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.

Tetrahedron Overlaps

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 S₁ⁿ ⇔ R 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 r ≠ r₀∈R, only if there exist r₁, r₂∈R, 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) = N_i 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?