TITLE: Simplest Proof of the Six Regular 4-Polytopes
QUESTION [8 upvotes]: I'm not necessarily looking for a rigorous proof, more an outline of how an undergrad could count the regular 4D polytopes (and perhaps investigate what they look like) as explicably as possible.

REPLY [3 votes]: Numberphile made a video on exactly this topic a while back. The idea is to consider the dihedral angle between adjacent faces for each of the five Platonic solids – a polychoron will consist of instances of one such solid. For the polychoron to be valid polychoron, at least three cells must meet at an edge and the sum of dihedral angles in 3-dimensional space must be strictly less than 360°.

Tetrahedron (dihedral angle 70.5°): three, four or five tetrahedra can share an edge in 3D without overlapping. Bent into 4D space, these configurations give the 5-cell, 16-cell and 600-cell respectively.
Cube (90°): three cubes around an edge yields the 8-cell or tesseract. Four around an edge, however, completely fills it up, yielding the ordinary cubic honeycomb and not any polychoron.
Octahedron (109.5°): only three octahedra can fit around an edge, yielding the 24-cell.
Dodecahedron (116.6°): the situation is the same as for the octahedron, and yields the 120-cell.
Icosahedron (138.2°): since this is larger than 120°, three of these cannot fit around an edge in the first place, so no regular polychoron has icosahedral cells.