Complexity of the Simplex

The n-simplex is the simplest n-polytope to describe.
At the same time, they're harder, in some ways, to get ones head around.

I'm not sure what we're going to do with them exactly. Definitely, we'll count their sub-simplexes. We'll construct regular n-simplexes inscribed in a unit (n-1)-sphere. We'll use the above construction to explore how spherical distance and Euclidean distance behave.

Not sure what else.

Image from: https://en.wikipedia.org/wiki/File:Simplexes.jpg