogue: the 24-cell. Yes, this is the kind of irregular stuff Ciro Santilli lives for.