## Monday, December 31, 2007

### Overlap Examples

Here are the 11 triangle overlaps. The top left one differs from the disjoint one by a single migration. I call this kind of migration, where a point jumps across a boundary, a point migration.

Here are three tetrahedron overlaps. The right two differ from the disjoint one by single migrations. The middle one is a point migration since it is a point jumping across a boundary. The rightmost one is a segment migration since it is a segment jumping across another segment.

## Linear Space Equivalence

Two linear spaces are equivalent iff they are isomorphic to the same finite set of nonparallel noncoincidental linear surfaces. Two finite sets of nonparallel noncoincidental linear surfaces are equivalent iff they are isomorphic to the same linear space. A finite set of nonparallel noncoincidental linear surfaces are said to be equivalent to a linear space iff they are isomorphic.

## Migration

In space Sn = (B,P,S,n), regions R1 and R2 are neighbors with respect to b0∈B iff R1(b) = R2(b) and R1(b0)R2(b0), for b∈B0, where B = B0∪{b0}. In space Sn = (B,P,S,n), region R2 is the opposite of region R1 iff for b∈B, R1(b)R2(b)R1 has a nonempty neighbor with respect to b. Two spaces S1n = (B,P1,S,n) and S2n = (B,P2,S,n) are migrations of each other iff for each region R in S1n and S2n, one or both of the following is true. R is empty in S1nR is empty in S2n. Or, R is empty the opposite of R is not empty.

## Example

Regions A, B, and C have empty opposites. Region A is the opposite of region E. Region E is the opposite of region D.

## Migration Theorem

If S1n and S2n are migrations of each other then, S1n is linear S2n is linear.

Without loss of generality, assume S1n is linear, R is empty in S2n,
and its opposite is not empty in S2n. Suppose S3n is any subspace of S2n. If R is not a region of S3n, then it is linear because its identical subspace in S1n is linear. If R is a region of S3n, then so is its opposite, so the number of nonempty regions in S3n is unaltered, so it is linear, and the theorem is proved.

## Continuity Theorem

There exists continuous L:R→{m linear surfaces in Rn}, such that L(r) is nonparallel and nonconicidental for rr0R, only if there exist r1, r2R, and linear spaces S1n and S2n, such that r1<r0<r2, L(r1)≡S1n, L(r2)≡S2n, and S1n and S2n are migrations of each other.

As a set of linear surfaces changes continuously, its regions cannot suddenly become empty. Before becoming empty, a region must become infinitesimally small. As a region becomes infinitesimally small, so does its opposite. Thus, as a region becomes empty, its opposite becomes nonempty.

## Simplex Overlap

A simplex overlap is two disjoint sets of boundaries, each of which is a simplex. The inside of a simplex overlap Sn = (B,P,S,n) is a space S1n = (B,P1,S,n), such that P1 is the union of the insides of the simplices. Two simplex overlaps, (M1,M2), and (N1,N2), where M1, M2, N1, N2 are the simplices of the overlaps, are equivalent iff there exists one to one mapping g:M1∪M2→N1∪N2, such that g(M1) = Ni for some i∈{1,2}, and for each b∈M1∪M2 the section of the inside of (M1,M2) by b is equivalent to the section of the inside of (N1,N2) by g(b). There are 11 equivalence classes of triangle overlap. How many equivalence classes of tetrahedron overlap are there?