Finite Regular Skew Polyhedra of 4-space
| A4 Coxeter plane projections | |
|---|---|
| {4, 6 | 3} | {6, 4 | 3} |
| Runcinated 5-cell (60 edges, 20 vertices) |
Bitruncated 5-cell (60 edges, 30 vertices) |
| F4 Coxeter plane projections | |
| {4, 8 | 3} | {8, 4 | 3} |
| Runcinated 24-cell (576 edges, 144 vertices) |
Bitruncated 24-cell (576 edges, 288 vertices) |
| Some of the 4-dimensional regular skew polyhedra fits inside the uniform polychora as shown in these projections. | |
Coxeter also enumerated the a larger set of finite regular polyhedra in his paper "regular skew polyhedra in three and four dimensions, and their topological analogues".
Just like the infinite skew polyhedra represent manifold surfaces between the cells of the convex uniform honeycombs, the finite forms all represent manifold surfaces within the cells of the uniform polychora.
A first form, {l, m | n}, repeats the five convex Platonic solids, and one nonconvex Kepler-Poinsot solid:
| {l, m | n} | Faces | Edges | Vertices | p | Polyhedron | Symmetry order |
|---|---|---|---|---|---|---|
| {3,3| 3} = {3,3} | 4 | 6 | 4 | 0 | Tetrahedron | 12 |
| {3,4| 4} = {3,4} | 8 | 12 | 6 | 0 | Octahedron | 24 |
| {4,3| 4} = {4,3} | 6 | 12 | 8 | 0 | Cube | 24 |
| {3,5| 5} = {3,5} | 20 | 30 | 12 | 0 | Icosahedron | 60 |
| {5,3| 5} = {5,3} | 12 | 30 | 20 | 0 | Dodecahedron | 60 |
| {5,5| 3} = {5,5/2} | 12 | 30 | 12 | 4 | Great dodecahedron | 60 |
The remaining solutions of the first form, {l, m | n} exist in 4-space. Polyhedra of the form {l, m | n} have a cyclic Coxeter group symmetry of, which reduces to the linear when m is 4, and when l=4. {4,4|n} produces a double n-prism, or n-n duoprism, and specifically {4,4|4} fits inside of a {4}x{4} tesseract. {a,4|b} is represented by the {a} faces of the bitruncated {b,a/2,b} uniform polychoron, and {4,a|b} is represented by square faces of the runcinated {b,a/2,b}.
| {l, m | n} | Faces | Edges | Vertices | p | Structure | Symmetry | Order | Related uniform polychora |
|---|---|---|---|---|---|---|---|---|
| {4,4| 3} | 9 | 18 | 9 | 1 | D3xD3 | ] | 18 | 3-3 duoprism |
| {4,4| 4} | 16 | 32 | 16 | 1 | D4xD4 | ] | 32 | 4-4 duoprism or tesseract |
| {4,4| 5} | 25 | 50 | 25 | 1 | D5xD5 | ] | 50 | 5-5 duoprism |
| {4,4| 6} | 36 | 72 | 36 | 1 | D6xD6 | ] | 72 | 6-6 duoprism |
| {4,4| n} | n2 | 2n2 | n2 | 1 | DnxDn | ] | 2n2 | n-n duoprism |
| {4,6| 3} | 30 | 60 | 20 | 6 | S5 | ] | 120 | Runcinated 5-cell |
| {6,4| 3} | 20 | 60 | 30 | 6 | S5 | ] | 120 | Bitruncated 5-cell |
| {4,8| 3} | 288 | 576 | 144 | 73 | ] | 1152 | Runcinated 24-cell | |
| {8,4| 3} | 144 | 576 | 288 | 73 | ] | 1152 | Bitruncated 24-cell |
| {l, m | n} | Faces | Edges | Vertices | p | Structure | Symmetry | Order | Related uniform polychora |
|---|---|---|---|---|---|---|---|---|
| {4,5| 5} | 90 | 180 | 72 | 10 | A6 | ] | 360 | Runcinated grand stellated 120-cell |
| {5,4| 5} | 72 | 180 | 90 | 10 | A6 | ] | 360 | Bitruncated grand stellated 120-cell |
| {l, m | n} | Faces | Edges | Vertices | p | Structure | Order |
|---|---|---|---|---|---|---|
| {4,5| 4} | 40 | 80 | 32 | 5 | ? | 160 |
| {5,4| 4} | 32 | 80 | 40 | 5 | ? | 160 |
| {4,7| 3} | 42 | 84 | 24 | 10 | LF(2,7) | 168 |
| {7,4| 3} | 24 | 84 | 42 | 10 | LF(2,7) | 168 |
| {5,5| 4} | 72 | 180 | 72 | 19 | A6 | 360 |
| {6,7| 3} | 182 | 546 | 156 | 105 | LF(2,13) | 1092 |
| {7,6| 3} | 156 | 546 | 182 | 105 | LF(2,13) | 1092 |
| {7,7| 3} | 156 | 546 | 156 | 118 | LF(2,13) | 1092 |
| {4,9| 3} | 612 | 1224 | 272 | 171 | LF(2,17) | 2448 |
| {9,4| 3} | 272 | 1224 | 612 | 171 | LF(2,17) | 2448 |
| {7,8| 3} | 1536 | 5376 | 1344 | 1249 | ? | 10752 |
| {8,7| 3} | 1344 | 5376 | 1536 | 1249 | ? | 10752 |
A final set is based on Coxeter's further extended form {q1,m|q2,q3...} or with q2 unspecified: {l, m |, q}.
| {l, m |, q} | Faces | Edges | Vertices | p | Structure | Order |
|---|---|---|---|---|---|---|
| {3,6|,q} | 2q2 | 3q2 | q2 | 1 | ? | 2q2 |
| {3,2q|,3} | 2q2 | 3q2 | 3q | (q-1)*(q-2)/2 | ? | 2q2 |
| {3,7|,4} | 56 | 84 | 24 | 3 | LF(2,7) | 168 |
| {3,8|,4} | 112 | 168 | 42 | 8 | PGL(2,7) | 336 |
| {4,6|,3} | 84 | 168 | 56 | 15 | PGL(2,7) | 336 |
| {3,7|,6} | 364 | 546 | 156 | 14 | LF(2,13) | 1092 |
| {3,7|,7} | 364 | 546 | 156 | 14 | LF(2,13) | 1092 |
| {3,8|,5} | 720 | 1080 | 270 | 46 | ? | 2160 |
| {3,10|,4} | 720 | 1080 | 216 | 73 | ? | 2160 |
| {4,6|,2} | 12 | 24 | 8 | 3 | S4xS2 | 48 |
| {5,6|,2} | 24 | 60 | 20 | 9 | A5xS2 | 120 |
| {3,11|,4} | 2024 | 3036 | 552 | 231 | LF(2,23) | 6072 |
| {3,7|,8} | 3584 | 5376 | 1536 | 129 | ? | 10752 |
| {3,9|,5} | 12180 | 18270 | 4060 | 1016 | LF(2,29)xA3 | 36540 |
Read more about this topic: Regular Skew Polyhedron
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