by wendy » Tue Jan 10, 2006 8:32 am
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