User:Tomruen/Regular product polytopes
- Duo-forms
Regular product forms exist as honeycombs on a 3-sphere: {p,2,q}, with dihedral cells and hosohedral vertex figures. They are related to the 4D duoprisms where the square faces are flattened into rectangles and degenerated into digons.
| Schläfli {p,2,q} |
Coxeter    
|
Cells {p,2}π/q |
Faces {p} |
Edges | Vertices | Vertex figure {2,q} |
Symmetry | Dual |
|---|---|---|---|---|---|---|---|---|
| {p,2,2} | 
|
2 {p,2} |
2 {p} |
p | p | {2,2}π/p | [p,2,2] | {2,2,p} |
| {2,2,2} |
|
2 {2,2} 
|
2 {2} |
2 | 2 | {2,2}π/2
|
[2,2,2] | Self-dual |
| {3,2,2} | 
|
2 {3,2} |
2 {3} |
3 | 3 | {2,2}π/3 | [3,2,2] | {2,2,p} |
| {p,2,p} |
|
p {p,2} |
p {p} |
p | p | {2,p}π/p | [p,2,p] | Self-dual |
| {3,2,3} |
|
3 {3,2} 
|
3 {3} |
3 | 3 | {2,3}π/3
|
[3,2,3] | Self-dual |
| {p,2,q} |
|
q {p,2} |
q {p} |
p | p | {2,q}π/p | [p,2,q] | {q,2,p} |
| {4,2,3} | 
|
3 {4,2} 
|
3 {4} |
4 | 4 | {2,3}π/4
|
[4,2,3] | {3,2,4} |
- {2,p,2} = {2,2,p}
| Schläfli {2,p,2} |
Coxeter
|
Cells {2,p}π/p |
Faces {2}π/p,π/2 |
Edges | Vertices | Vertex figure {p,2} |
Symmetry | Dual |
|---|---|---|---|---|---|---|---|---|
| {2,p,2} |
|
p {2,p} |
p {2} |
2 | 2 | {p,2} | [2,p,2] | Self-dual |
| {2,3,2} |
|
3 {2,3} |
3 {2} |
2 | 2 | {3,2} | [2,3,2] | Self-dual |
- Ditopes and hosotopes
Regular di-4-topes (2 facets) include: {3,3,2}, {3,4,2}, {4,3,2}, {5,3,2}, {3,5,2}, and their hoso-4-tope duals (2 vertices): {2,3,3}, {2,4,3}, {2,3,4}, {2,3,5}, {2,5,3}.
| Schläfli {2,p,q} |
Coxeter
|
Cells {2,p}π/q |
Faces {2}π/p,π/q |
Edges | Vertices | Vertex figure {p,q} |
Symmetry | Dual |
|---|---|---|---|---|---|---|---|---|
| {2,3,3} |
|
4 {2,3}π/3
|
6 {2}π/3,π/3 |
4 | 2 | {3,3}
|
[2,3,3] | {3,3,2} |
| {2,4,3} |
|
6 {2,4}π/3 
|
12 {2}π/4,π/3 |
8 | 2 | {4,3}
|
[2,4,3] | {3,4,2} |
| {2,3,4} |
|
8 {2,3}π/4
|
12 {2}π/3,π/4 |
6 | 2 | {3,4}
|
[2,4,3] | {4,3,2} |
| {2,5,3} | 
|
12 {2,5}π/3
|
30 {2}π/5,π/3 |
20 | 2 | {5,3}
|
[2,5,3] | {3,5,2} |
| {2,3,5} |
|
20 {2,3}π/5 
|
30 {2}π/3,π/5 |
12 | 2 | {3,5}
|
[2,5,3] | {5,3,2} |