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 ofEach 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.
- Regular polytope | 5, 256, 2
- Polytope | 0, 256, 3
- Geometry | 0, 305, 8
- Mathematics | 17, 10k, 239
- Ciro Santilli's Homepage | 238, 147k, 2k
- Platonic solid | 38
- The beauty of mathematics | 257, 548, 4