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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.

Regular degenerate product polytopes as 3-sphere honeycombs
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}
Regular degenerate product polytopes as 3-sphere honeycombs
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}.

Regular hoso-4-topes as 3-sphere honeycombs
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}