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
[edit]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:
- Products are orthotopes (prisms) with 2n coordinates (±1,±1,±1,...,±1), being a hypercube.
- 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).
- Products and sums are Hanner polytopes (list)
- 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.
1D
[edit]| Construction | Name | Symmetry | Order | Coordinates | Vertices | f0 | Dual | |
|---|---|---|---|---|---|---|---|---|
| { } | Segment | [ ] | 2 | (±1) | 2 | 2 | 2 | Self-dual |
| h{4} = {2} | Segment | [2] | 4 | ±(1,1) | 2 | 2 | 2 | Self-dual |
| ( )∨( ) | Segment | [1]+ | 1 | (-1,-1), (+1,+1) | 1+1 | 1 | Self-dual | |
| { } | Segment | [1] | 1 | ([1,0]) | 2 | 1 | Self-dual |
2D
[edit]| Construction | Name | Symmetry | Order | Coordinates | Vertices | f0 | f1 | Dual |
|---|---|---|---|---|---|---|---|---|
| { }+{ } or r({ }×{ }) = {4} | Diamond | [4] | 8 | (±1,0), (0,±1) = ([±1,0]) | 2+2 | 4 | 4 | Square |
| { }×{ } or r({ }+{ }) = {4} | Square | [4] | 8 | (±1,±1) | 2×2 | 4 | 4 | Diamond |
| { }∨( ) | Triangle | [1,1] | 2 | ([1,0],0), (0,0,+1) | 2+1 | 3 | 3 | Self-dual |
| ( )∨( )∨( ) = {3} | Triangle | [3,1] | 2 | ([1,0,0]) | 3 | 3 | 3 | Self-dual |
3D
[edit]| Construction | Name | Symmetry | Order | Coordinates | Vertices | f0 | f1 | f2 | Dual |
|---|---|---|---|---|---|---|---|---|---|
| { }+{ }+{ } or {3,4} | Octahedron | [4,3] | 48 | ([±1,0,0]) | 2×3 | 6 | 12 | 8 | Cube |
| r{4,3} or r{3,4} | Cuboctahedron | [4,3] | 48 | ([±1,±1,0]) | 12 | 12 | 24 | 14 | Rhombic dodecahedron |
| { }×{ }×{ } or {4,3} | Cube | [4,3] | 48 | (±1,±1,±1) | 23 | 8 | 12 | 6 | Octahedron |
| h{4,3} | Tetrahedron | [3,3] | 24 | half (±1,±1,±1) | 4 | 4 | 6 | 4 | Self-dual |
| s{2,4} or h{4,3} = {3,3} | Rhombic disphenoid | [2+,4] or [3,3] | 8 or 24 | (±(1,1),-1), (±(1,-1),1) | 4 | 4 | 6 | 4 | Self-dual |
| ({ }×{ })∨( ) or {4}∨( ) | Square pyramid | [4,1] | 8 | (±1,±1,0), (0,0,+1) | 2×2+1 | 5 | 8 | 5 | Self-dual |
| { }∨( )∨( ) | Triangle pyramid | [1,1,1] | 2 | ([1,0],0,0), (0,0,1,0), (0,0,0,1) | 2+1+1 | 4 | 6 | 4 | Self-dual |
| { }∨{ } | Digonal disphenoid | [2,2,1] | 8 | ([1,0],0,0), (0,0,[1,0]) | 2+2 | 4 | 6 | 4 | Self-dual |
| ( )∨( )∨( )∨( ) = {3}∨( ) | Triangular pyramid | [3,1,1] | 6 | ([1,0,0],0), (0,0,0,1) | 3+1 | 4 | 6 | 4 | Self-dual |
| ( )∨( )∨( )∨( ) = {3,3} | tetrahedron | [3,3,1] | 24 | ([1,0,0,0]) | 4 | 4 | 6 | 4 | Self-dual |
4D
[edit]| Construction | Name | Symmetry | Order | Coordinates | Vertices | f0 | f1 | f2 | f3 | Dual |
|---|---|---|---|---|---|---|---|---|---|---|
| { }+{ }+{ }+{ } or {3,3,4} | 16-cell | [4,3,3] | 384 | ([±1,0,0,0]) | 2×4 | 8 | 24 | 32 | 16 | Tesseract |
