There are a number of things to measure about the regular simplex of n dimensions and a variety of ways to do so. These various methods may indeed make some new connections between mathematical subjects. Sometimes we resort to studying general simplexes too.
The regular 2D simplex is the equilateral triangle, with angles of π/3. The regular 3D simplex is a tetrahedron, with dihedral angles of cos−1(1/3) and solid angles, or trihedral angles at each vertex of 3cos−1(1/3) − π. The regular n dimensional simplex has n−1 different angles associated with it — j-hedral angles, αnj, for 1 < j < n+1. I find these numbers popping up in several contexts and need to find their values and understand the n dimensional generalizations.
Here is a definition of k-hedral angles for n dimensional polytopes, including simplexes and especially regular simplexes.
α22 is the angle of an equilateral triangle. It is 1/6 of a circle or π/3 radians. α32 is the dihedral angle of the tetrahedron (= cos−1(1/3)) and α33 is the trihedral angle at the vertex of the tetrahedron. An0 = 1 and An1 = 1/2. αn2 = cos−1(1/n) which is Coxeter’s result which we hope to confirm here. Ann+1 = 0.
Here I use several well known properties of normal distributions to provide Monte Carlo estimates of the k-hedral angles. Here are more Monte Carlo schemes for finding the magnitude for A2424. Here we seek an extension to the classic spherical triangle area formula by which we computed A33.
I show how to compute the normal vectors to the faces of an arbitrary simplex given its edge lengths. The angles between the face normals is the dihedral angle between the faces. One can calculate trihedral angles from 3 normals. This provides αn3 for general simplexes. See this program.
more simplexes stuff