Classification of regular polytopes

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 per dimension are:

The cool thing is that the 3 that exist in 5+ dimensions are all of one of the three families: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.

Triangle, tetrahedron.

The name does not imply regular by default. For regular ones, you should say "regular polytope".

Non-regular description: take convex hull take D + 1 vertices that are not on a single D-plan.

square, cube. 4D case known as tesseract.

Convex hull of all $\{-1,1{\}}^D$ (Cartesian product power) D-tuples, e.g. in 3D:

From this we see that there are $2^D$ vertices.

Two vertices are linked iff they differ by a single number. So each vertex has D neighbors.

The non-regular version of the hypercube.

Examples: square, octahedron.

Take $(0,0,0,\dots ,0)$ and flip one of 0's to $\pm 1$. Therefore has $2\times D$ vertices.

Each edge E is linked to every other edge, except it's opposite -E.