| r{3,3,4} | Rectified 16-cell or 24-cell | [4,3,3] | 384 | ([±1,±1,0,0]) | 4×6 | 24 | 96 | 96 | 24 | Self-dual |
| r{4,3,3} | Rectified tesseract | [4,3,3] | 384 | ([±1,±1,±1,0]) | 8×4 | 32 | 96 | 88 | 24 | |
| { }×{ }×{ }×{ } or {4,3,3} | Tesseract | [4,3,3] | 384 | (±1,±1,±1,±1) | 24 | 16 | 32 | 24 | 8 | 16-cell |
| h{4,3,3} | 16-cell | [31,1,1] | 192 | half (±1,±1,±1,±1) | 8 | 8 | 24 | 32 | 16 | Tesseract |
| ({ }+{ }+{ })×{ } or {3,4}×{ } | Octahedral prism | [4,3,2] | 96 | ([±1,0,0]; ±1) | 8×2 | 12 | 30 | 28 | 10 | cubical bipyramid |
| r{3,4}×{ } | Cuboctahedral prism | [4,3,2] | 96 | ([±1,±1,0]; ±1) | 12×2 | 24 | 60 | 52 | 16 | rhombic dodedecahedral bipyramid |
| r{3,4}+{ } | Cuboctahedral bipyramid | [4,3,2] | 96 | ([±1,±1,0]; 0), (0,0,0; ±1) | 12+2 | 14 | rhombic dodedecahedral prism | |||
| { }×{ }×{ }+{ } or {4,3}+{ } | Cubical bipyramid | [4,3,2] | 96 | ([±1,±1,±1]; 0), (0,0,0; ±1) | 8+2 | 10 | 28 | 30 | 12 | octahedral prism |
| {3,4}∨( ) | Octahedral pyramid | [4,3,1] | 48 | ([±1,0,0]; 0), (0,0,0; 1) | 6+1 | 7 | 18 | 20 | 9 | cubical pyramid |
| r{3,4}∨( ) | Cuboctahedral pyramid | [4,3,1] | 48 | ([±1,±1,0]; 0), (0,0,0; 1) | 12+1 | 13 | 36 | 38 | 15 | rhombic dodecahedral pyramid |
| {4,3}∨( ) | Cubic pyramid | [4,3,1] | 48 | ([±1,±1,±1]; 0), (0,0,0; 1) | 8+1 | 9 | 20 | 18 | 7 | octahedral pyramid |
| {4}∨{ } | Square-segment disphenoid | [4,2,1] | 16 | ([±1,±1],0,0), (0,0,±1,1) | 4+2 | 6 | 13 | 13 | 6 | Self-dual |
| ( )∨( )∨( )∨( )∨( ) = { }∨{ }∨( ) | Digonal disphenoid pyramid = 5-cell | [2,2,1,1] | 8 | ([1,0],0,0,0), (0,0,[1,0],0), (0,0,0,0,1) | 2+2+1 | 5 | 10 | 10 | 5 | Self-dual |
| ( )∨( )∨( )∨( )∨( ) = {3}∨{ } | Triangle-segment disphenoid = 5-cell | [3,2,1,1] | 12 | ([1,0,0]; 0,0), (0,0,0; [1,0]) | 3+2 | 5 | 10 | 10 | 5 | Self-dual |
| ( )∨( )∨( )∨( )∨( ) = {3,3}∨( ) | Tetrahedral pyramid = 5-cell | [3,3,1,1] | 24 | ([1,0,0,0]; 0), (0,0,0,0; 1) | 4+1 | 5 | 10 | 10 | 5 | Self-dual |
| ( )∨( )∨( )∨( )∨( ) = {3,3,3} | 5-cell | [3,3,3,1] | 120 | ([1,0,0,0,0]) | 5 | 5 | 15 | 10 | 5 | Self-dual |
5D
[edit]| Construction | Name | Symmetry | Order | Coordinates | Vertices | f0 | f1 | f2 | f3 | f4 | Dual |
|---|---|---|---|---|---|---|---|---|---|---|---|
| { }+{ }+{ }+{ }+{ } or {3,3,3,4} | 5-orthoplex | [4,3,3,3] | 3840 | ([±1,0,0,0,0]) | 2×5 | 10 | 40 | 80 | 80 | 32 | 5-cube |
| r{3,3,3,4} | rectified 5-orthoplex | [4,3,3,3] | 3840 | ([±1,±1,0,0,0]) | 4×10 | 40 | 240 | 400 | 240 | 42 | |
| 2r{3,3,3,4} = 2r{4,3,3,3} | birectified 5-orthoplex | [4,3,3,3] | 3840 | ([±1,±1,±1,0,0]) | 8×10 | 80 | 480 | 640 | 280 | 42 | |
| 3r{3,3,3,4} = r{4,3,3,3} | birectified 5-cube | [4,3,3,3] | 3840 | ([±1,±1,±1,±1,0]) | 16×5 | 80 | 320 | 400 | 200 | 42 | |
