Coordinates of 120-cell vertices

[quickfur]

Hi, does anyone know where I can get a list of the vertices of the 120-cell in precise, algebraic format? (I.e., not just a list of points with floating-point numbers, but something expressed as algebraic numbers, like (1,0,0,1+sqrt 5).) Thanks!

[wendy]

Let's see

1. Let f=1.61803398875&c, fN = f**N

2. Let v = 1/f amd vN = v**N

3. Let r5 = sqrt(5)

The following are single vertices in the pyrochoral symmetry, that is, even change of sign, and all permutations of coordinates. The first two rows belong to an inscribed {3,3,5}

row 1
2 2 0 0 = 24 v

row 2
f2 v v v = 32+32 v
v2 f f f = 32 +32 v
-r5 1 1 1 = 32 +32 v

row 3
f2 1 v2 0, = 96 v
r5 f v 0, = 96 v
2 f 1 v, = 192 v

being 192, 96, 96 v zS 384 +96 from doubling row 2, + 24,
makes 600 all together.

For the {3,3,5} it's all permutations, all change of sign
For the {5,3,3} it's even permutations, all change of sign

W

[coordinates from HSM Coxeter 'regular polytopes'.

W

Edited to make the coordinates easier to spot, plus correct the count of verticies derived from this random point.

W

[quickfur]

Let's see

1. Let f=1.61803398875&c, fN = f**N

Wait, is f = (1+sqrt 5)/2 ?

[wendy]

of the numbers mentioned

r5 = sqrt(5)

f = (1+r5)/2 = 0.5+0.5 r5

f2 = f+1 = 1.5 + 0.5 r5

v = f-1 = -0.5 + 0.5 r5

v2 = 2-v1 = 1.5 - 0.5 r5

W