Vertices coordinates for the regular 4-polytopes in 4d

Question

Hi there I'm looking for the coordinates for each point of the 4 regular polytopes in 4d (we can use side=1 for simplicity). I have found them for the tesseract. But I don't know how to obtain the rest.

Answer 1

You can get these from the pertinent Wikipedia articles. Note that there are six, not four; see https://en.wikipedia.org/wiki/Regular_4-polytope#Regular_convex_4-polytopes .

For example, for the 24-cell, the article states:

The 24-cell is the convex hull of its vertices which can be described as the 24 coordinate permutations of: $$ (\pm 1,\pm 1,0,0)\in \mathbb {R} ^{4}.$$

(https://en.wikipedia.org/wiki/24-cell#Coordinates)

This means that the vertices are at any arrangement of two zeroes, and two plus or minus ones, in any order, including: $(0,0,1,1), (0,0,1,-1), … (-1,0,1,0), … (-1, -1, 0, 0)$.

A little later the article gives two other possible sets of coordinates, corresponding to different positions of the polytope in space.

For the 120-cell:

The 600 vertices of a 120-cell with an edge length of ${2\over φ^2} = 3−√5$ include all permutations of:

$$ (0, 0, ±2, ±2) \\ (±1, ±1, ±1, ±√5) \\ (±φ^{−2}, ±φ, ±φ, ±φ) \\ (±φ^{−1}, ±φ^{−1}, ±φ^{−1}, ±φ^2) $$

and all even permutations of

$$ (0, ±φ^{−2}, ±1, ±φ^2) \\ (0, ±φ^{−1}, ±φ, ±√5) \\ (±φ^{−1}, ±1, ±φ, ±2) $$ where $φ$ (also called τ) is the golden ratio, $1 + √5\over2$.

https://en.wikipedia.org/wiki/120-cell#Cartesian_coordinates

An “even” permutation means that you take one of the listed sets of four coordinates, and then switch any two of them an even number of times.