| { }×{ }×{ }×{ }×{ } or {4,3,3,3} | 5-cube | [4,3,3,3] | 3840 | (±1,±1,±1,±1,±1) | 25 | 32 | 80 | 80 | 40 | 10 | 5-orthoplex |
| h{4,3,3,3} | 5-demicube | [3,3,31,1] | 1920 | half (±1,±1,±1,±1,±1) | 24 | 16 | 80 | 160 | 120 | 26 | |
| ({ }+{ }+{ }+{ })×{ } or {3,3,4}×{ } | 16-cell prism | [4,3,3,2] | 768 | ([±1,0,0,0]; ±1) | 8×2 | 16 | 56 | 88 | 64 | 18 | tesseract bipyramid |
| { }×{ }×{ }×{ }+{ } or {4,3,3}+{ } | tesseract bipyramid | [4,3,3,2] | 768 | ([±1,±1,±1,±1]; 0), (0,0,0,0,±1) | 16+2 | 18 | 64 | 88 | 56 | 16 | tesseract 16-cell |
| {4,3}+{4} | cube-square duopyramid | [4,3,2,4] | 384 | ([±1,±1,±1]; 0,0), (0,0,0; ±1,±1) | 8+4 | 12 | 48 | 86 | 72 | 24 | octahedron,square double-prism |
| r{4,3}+{4} | cuboctahedron-square duopyramid | [4,3,2,4] | 384 | ([±1,±1,0]; 0,0), (0,0,0; ±1,±1) | 12+4 | 16 | rhombic-dodecahedron-square duoprism | ||||
| r{4,3}×{4} | cuboctahedron-square prism | [4,3,2,4] | 384 | ([±1,±1,0]; ±1,±1) | 12×4 | 48 | rhombic-dodecahedron-square double-pyramid | ||||
| {3,4}×{4} | octahedron-square duoprism | [4,3,2,4] | 384 | ([±1,0,0]; ±1,±1) | 6×4 | 24 | 72 | 86 | 48 | 12 | cube-square double-pyramid |
| {3,4}×{ }+{ } | octahedral prism bipyramid | [4,3,2,4] | 384 | ([±1,0,0]; ±1; 0), (0,0,0; 0; ±1) | 6×2+2 | 14 | 54 | 88 | 66 | 20 | cubic bipyramid prism |
| r{4,3}×{ }+{ } | cuboctahedral prism bipyramid | [4,3,2,4] | 384 | ([±1,±1,0]; ±1; 0), (0,0,0; 0; ±1) | 12×2+2 | 26 | rhombic-dodecahedral bipyramid prism | ||||
| ({4,3}+{ })×{ } | cubic bipyramid prism | [4,3,2,4] | 384 | ([±1,±1,±1]; 0; ±1), (0,0,0; ±1; ±1) | (8+2)×2 | 20 | 66 | 88 | 54 | 14 | octahedral prism bipyramid |
| {3,4}∨{ } | octahedron-segment disphenoid | [4,3,2,1] | 96 | ([±1,0,0]; 0; -1), (0,0,0; ±1; +1]) | 6+2 | 8 | 10 | cube-segment pyramid | |||
| r{4,3}∨{ } | cuboctahedron-segment disphenoid | [4,3,2,1] | 96 | ([±1,±1,0]; 0; -1), (0,0,0; ±1; +1]) | 12+2 | 14 | rhombic-dodecahedron-segment pyramid | ||||
| {4,3}∨{ } | cube-segment disphenoid | [4,3,2,1] | 96 | ([±1,±1,±1]; 0; -1), (0,0,0; ±1; +1) | 8+2 | 10 | 8 | octahedron-segment pyramid | |||
| {3,4}×{ }+{ } | octahedron-prism bipyramid | [4,3,2,2] | 192 | ([±1,0,0]; ±1; 0), (0,0,0; 0; ±1) | 6×2+2 | 14 | 20 | cube bipyramid prism | |||
| r{3,4}×{ }+{ } | cuboctahedron-prism bipyramid | [4,3,2,2] | 192 | ([±1,±1,0]; ±1; 0), (0,0,0; 0; ±1) | 12×2+2 | 26 | rhombic-dodecahedral bipyramid prism | ||||
| ({4,3}+{ })×{ } | cube bipyramid prism | [4,3,2,2] | 192 | ([±1,±1,±1]; 0; ±1), (0,0,0; ±1; ±1) | (8+2)×2 | 20 | 14 | octahedron-prism bipyramid | |||
| {3,3,4}∨( ) | 16-cell pyramid | [4,3,3,1] | 384 | ([±1,0,0,0]; 0), (0,0,0,0; 1) | 8+1 | 9 | 17 | tesseract pyramid | |||
| r{3,3,4}∨( ) | Rectified 16-cell pyramid | [4,3,3,1] | 384 | ([±1,±1,0,0]; 0), (0,0,0,0; 1) | 12+1 | 13 | |||||
