User:Tomruen/triaprism
| Type | Uniform 6-polytope |
| Schläfli symbol | {p}×{q}×{r} |
| Coxeter diagram |       or    
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| 5-faces | 9: 3 {p}×{q}×{ } 3 {p}×{r}×{ } 3 {q}×{r}×{ } |
| 4-faces | pq+pr+qr+p+q+r |
| Cells | pqr+2(pq+pr+qr) |
| Faces | 3pqr+pq+pr+qr |
| Edges | 3pqr |
| Vertices | pqr |
| Vertex figure | 5-simplex = { }∨{ }∨{ } |
| Symmetry | [p,2,q,2,r], order 8pqr |
| Dual | p-q-r pyramid |
| Properties | convex, vertex-uniform, facet-transitive |
A triaprism is a 6-polytope constructed as the product of 3 orthogonal polygons.
A uniform triaprism is the product of three regular polygons, {p}×{q}×{r}, form an infinite class for all integers p,q,r>2.
The smallest triaprism is a 3-3-3 prism. {4}×{4}×{4} makes a lower symmetry form of the 6-cube.
A p-q-r prism is a real representation of the set of complex polytopes 
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| Type | Uniform 6-polytope |
| Schläfli symbol | {p}×{p}×{p} = {p}3 |
| Coxeter diagram | or
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| 5-faces | 3p {p}×{p}×{ } |
| 4-faces | 3p2 {p}×{4} 3p {p}2 |
| Cells | 6p2 {p}×{ } p3 {4}2 |
| Faces | 3p2 {p} 3p3 {4} |
| Edges | 3p3 |
| Vertices | p3 |
| Vertex figure | 5-simplex = { }∨{ }∨{ } |
| Symmetry | [3[p,2,p,2,p]], order 48p3 |
| Dual | p-p-p pyramid |
| Properties | convex, vertex-uniform, facet-transitive |
p-p-p prisms
[edit]A p-p-p prism or p-gonal triple prism or p-gonal triaprism, {p}×{p}×{p} or {p}3, has extended symmetry [3[p,2,p,2,p]], order 48p3. A 4-4-4 prism is also a 6-cube, extends symmetry order from 3072 to 266! or 46080. Therefore [4]3 is an index 15 subgroup of [4,3,3,3,3].
A p-p-p prism is a real representation of the set of complex polytope generalized cubes 
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| Name | {3}3 | {4}3 | {5}3 | {6}3 | {7}3 | {8}3 |
|---|---|---|---|---|---|---|
| Vertices | 27 | 64 | 125 | 216 | 343 | 512 |
| Edge | 81 | 192 | 375 | 648 | 1029 | 1536 |
| Symmetry Order |
1296 | 3072 | 6000 | 10364 | 16464 | 24576 |
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