List of convex regular-faced polyhedra (Meta, 13)
From Hi.gher. Space
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<td>[[Pentagonal trapezohedron]]</td><td><tt>---</tt></td><td>''Unknown''</td> | <td>[[Pentagonal trapezohedron]]</td><td><tt>---</tt></td><td>''Unknown''</td> | ||
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| + | The ''utility prisms'' are so named because they are the only prisms which appear as cells in higher-dimensional uniform polytopes. The antiprisms of the same symmetry groups are included for completeness. | ||
[[Category:CRF polyhedra|$]] | [[Category:CRF polyhedra|$]] | ||
Revision as of 13:18, 5 December 2010
Three classes used in the table below:
- B = Brick: point (x, y, z) implies point (±x, ±y, ±z)
- S = Simple: exactly three faces meet at each vertex
- Z = Zonotope: each face has point symmetry
| Index | Parent | Elements | Dual | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Name | Classes | SSC2 | Faces | Edges | Vertices | Name | Classes | SSC2 | |
| Platonic solids (total 5) | |||||||||
| 1 | Tetrahedron | -S- | Kt1 | 4 | 6 | 4 | Self-dual | ||
| 2 | Cube | BSZ | Ko1 | 6 | 12 | 8 | 3. Octahedron (below) | ||
| 3 | Octahedron | B-- | Ko4 | 8 | 12 | 6 | 2. Cube (above) | ||
| 4 | Dodecahedron | BS- | Ki1 | 12 | 30 | 20 | 5. Icosahedron (below) | ||
| 5 | Icosahedron | B-- | Ki4 | 20 | 30 | 12 | 4. Dodecahedron (above) | ||
| Archimedean solids (total 13) | |||||||||
| 6 | Tetrahedral truncate | -S- | Kt3 | 8 | 18 | 12 | Triakis tetrahedron | --- | Unknown |
| 7 | Cuboctahedron | B-- | Ko2 | 14 | 24 | 12 | Rhombic dodecahedron | B-Z | Unknown |
| 8 | Cubic truncate | BS- | Ko3 | 14 | 36 | 24 | Triakis octahedron | B-- | Unknown |
| 9 | Octahedral truncate | BS- | Ko6 | 14 | 36 | 24 | Tetrakis hexahedron | B-- | Unknown |
| 10 | Cuboctahedral rectate | B-- | Ko5 | 26 | 48 | 24 | Deltoidal icositetrahedron | --- | Unknown |
| 11 | Cuboctahedral truncate | BSZ | Ko7 | 26 | 72 | 48 | Disdyakis dodecahedron | B-- | Unknown |
| 12 | Cubic snub | --- | Ko0 | 38 | 60 | 24 | Pentagonal icositetrahedron | --- | Unknown |
| 13 | Icosidodecahedron | B-- | Ki2 | 32 | 60 | 30 | Rhombic triacontahedron | B-Z | Unknown |
| 14 | Dodecahedral truncate | BS- | Ki3 | 32 | 90 | 60 | Triakis icosahedron | B-- | Unknown |
| 15 | Icosahedral truncate | BS- | Ki6 | 32 | 90 | 30 | Pentakis dodecahedron | B-- | Unknown |
| 16 | Icosidodecahedral rectate | B-- | Ki5 | 62 | 120 | 60 | Deltoidal hexecontahedron | --- | Unknown |
| 17 | Icosidodecahedral truncate | BSZ | Ki7 | 62 | 180 | 120 | Disdyakis triacontahedron | B-- | Unknown |
| 18 | Dodecahedral snub | --- | Ki0 | 92 | 150 | 60 | Pentagonal hexecontahedron | --- | Unknown |
| Johnson solids (total 92) | |||||||||
| 19 = J1 | Square pyramid | --- | &G4 | 5 | 8 | 5 | Self-dual | ||
| 20 = J2 | Pentagonal pyramid | --- | &G5 | 6 | 10 | 6 | Self-dual | ||
| 21 = J3 | Triangular cupola | --- | G3$H6 | 8 | 15 | 9 | Unknown | ||
| 22 = J4 | Square cupola | --- | G4$H8 | 10 | 20 | 12 | Unknown | ||
| 23 = J5 | Pentagonal cupola | --- | G5$H10 | 12 | 25 | 15 | Unknown | ||
| 24 = J6 | Pentagonal rotunda | --- | G5$H5$H10 | 17 | 35 | 20 | Unknown | ||
| Utility prisms (total 7) | |||||||||
| 111 | Triangular prism | -S- | +G3 | 5 | 9 | 6 | 30. Triangular bipyramid (above) | ||
| 112 | Hexagonal prism | BSZ | +G6 | 8 | 18 | 12 | Hexagonal bipyramid | B-- | G6vM1 |
| 113 | Octagonal prism | BSZ | +G8 | 10 | 24 | 16 | Octagonal bipyramid | B-- | G8vM1 |
| 114 | Square antiprism | --- | G4$H4 | 10 | 16 | 8 | Tetragonal trapezohedron | --- | Unknown |
| 115 | Pentagonal prism | -S- | +G5 | 7 | 15 | 10 | 31. Pentagonal bipyramid (above) | ||
| 116 | Decagonal prism | BSZ | +G10 | 12 | 30 | 20 | Decagonal bipyramid | B-- | G10vM1 |
| 117 | Pentagonal antiprism | --- | G5$H5 | 12 | 20 | 10 | Pentagonal trapezohedron | --- | Unknown |
The utility prisms are so named because they are the only prisms which appear as cells in higher-dimensional uniform polytopes. The antiprisms of the same symmetry groups are included for completeness.
