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:

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 $(−1,1)_{D}$ (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 $(±1,0,0,…,0)$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.

- Regular polytope | 5, 273, 2
- Polytope | 0, 273, 3
- Geometry | 0, 2k, 34
- Mathematics | 17, 13k, 336
- Ciro Santilli's Homepage | 262, 182k, 3k

- Platonic solid | 40
- The beauty of mathematics | 259, 552, 4