Let $d\Delta$ be the simplex that's the convex hull of $(0, 0, 0, 0), (d, 0, 0, 0), (0, d, 0, 0), (0, 0, d, 0), (0, 0, 0, d)$. A unimodular triangulation of $d\Delta$ is a subdivision of it into smaller simplices with volume $1/d^4$ times the total volume of the simplex. What is the number of total edges, in terms of $d$, of such a triangulation?
I am able to find the number of simplices, tetrahedra, boundary triangles, and boundary edges, but I still need to find the number of interior triangles and edges. Once I have one of the two, I can get the other by using the fact that the Euler Characteristic of the polytope is $1$.