A composite polytope is a polytope that can can be decomposed into orthogonal elements. Examples include prisms, duoprisms, pyramids, bipyramids, duopyramids.
Four operators
editThere 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.
| Operator names | Symbols | Powers | f-vector | Polytope names |
|---|---|---|---|---|
| Join[1][2] Pyramid product[3][4] |
A ∨ B A ⋈ B A ×1,1 B |
n ⋅ A n ⋅ ( ) |
(1,fA,1)*(1,1) (1,fA,1)*(1,2,1) (1,fA,1)*(1,fB,1) (1,fA,1)*(1,fB,1)*(1,fC,1) (1,fA,1)n (1,1)n |
A ∨ ( ) = Pyramid A ∨ { } = Wedge A ∨ B = Duowedge A ∨ B ∨ C = Tri-wedge n ⋅ A = A-topal n-wedge (n+1) ⋅ ( ) = n-simplex, αn |
| "Rhombic sum"[1] Direct sum[2] Tegum product[3] |
A + B A ⊕ B A ×1,0 B |
n A n { } |
(1,fA)*(1,2) (1,fA)*(1,fB) (1,fA)*(1,fB)*(1,fC) (1,fA)n (1,p,p)n (1,2)n |
A + { } = Fusil or bipyramid A + B = Duofusil or duopyramid A + B + C = Tri-fusil n A = A-topal n-fusil n { } = n-fusil, n-orthoplex, βn n {p} = p-gonal n-fusil |
| Rectangular product[1] Cartesian product[2] Prism product[3] |
A×B A ×0,1 B |
An { }n |
(fA,1)*(2,1) (fA,1)*(fB,1) (fA,1)*(fB,1)*(fC,1) (fA,1)n (2,1)n (p,p,1)n |
A×{ } = Prism A×B = Duoprism, duoprism A×B×C = Tri-prism An = A-topal n-prism { }n = n-prism, n-cube, γn {p}n = p-gonal n-prism |
| Meet Topological product[2] Honeycomb[3] Comb product[4] Torus product |
A ∧ B A □ B A ×0,0 B |
A(n) { }(n) |
(fA)*(2) = (2fA) (fA)*(fB) = (fA*fB) (fA)*(fB)*(fC) = (fA*fB*fC) (fA)n = (fAn) (p,p)n = p(1,1)n (∞,∞)n = ∞(1,1)n (2)n = (2n) |
A ∧ { } = Skew meet A ∧ B = Skew duomeet A ∧ B ∧ C = Skew tri-meet A(n) = Skew A-topal n-meet {p}(n) = Reg. skew p-gonal n-meet {∞}(n) = cubic n-comb, δn+1 { }(n) = Skew dionic n-meet |
| Operator names | Symbols | Powers | f-vector | Polytope names |
|---|---|---|---|---|
| Prism-meet | A ⋋ B A ⋌ B |
(fA,1)*(2) (2,1)*(fA) (fA,1)*(fB) |
A ⋋ { } = Semi-prism { } ⋋ A = Open-prism A ⋋ B = Semi-duoprism |
Examples
edit| Operator names |
Symbols f-vector |
Rank | Polytope names |
Examples | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Join[1] Join product[2] Pyramid product[3][4] |
A ∨ B A ⋈ B A ×1,1 B (1,fA,1) * (1,fB,1) |
Rank(A) + Rank(B)+1 | A ∨ ( ) = pyramid A ∨ { } = wedge A ∨ B = duowedge A ∨ B ∨ C = tri-wedge |
3D polyhedron { } ∨ { } or {3,3} [2,1] or [4,2+] Dionic wedge  (1,2,1)*(1,2,1) = (1,4,6,4,1) |
4D skew polyhedron 3{ } ∨ { } [3,2,1] Trionic wedge  (1,3,1)*(1,2,1) = (1,5,8,5,1) |
4D polychoron {3} ∨ { } or {3,3,3} [3,2,1] Triangular wedge  (1,3,3,1)*(1,2,1) =(1,5,10,10,5,1) |