| 2r{3,3,4}∨( ) | Rectified tesseract pyramid | [4,3,3,1] | 384 | ([±1,±1,±1,0]; 0), (0,0,0,0; 1) | 24+1 | 25 | |||||
| {4,3,3}∨( ) | tesseract pyramid | [4,3,3,1] | 384 | ([±1,±1,±1,±1]; 0), (0,0,0,0; 1) | 16+1 | 17 | 9 | 16-cell pyramid | |||
| ({3,4}∨( ))+{ } | octahedral pyramid bipyramid | [4,3,2,1] | 96 | ([±1,0,0]; 0; 0), (0,0,0; 1; 0), (0,0,0; 0; ±1) | 6+1+2 | 9 | 18 | cube pyramid prism | |||
| (r{3,4}∨( ))+{ } | cuboctahedral pyramid bipyramid | [4,3,2,1] | 96 | ([±1,±1,0]; 0; 0), (0,0,0; 1; 0), (0,0,0; 0; ±1) | 12+1+2 | 15 | rhombic-dodecahedral pyramid prism | ||||
| ({4,3}∨( ))×{ } | cube pyramid prism | [4,3,2,1] | 96 | ([±1,±1,±1]; 0; ±1), (0,0,0; ±1; 0) | (8+1)×2 | 18 | 9 | octahedral pyramid bipyramid | |||
| {4}∨{4} | square-square disphenoid | [4,2,4,1] | 64 | ([±1,±1],0,0,-1), (0,0,[±1,±1],+1) | 4+4 | 8 | 8 | self-dual | |||
| {3,4}×{3} | octahedron-triangle prism | [4,3,2,3,1] | 384 | ([±1,0,0]; [1,0,0]) | 6×3 | 18 | 11 | cube-triangle bipyramid | |||
| r{3,4}×{3} | cuboctahedron-triangle prism | [4,3,2,3,1] | 384 | ([±1,±1,0]; [1,0,0]) | 12×3 | 36 | 11 | rhombic-dodecahedral-triangle bipyramid | |||
| {4,3}+{3} | cube-triangle duopyramid | [4,3,2,3,1] | 384 | ([±1,±1,±1]; 0,0,0), (0,0,0, [1,0,0]) | 8+3 | 11 | 18 | octahedron-triangle prism | |||
| {3,3}×{4} | tetrahedron-square prism | [3,3,2,4,1] | 192 | ([1,0,0,0]; ±1; ±1) | 12×4 | 16 | 8 | tetrahedron-square bipyramid | |||
| {3,3}+{4} | tetrahedron-square duopyramid | [3,3,2,4,1] | 192 | ([1,0,0,0]; 0,0), (0,0,0,0, ±1; ±1) | 4+4 | 8 | 16 | tetrahedron-square prism | |||
| {3,3,3}×{ } | 5-cell-segment prism | [3,3,3,2,1] | 240 | ([1,0,0,0,0]; ±1) | 5×2 | 10 | 7 | 5-cell-segment bipyramid | |||
| {3,3,3}+{ } | 5-cell-segment bipyramid | [3,3,3,2,1] | 240 | ([1,0,0,0,0]; 0), (0,0,0,0,0; ±1) | 5+2 | 7 | 10 | 5-cell-segment prism | |||
| {4}∨{3} | square-triangle disphenoid | [4,2,3,1,1] | 48 | ([±1,±1],0,0,0,-1), (0,0,[1,0,0],+1) | 4+3 | 7 | 19 | 26 | 19 | 7 | self-dual |
| {3,3,3,3} | 5-simplex | [3,3,3,3,1] | 720 | ([1,0,0,0,0,0]) | 6 | 6 | 15 | 20 | 15 | 6 | self-dual |
| {3,3,3}∨( ) | 5-cell pyramid | [3,3,3,1,1] | 120 | ([1,0,0,0,0],0), (0,0,0,0,0,1) | 5+1 | 6 | 15 | 20 | 15 | 6 | self-dual |
| {3,3}∨{ } | tetrahedron-segment disphenoid | [3,3,2,1,1] | 48 | ([1,0,0,0],0,0), (0,0,0,0,[1,0]) | 4+2 | 6 | 15 | 20 | 15 | 6 | self-dual |
| {3}∨{3} | triangle-triangle disphenoid | [3,2,3,1,1] | 36 | ([1,0,0],0,0,0), (0,0,0,[1,0,0]) | 3+3 | 6 | 15 | 20 | 15 | 6 | self-dual |
| { }∨{ }∨{ } | segment-segment-segment trisphenoid | [2,2,2,1,1] | 16 | ([1,0],0,0,0,0), (0,0,[1,0],0,0), (0,0,0,0,[1,0]) | 2+2+2 | 6 | 15 | 20 | 15 | 6 | self-dual |
References
[edit]- Beeler, Katy, Polytopes with few coordinate values 2017