In n dimensions a n-hedron is a set of oriented (n−1)-simplexes (called faces here) such that each (n−2)-simplex of one face is also a (n−2)-simplex of just one other face, but with the opposite orientation. This boundary is a simplicial complex. Note that the simplexes need not be disjoint and the union of them need not be connected. A hedron defines a winding number which maps points in the n-space to integers. The winding number is not defined within the faces proper. If the faces are disjoint and their union is connected then the winding numbers are either 0 and 1, or 0 and −1.

We shall show that if k < n the intersection of a n-hedron and a k dimensional subspace is a k-hedron. Within the subspace, the winding number defined by the k-hedron is the same as the n-hedron We consider how to compute this intersection for k=n−1.

A (n−1)-subspace divides the vertices of a face into two sets. For each face of the n-hedron with i vertices in one set and n+1−i vertices in the other, the intersection of that face and the subspace is a set of points that is the union of interiors of xx (n−1)-simplexes in the subspace. The boundary of the intersection is ...

Here is a program to find repeated (n−2)-simplexes.