Friday, January 16, 2015
Consider an m-section of an n-space. It is a coincidence for more than n-m boundaries in the n-space to contain the m-section. An m-section in an n-space is isomorphic to a 0-section in an (n-m)-space; just take any (n-m)-section by boundaries not coincident with the m-section. Consider the subspace of an n-space consisting of just the boundaries coincident at a 0-section. This subspace is isomorphic to an (n-1)-dimensional round sidedness space defined as follows. The number of regions in every subspace of a constructable round sidedness space is 2*f(n,m-1), where n is the dimension, m is the number of boundaries, and f is the number of regions in a flat sidedness space. Every region in a round sidedness space is an inside region and has a complement. To convert between flat and round sidedness spaces, construct the planes through the flat space boundaries and the center of the sphere. And construct the plane through the center parallel to the flat space. Since the round space has one more boundary than the flat space, there is a flat space for each boundary in the round space. Thus, flat sidedness spaces of n dimensions and m boundaries are classified into disjoint sets, each corresponding to a different one of all possible kinds of coincidence between m+1 boundaries in (n+1)-dimensional space. Spaces in the same class are called rotations of one another. A set of boundaries B in an n-space can collapse to coincide at a 0-section iff none of the inside regions of the B-subspace are divided. Then a new n-space may be found by finding a new B-subspace with the same outside regions. This is a generalization of migration defined in previous posts.