# Talk:3 21 polytope

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Field: Geometry

## Coordinates

The 56 vertices can be most simply represented in 8-dimensional space, obtained by the 28 permutations of the coordinates and their opposite: ± (-3, -3, 1, 1, 1, 1, 1, 1)

Why "-3" rather than "3" here? —Tamfang (talk) 22:57, 25 July 2010 (UTC)

That's what Coxeter uses - I can't explain. Is it because its a 7-subspace of 8-space?! Tom Ruen (talk) 19:50, 26 July 2010 (UTC)
Consider an equilateral triangle in 3-space, at (1,0,0), (0,1,0), (0,0,1). (as the facet of an octahedron) You then can center it to the origin by subtracting (1/3,1/3,1/3) to (2/3,-1/3,-1/3), ..., and rescale to whole numbers as (2,-1,-1), (-1,2,-1), (-1,-1,2). That's the same issue as the 3_21 polytope in 7-space. The origin center demands the sign reversal. Tom Ruen (talk) 06:42, 27 July 2010 (UTC)
If you could see me do it I'd slap my forehead. —Tamfang (talk) 08:06, 28 July 2010 (UTC)