Intersection of Boundaries

I don’t know that we must characterize the intersection of the boundaries of our hedrons, but here goes anyway.

In 2D the boundaries are 1D and their intersection is 0D—a set of oriented points. (At some points polygon X enters Y and Y leaves X.) In n dimensions the intersection is a (n−2)-D complex.

In 3D the intersection of boundaries is a 1D complex whose vertices are at the piercings of a 2D face (boundary element) by a 1D edge of the other hedron. These do not alternate.

I stretch my intuition to 4D. The boundary intersection is 2D and its vertices (0D elements) are:

piercings of hedron X’s 3D faces with 1D edges of hedron Y,
Intersections of the 2D complexes of the mutual boundaries,
piercings of hedron Y’s 3D faces with 1D edges of hedron X.

The 1D elements of the boundary intersection are:

Intersections of hedron X’s 3D faces with 2D simplex in hedron Y’s boundary,
Vice, versa

I think I can generalize: Given hedrons X and Y in n dimensions, the intersection of their (n−1) dimensional boundaries, Xb and Yb is a (n−2) complex BI. (The boundary of BI is empty.) I think that BI is a simplicial complex. For each simplex Bs in BI:

there are k and m such that 0≤k≤m≤n−2,
there is a k-simplex Xs of Xb and a (n−k−m)-simplex Ys of Yb,
Bs = Xs ∩ Ys is a m-simplex.

This needs clarification for clearly the intersection of simplexes is seldom a simplex. The intersection of a k-simplex and (n−k−m)-simplex in n dimensions, is however, an m-simplex, I think. It is not generally a sub-simplex of either of the two given simplexes, however.

Alas this is false. In 4D consider the two tetrahedra which are the convex hulls of
{[1, 1, 1, 1], [1, −1, −1, 1], [−1, 1, −1, −1], [−1, −1, 1, −1]} and
{[1, −1, 1, −1], [1, 1, −1, −1], [−1, 1, 1, 1], [−1, −1, −1, 1]}. If the coordinates are called [w, x, y, z] note that w=z in the first simplex and w=−z in the second. This must thus be true as well of the respective hulls and thus w=z=0 for their intersection. The points [0, 1, 0, 0], [0, 0, 1, 0], [0, −1, 0, 0], [0, 0, −1, 0] are mid points of edges of each of the two hulls. The square that is the convex hull of these four points is thus the intersection of the two original tetrahedra. Bahh.