5D polyteron {3,3} ∨ { } or {3,3,3,3} [3,3,2,1] Tetrahedral wedge  (1,4,6,4,1)*(1,2,1) =(1,6,15,20,15,6,1) |
5D skew polyhedron 3{ } ∨ 3{ } [3,2,3,1] or [[3,2,3],1] Trionic duowedge  (1,3,1)*(1,3,1) = (1,6,11,6,1) |
5D skew polychoron {3} ∨ 3{ } [3,2,3,1] Triangle-trion duowedge  (1,3,3,1)*(1,3,1) =(1,6,13,13,6,1) |
5D polyteron {3} ∨ {3} = 2⋅{3} or {3,3,3,3} [3,2,3,1] or [[3,2,3],1] Triangular duowedge  (1,3,3,1)*(1,3,3,1) =(1,6,15,20,15,6,1) |
6D polypeton {3,3} ∨ {3} or {3,3,3,3,3} [3,3,2,3,1] Tetrahedron-triangle duowedge  (1,4,6,4,1)*(1,3,3,1) =(1,7,21,35,35,21,7,1) |
7D polyexon {3,3} ∨ {3,3} = 2⋅{3,3} or {3,3,3,3,3,3} [3,3,2,3,3,1] Tetrahedral duowedge  (1,4,6,4,1)*(1,4,6,4,1) =(1,8,28,56,70,56,28,8,1) |
| "Rhombic sum"[1] Direct sum[2] Tegum product[3] |
A + B A ⊕ B A ×1,0 B (1,fA) * (1,fB) |
Rank(A) + Rank(B) | A + { } = fusil or bipyramid A + B = duofusil or duopyramid A + B + C = tri-fusil |
2D polygon { } + { } = 2{ } or {4} [ ]2 = [2] or [4] Dionic fusil  (1,2)*(1,2) = (1,4,4) |
3D skew polygon 3{ } + { } [3,2] Trionic fusil  (1,3)*(1,2) = (1,5,6) |
3D polyhedron {3} + { } [3,2] Triangular fusil  (1,3,3)*(1,2) = (1,5,9,6) |
4D polychoron {3,3} + { } [3,3,2] Tetrahedral fusil  (1,4,6,4)*(1,2) = (1,6,14,16,8) |
4D skew polygon 3{ } + 3{ } = 23{ } [3,2,3] or [[3,2,3]] Trionic duofusil  (1,3)*(1,3) = (1,6,9) |
4D skew polyhedron {3} + 3{ } [3,2,3] Triangle-trion duofusil  (1,3,3)*(1,3) = (1,6,12,9) |
4D polychoron {3} + {3} or 2{3} [3,2,3] or [[3,2,3]] Triangular duofusil (1,3,3)*(1,3,3) = (1,6,15,18,9) |
5D polyteron {3,3} + {3} [3,3,2,3] Tetrahedron-triangle duofusil (1,4,6,4)*(1,3,3) = (1,7,21,34,30,12) |
6D polypeton {3,3} + {3,3} = 2{3,3} [3,3,2,3,3] Tetrahedral duofusil (1,4,6,4)*(1,4,6,4) = (1,8,28,56,68,48,16) |
| Rectangular product[1] Cartesian product[2] Prism product[3] |
A × B A ×0,1 B (fA,1) * (fB,1) |
Rank(A) + Rank(B) | A × { } = prism A × B = duoprism A × B × C = tri-prism |
2D polygon { } × { } = { }2 or {4} [ ]2 = [2] or [4] Dionic prism  (2,1)*(2,1) = (4,4,1) |
3D skew polygon 3{ } × { } [3,2] Trionic prism  (3,1)*(2,1) = (6,5,1) |
3D polyhedron {3} × { } [3,2] Triangular prism  (3,3,1)*(2,1) = (6,9,5,1) |
4D polychoron {3,3} × { } [3,3,2] Tetrahedral prism  (4,6,4,1)*(2,1) = (8,16,14,6,1) |
4D skew polygon 3{ } × 3{ } = 3{ }2 [3,2,3] or [[3,2,3]] Trionic duoprism  (3,1)*(3,1) = (9,6,1) |
4D skew polyhedron {3} × 3{ } [3,2,3] Triangle-trion duoprism  (3,3,1)*(3,1) = (9,12,6,1) |
4D polychoron {3} × {3} or {3}2 [3,2,3] or [[3,2,3]] Triangular duoprism  (3,3,1)*(3,3,1) = (9,18,15,6,1) |
