In another problem we showed an analogue of this occurs in infinitely many higher dimensions. Ie It is possible to make a regular n-simplex who's vertices all lie on the vertices of a regular n-hypercube. Specifically, one can show that this happens for 7-simplices and 7-hypercubes. I have 2 questions that I have tried to answer but would like to see if anyone gets the same result. SO here are the questions.
How many distinct 7-simplices that have vertices on a 7-hypercube all share at least one common vertex?
How many distinct 7-simplices in total have vertices on a common 7-hypercube?
What's the size of an orbit of a single 7-simplex in the set of all the 7-simplices under the group of the symmetries of the 7-hypercube?
Details and Assumptions:
A regular -dimensional simplex in has vertices that are all an equal distance apart. (It's like an -dimensional version of an equilateral triangle!)
Here is the Wikipedia article on hypercubes. (It's like an -dimensional version of a square!)