Coordinates of Oca

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Coordinates of Oca

Postby Mecejide » Sun Apr 26, 2020 10:44 pm

The coordinates of a unit octaexon centered on the origin can be given as:
{1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4}
{1/4, 1/4, 1/4 ,-1/4, -1/4, -1/4, -1/4}
{1/4, -1/4, -1/4, -1/4, -1/4, 1/4, 1/4}
{1/4, -1/4, -1/4, 1/4, 1/4, -1/4, -1/4}
{-1/4, 1/4, -1/4, 1/4, -1/4, 1/4, -1/4}
{-1/4, 1/4, -1/4, -1/4, 1/4, -1/4, 1/4}
{-1/4, -1/4, 1/4, 1/4, -1/4, -1/4, 1/4}
{-1/4, -1/4, 1/4, -1/4, 1/4, 1/4, -1/4}
Is this known?
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Re: Coordinates of Oca

Postby URL » Mon Apr 27, 2020 5:38 am

I don't think so, I haven't found similar coordinates for the 7-simplex elsewhere.

How did you arrive at these coordinates?
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Re: Coordinates of Oca

Postby URL » Mon Apr 27, 2020 6:41 am

Actually, there’s a pretty cool way to derive these coordinates.

Consider the Fano plane, and enumerate it’s vertices and lines arbitrarily. Then, consider seven 7D points, and set the coordinate i of point j to -1 if the i-th point of the Fano plane lies in line j, and 1 otherwise. Every two points will have three coordinates in common, and three negative signs. Adding the point (1,1,1,1,1,1,1) completes the construction of a regular 7-simplex with edge length 4.

Unfortunately, I’m not sure if this generalizes.
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