User:Tomruen/Regular product polytopes
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- 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} |