## Monday, December 31, 2007

### Overlap Examples

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.

## Saturday, December 8, 2007

### 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 ** S**, regions

^{n}= (B,P,S,n)

**and**R

_{1}

**are neighbors with respect to**R

_{2}

**iff**b

_{0}∈B

**and**R = R

_{1}(b)

_{2}(b)

**, for**R ≠ R

_{1}(b

_{0})

_{2}(b

_{0})

**, where**b∈B

_{0}

**. In space**B = B

_{0}∪{b

_{0}}

**, region**S = (B,P,S,n)

^{n}

**is the opposite of region**R

_{2}

**iff for**R

_{1}

**,**b∈B

**has a nonempty neighbor with respect to**R ≠ R ⇔ R

_{1}(b)

_{2}(b)

_{1}

**. Two spaces**b

**and**S = (B,P

_{1}

^{n}

_{1},S,n)

**are migrations of each other iff for each region**S = (B,P

_{2}

^{n}

_{2},S,n)

**in**R

**and**S

_{1}

^{n}

**, one or both of the following is true.**S

_{2}

^{n}

**is empty in**R

**is empty in**S ⇔ R

_{1}

^{n}

**. Or,**S

_{2}

^{n}

**is empty**R

**⇔**the opposite of

**is not empty.**R

Example

Regions ** A**,

**, and**B

**have empty opposites. Region**C

**is the opposite of region**A

**. Region**E

**is the opposite of region**E

**.**D

Migration Theorem

If ** S** and

_{1}

^{n}

**are migrations of each other then,**S

_{2}

^{n}

**is linear**S

_{1}

^{n}

**⇔**S is linear.

_{2}

^{n}

Without loss of generality, assume ** S** is linear,

_{1}

^{n}

**is empty in**R

**,**S

_{2}

^{n}

and its opposite is not empty in

**. Suppose**S

_{2}

^{n}

**is any subspace of**S

_{3}

^{n}

**. If**S

_{2}

^{n}

**is not a region of**R

**, then it is linear because its identical subspace in**S

_{3}

^{n}

**is linear. If**S

_{1}

^{n}

**is a region of**R

**, then so is its opposite, so the number of nonempty regions in**S

_{3}

^{n}

**is unaltered, so it is linear, and the theorem is proved.**S

_{3}

^{n}

Continuity Theorem

There exists continuous linear surfaces in

*L*:

*R*→{m

**, such that**

*R*

^{n}}

**is nonparallel and nonconicidental for**

*L*(

*r*)

**, only if there exist** ≠

*r*

*r*

_{0}∈

*R*

**,**

*r*

_{1}

**, and linear spaces**

*r*

_{2}∈

*R*

**and**S

_{1}

^{n}

**, such that**S

_{2}

^{n}

**,**

*r*

_{1}<

*r*

_{0}<

*r*

_{2}

**,**

*L*(

*r*

_{1})≡S

_{1}

^{n}

**, and**

*L*(

*r*

_{2})≡S

_{2}

^{n}

**and**S

_{1}

^{n}

**are migrations of each other.**S

_{2}

^{n}

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 ^{n}_{1}^{n}_{1},S,n)_{1} is the union of the insides of the simplices. Two simplex overlaps, ** (M**, and

_{1},M

_{2})

**, where**(N

_{1},N

_{2})

**,**M

_{1}

**,**M

_{2}

**,**N

_{1}

**are the simplices of the overlaps, are equivalent iff there exists one to one mapping**N

_{2}

**, such that**g:M

_{1}∪M

_{2}→N

_{1}∪N

_{2}

**for some**g(M = N

_{1})

_{i}

**, and for each**i∈{1,2}

**M**b∈

_{1}∪M

_{2}the section of the inside of (M

_{1},M

_{2}) by

**is equivalent to the section of the inside of (N**b

_{1},N

_{2}) by

**. There are 11 equivalence classes of triangle overlap. How many equivalence classes of tetrahedron overlap are there?**g(b)