The Simplex

The simplex is a generalization to n-dimensions of the triangle in 2D and the tetrahedron in 3D. An n-simplex is the convex hull of n+1 points that do not all lie in some n–1 dimensional subspace. We write H(x) for the convex hull of the set of points x. For n from 0 to 3 the n-simplexes are:
nn-simplex
0point
1line segment
2triangle
3tetrahedron
It goes on. We say that the n+1 points span the n-simplex.

If V is the set of vertices of a simplex then H(V) is the simplex itself. For each of the 2n+1 subsets v of V the simplex H(v) is a subsimplex of H(V). That includes the –1-simplex which is the null set (Ø) which is spanned by Ø. The null simplex will not much concern us. The subsimplexes with the same number of vertices are generally considered together in proofs.

See Wolfram’s Simplexes.

Within an n-simplex there is a notion of two sub simplexes opposite each other. If v1 ∩ v1 = Ø and v1 ∪ v2 = V then H(v1) and H(v2) are opposite sub-simplexes. For instance in a triangle there is the side opposite a vertex and vice-versa. In a tetrahedron there is a face opposite a vertex, an edge opposite another edge.

The the angles of a simplex. Barycentric coordinates for a simplex, An Application to Physics
The following article may have answered all of my questions: Affentranger and Schneider, titled "Random Projections of Regular Simplices", in Discrete and Computational Geometry, vol.7, no.3, 1992, pp. 219-226
Don Chakerian told me of this article.