5D polyteron {3,3} × {3} [3,3,2,3] Tetrahedron-triangle duoprism (4,6,4,1)*(3,3,1) = (12,30,34,21,7,1) |
6D polychoron {3,3} × {3,3} = {3,3}2 [3,3,2,3,3] Tetrahedral duoprism  (4,6,4,1)*(4,6,4,1) = (16,48,68,56,28,8,1) |
| Meet Topological product[2] Honeycomb[3] Comb product[4] Torus product Skew product |
A ∧ B A □ B A ×0,0 B fA * fB |
Rank(A) + Rank(B)-1 | A ∧ { } = meet A ∧ B = duomeet A ∧ B ∧ C = tri-meet |
2D skew 1-polytope { } ∧ { } = 2,2{ } or 4{ } [ ]2 = [2] or [4] Dionic meet  (2)*(2) = (4) |
3D skew 1-polytope 3{ } ∧ { } = 3,2{ } [3,2] Trionic meet  (3)*(2) = (6) |
3D skew polygon {3} ∧ { } [3,2] Triangular meet  (3,3)*(2) = (6,6) |
4D skew polyhedron {3,3} ∧ { } [3,3,2] Tetrahedral meet  (4,6,4)*(2) = (8,12,8) |
4D skew 1-polytope 3{ } ∧ 3{ } = 3,3{ } = 3{ }(2) [3,2,3] or [[3,2,3]] Trionic duomeet  (3)*(3) = (9) |
4D skew polygon {3} ∧ 3{ } [3,2,3] Triangle-trion duomeet  (3,3)*(3) = (9,9) |
4D skew polyhedron {3} ∧ {3} = {4,4|3} [3,2,3] or [[3,2,3]] Triangular duomeet  (3,3)*(3,3) = (9,18,9) |
5D skew polychoron {3,3} ∧ {3} [3,3,2,3] Tetrahedron-triangle duomeet  (4,6,4)*(3,3) = (12,30,30,12) |
6D skew polyteron {3,3} ∧ {3,3} = {3,3}(2) [3,3,2,3,3] Tetrahedral duomeet (4,6,4)*(4,6,4) = (16,48,68,48,16) |
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)
Examples
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Skew 1-polytopes
editSkew 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.
| Notation | { }∧{ } = 2,2{ } = 4{ } = ~{4} |
3{ }∧{ } = 3,2{ } = ~~{3}×{ } |
{ }∧{ }∧{ }= 2,2,2{ } = ~~{4,3} |
{ }∧{ }∧{ }∧{ }= 2,2,2,2{ } = ~~~{4,3,3} |
3{ }∧3{ }=3,3{ } = ~~~{3}×{3} |
3{ } = ~{3} | 4{ } = ~{4} | 5{ } = ~{5} | 6{ } = ~{6} |
|---|---|---|---|---|---|---|---|---|---|
| Name | Tetron | Hexon | Octon | Hexadekon | Enneon | Trion | Tetron | Penton | Hexon |
| Dimension | 2 | 3 | 4 | 4 | 2 | 2 | 2 | 2 | 2 |
| f-vector | (2)*(2)=(4) | (3)*(2)=(6) | (2)3=(8) | (2)4=(16) | (3)2=(9) | (3) | (4) | (5) | (6) |
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Prisms and meets
editPair composites
editFor example { }×{ }, is a topological square.
| Rank | Operator | Name | f-vector | Vertices | Edges | χ | Image | Dual |
|---|---|---|---|---|---|---|---|---|
| 2 | { }×{ } Regular {4} |
Dion prism Square |
(2,1)*(2,1) =(4,4,1) |
4 | 4 | 0 |  |
 { }+{ } |
| 1 | { }∧{ } =2,2{ } Regular 4{ } |
Dion meet Tetron |
(2)*(2) =(4) |
4 | 4 | |
Self-dual |
Triple composites
editFor 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{ }.
