User:Tomruen/0/1/2-polytope

A 0/1/2-polytope or ternary polytope is the convex hull of a set of vertices {-1,0,+1}d. A d-dimensional polytope requires at least d+1 vertices, and can not all exist in the same hyperplane.

Regular polytope examples are family of hypercube and dual orthoplex. Taking (n+1) of 2n vertices of the n-cube makes a simplex.

Hanner polytopes are examples that recursively mix prism and bipyramid operators.

A subset include the 0/1-polytopes with coordinates {0,1}d.

Operator polytopes
A recursive class of 0/1/2 polytopes can be made by recursive operators of products, sums, and joins. Johnson defined product, sum, and join operators for constructing higher dimensional polytopes from lower. Johnson defines as a point (0-polytope), { } is a line segment defined between two points (1-polytope). Many vertex figures for uniform polytopes can be expressed with these operators.

A product operator, ×, defines rectangles and prisms with independent proportions. dim(A×B) = dim(A)+dim(B).

For instance { }×{ } is a rectangle, symmetry [2], (a lower symmetry form of a square), and {4}×{ } is a square prism, symmetry [4,2] (a lower symmetry form of a cube), and {4}×{4} is called a duoprism in 4-dimensions, symmetry [4,2,4] (a lower symmetry form of a tesseract).

A sum operator, +, makes duals to the prisms. dim(A+B) = dim(A)+dim(B).

For instance, { }+{ } is a rhombus or fusil in general, symmetry [2], {4}+{ } is a square bipyramid, symmetry [4,2] (lower symmetry form of a regular octahedron), and {4}+{4} is called a duopyramid in 4-dimensions, symmetry [4,2,4] (a lower symmetry form of the 16-cell).

The product and sum operators are related by duality: !(A×B)=!A+!B and !(A+B)=!A×!B, where !A is dual polytope of A.

A join operator, ∨, makes pyramidal composites, orthogonal orientations with an offset direction as well, with edges between all pairs of vertices across the two. dim(A∨B) = dim(A)+dim(B)+1.

The isosceles triangle can be seen as ∨{ }, symmetry [ ], and tetragonal disphenoid is { }∨{ }, symmetry [2]. A square pyramid is {4}∨, symmetry [4,1]. A 1 branch is symbolic, representing [4,2,1+], or, having an orthogonal mirror inactivated by an alternation.

The join operator is self-related by duality: !(A∨B)=!A∨!B. More generally any expression of these operators can be dualed by replacing polytopes by dual, and swapping product and sum operators.

Polytopes can be constructed with:
 * 1) Products are orthotopes (prisms) with 2n coordinates (±1,±1,±1,...,±1), being a hypercube.
 * 2) Sums are orthoplexes (bipyramids, duopyramids or fusils) Coordinates ([±1,0,0,...]) with 2n vertices: (±1,0,0,...,0), (0,±1,0,...,0), (0,0,±1,0,...,0), ... (0,...,0,0,±1).
 * 3) Products and sums are Hanner polytopes (list)
 * 4) Join are simplexes (pyramids and disphenoids). Regular simplices exists as integer coordinates going up one dimension. ([1,0,0,...,0]) means n vertices: (1,0,0,...,0), (0,1,0,...,0), (0,0,1,0...,0), ... (0,...0,0,1), being one nonnegative simplex facet of a n-orthoplex.

As well alternation cuts vertices in half for polytopes with even-sided faces.rectification creates new vertices mid-edge. However the duals of these polytopes are not generally 0/1/2 polytopes.