User:Tomruen/Composite polytope

A composite polytope is a polytope that can can be decomposed into orthogonal elements. Examples include prisms, duoprisms, pyramids, bipyramids, duopyramids.

Four operators
There are four classes that can be expressed as product operators on f-vectors.

The join, with descending wedge symbol ∨, include both f-1, and fn. The rhombic sum only includes f-1, and its dual rectangular product only includes fn. The meet includes neither and only applies to flat elements.

For instance a triangle has f-vector (3,3), with 3 vertices (f0) and 3 edges (f1). Extended with the nullitope (f-1) gives f=(1,3,3), extending with the (polygonal interior) body (f2) gives f=(3,3,1), while extending both is f=(1,3,3,1).

The rhombic sum and rectangular product are dual operators, with f-vectors reversed. The join and rhombic sums shares vertex counts, summing vertices in elements. The rectangular and meet products also share vertex counts, being the product of the element vertex counts.

The meet product, with wedge symbol ∧, is the same as Cartesian product if elements are infinite. Meets are not connected unless polytopes are polygons or higher.

An n-polytope existing in a space higher than n-dimensions, it can be categorized as skew. It does not have a well-defined interior.

Examples
A product A*B, with f-vectors fA and fB, fA∨B=fA*fB is computed like a polynomial multiplication polynomial coefficients.

For example for join of a triangle and dion, {3} ∨ { }:
 * fA(x) = (1,3,3,1) = 1 + 3x + 3x2 + x3 (triangle)
 * fB(x) = (1,2,1) = 1 + 2x + x2 (dion)
 * fA∨B(x) = fA(x) * fB(x)
 * = (1 + 3x + 3x2 + x3) * (1 + 2x + x2)
 * = 1 + 5x + 10x2 + 10x3 + 5x4 + x5
 * = (1,5,10,10,5,1) (triangle ∨ dion = 5-cell)

Skew 1-polytopes
Skew polytopes can be topologically connected or unconnected. Skew 1-polytopes can be drawn disconnected, but when part of a k-face of a larger polytope, the interior can be filled to show their relatedness.

Meet operators allow polytopes to be defined by "polytope holes" so can produce skew polytopes with prism ridges as facets.

As well, an operator ~P implies a polytope P is reduced by rank by one, making a complex skew polytope.

1-polytopes are self-dual.

Pair composites
For example { }×{ }, is a topological square.

Triple composites
For example { }×{ }×{ }, is a topological cube has three skew version with meet operators. There are 2 skew polygons and 1 skew 1-polytope sharing all the vertices.

All the skew forms are vertex-transitive, while 2 can be considered regular: polyhedron {4,3}, skew polygon 4{ }×{ } and skew 1-polytope { }∧{ }∧{ } = 2,2,2{ }.

Quadruple composites
For example { }×{ }×{ }×{ }, is a topological tesseract. There are 4 skew polyhedra and 4 skew polygons sharing all the vertices, and finally one skew 1-polytope with 16 points.

All the skew forms are vertex-transitive, while 5 can be considered regular: 4-polytope {4,3,3}, skew polyhedron {4}∧{4} = 2{4}, skew polygon, 4}∧4{ }, and skew 1-polytope { }∧{ }∧{ }∧{ } = 2,2,2,2{ } = 4,4{ }.