| Rank | Operator | Name | f-vector | Vertices | Edges | Faces | χ | Image | Dual |
|---|---|---|---|---|---|---|---|---|---|
| 3 | { }×{ }×{ } = { }3 {4}×{ } Regular {4,3} |
Dion triprism Square prism Cube |
(2,1)*(2,1)*(2,1) =(8,12,6,1) |
8 | 12 | 6 {4} | 2 |  |
 { }+{ }+{ } |
| 2 | ({ }×{ })∧{ } Regular {4}∧{ } |
Rectangle meet Square meet |
(4,4)*(2) =(8,8) |
8 | 8 | 0 |  |
Self-dual | |
| 2 | ({ }∧{ })×{ } 4{ }×{ } |
Dion meet prism Tetron prism |
(4,1)*(2,1) =(8,6,1) |
8 | 6 4 {}, 2 4{} |
2 |  |
 ({ }∧{ })+{ } | |
| 1 | { }∧{ }∧{ } = { }(3) Regular 2,2,2{ } |
Dion tri-meet | (2)*(2)*(2) =(8) |
8 | 8 | |
Self-dual |
Quadruple composites
editFor 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{ }.
| Rank | Operator | Name | f-vector | Vertices | Edges | Faces | Cells | χ | Image | Dual |
|---|---|---|---|---|---|---|---|---|---|---|
| 4 | { }×{ }×{ }×{ } = { }4 {4}×{ }×{ } {4}×{4}={4}2 Regular {4,3,3} |
Dion prism prism prism Square prism prism Square duoprism Tesseract, tes |
(2,1)4 =(4,4,1)2 =(16,32,24,8,1) |
16 | 32 | 24 | 8 | 0 |  |
 { }+{ }+{ }+{ } |
| 3 | ({ }×{ })∧({ }×{ }) = ({ }2)(2) {4}∧({ }×{ }) {4}∧{4} Regular 2{4} ={4,4|4} |
Rectangle duomeet Square-rectangle duomeet Square duomeet |
(4,4)2 =(16,32,16) |
16 | 32 | 16 | 0 |  |
Self-dual | |
| 3 | ({ }×{ }×{ })∧{ } ({4}×{ })∧{ } {4,3}∧{ } |
Cuboid meet Square prism meet Cubic meet |
(8,12,6)*(2) =(16,24,12) |
16 | 24 | 12 | 4 |  |
{3,4}∧{ } | |
| 3 | (({ }×{ })∧{ })×{ } ({4}∧{ })×{ } |
Rectangle meet prism Square meet prism |
(8,8,1)*(2,1) =(16,24,10,1) |
16 | 24 | 10 | 2 | ({4}∧{ })+{ } | ||
| 3 | { }×{ }×({ }∧{ }) {4}×({ }∧{ }) {4}×4{ } |
Rectangle-(dion meet) duoprism Square-(dion meet) duoprism Square-tetron duoprism |
(4,4,1)*(4,1) =(16,20,8,1) |
16 | 20 | 8 | 4 | {4}+4{ } | ||
| 2 | ({ }×{ })∧{ }∧{ } {4}∧{ }∧{ } Regular {4}∧4{ } |
Rectangle meet meet Square meet meet Square-tetron duomeet |
(4,4)*(2)2 =(16,16) |
16 | 16 | 0 | Self-dual | |||
| 2 | (({ }∧{ })×{ })∧{ } (4{ }×{ })∧{ } |
Tetron prism meet | (8,6)*(2) =(16,12) |
16 | 12 | 4 | (4{ }+{ })∧{ } | |||
| 2 | ({ }∧{ }∧{ })×{ } 2,2,2{ }×{ } = 2,4{ }×{ } |
Dion tri-meet prism Octon prism |
(8,1)*(2,1) =(16,10,1) |
16 | 10 | 6 |  |
2,4{ }+{ } | ||
| 2 | ({ }∧{ })×({ }∧{ }) 4{ }×4{ }=4{ }2 |
Dion meet duoprism Tetron duoprism |
(4,1)2 =(16,8,1) |
16 | 8 | 8 |  |
({ }∧{ })+({ }∧{ }) | ||
| 1 | { }∧{ }∧{ }∧{ } = { }(4) 4{ }∧4{ } = 4{ }(2) Regular 4,4{ } or 2,2,2,2{ } |
Tetron duomeet Dion tetra-meet |
(2)4 =(16) |
16 | 16 | |
Self-dual |
References
edit- ^ a b c d e f g h i Geometries and TransformationNorman Johnson, 2018, 11.3 Pyramids, Prisms, and Antiprisms, p.163
- ^ a b c d e f g h i j k l Products of abstract polytopes Ian Gleason and Isabel Hubard, 2016
- ^ a b c d e f g h i j k l https://bendwavy.org/klitzing/explain/product.htm
- ^ a b c d http://www.os2fan2.com/gloss/polytope.pdf