Dual of 7 dimensional regular simplex

Question

I have the vertices of a 7-simplex that can be inscribed in a 7-cube. Given this information, how can I find the vertices of its dual simplex? (Note that the vertices of this dual should be the vertices of the cube)

Answer 1

A dual of a simplex is produced by central symmetry. Just invert all coordinates, and that would be it.

This is not necessarily the case with all other self-dual polytopes (certainly not with $F_4$, for example), but that's another story.

How did I know that? Well, let's say our n-hypercube has all coordinates +1 or -1, with center at 0. A dual of an n-simplex must be another simplex, which sits in such a position that its vertices are at the centers of (n-1)-hyperfaces of our simplex, scaled accordingly. What are those hyperfaces? Why, that's simple: each of them contains all but one vertices of the simplex. What is the sum of coordinates of all but one vertices? Again, that's simple: just the coordinates of the remaining vertex, taken with the opposite sign.

So it goes.