# Talk:Birkhoff polytope

## what are the coords of Bn?

H A M 1 (talk) 02:54, 2 January 2023 (UTC)

It's literally stated in the page --Galoomba (talk) 08:08, 2 January 2023 (UTC)
the coordinates are matrices, not regular coords like cube, H A M 1 (talk) 17:06, 2 January 2023 (UTC)
in Triangular duotegum H A M 1 (talk) 17:20, 2 January 2023 (UTC)
In the first paragraph of the article, it says that matrices are converted to points by unraveling the entries in reading order. Vel (talk) 17:43, 2 January 2023 (UTC)

what's B4 H A M 1 (talk) 20:53, 2 January 2023 (UTC)

I've added some info, but I would be very interested to see a visualization of some kind. Vel (talk) 06:39, 3 January 2023 (UTC)

## which plane a birkhoff polytope lies in

H A M 1 (talk) 18:55, 1 August 2023 (UTC)

The plane passes through the matrix ${\displaystyle B_{\star }}$ all of whose entries are 1/n. If the polytope is translated to center on the origin, it's contained in the linear subspace spanned by the n! vectors ${\displaystyle B_{i}-B_{\star }}$ where each B_i is a permutation matrix. I'm not certain how to whittle this spanning set down to an actual basis but this may be relevant: https://math.stackexchange.com/questions/70569/span-of-permutation-matrices. Vel (talk) 19:13, 1 August 2023 (UTC)
i mean the minimum flat (it's 4d in b3)H A M 1 (talk) 02:33, 2 August 2023 (UTC)
coordinates of dual of Bn? H A M 1 (talk) 02:35, 2 August 2023 (UTC)