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In geometry, a semiregular 4-polytope (or polychoron) is a 4-dimensional polytope which is vertex-transitive (i.e. the symmetry group of the polytope acts transitively on the vertices) and whose cells are regular polyhedra. These represent a subset of the uniform polychora which are composed of both regular and uniform polyhedra cells.## Summary

Polychora## Regular polytopes

## Semiregular polytopes

## Infinite tessellations (Honeycombs)

## Dual honeycombs

## Existence enumeration by edge configurations

where φ = (1 + √5)/2 is the golden mean.## See also

## External links

A further constraint can require edge-transitivity. Polychora that fail this contraint are listed and noted as such. The regular and semiregular honeycombs, and regular polychora are also listed here for completeness.

- 6 regular polychora
- 2 Vertex-transitive AND edge-transitive: rectified 5-cell, Rectified 600-cell
- 1 Vertex-transitive: snub 24-cell

Honeycombs

- 1 Regular honeycomb: cubic honeycomb
- 1 Vertex-transitive AND edge-transitive: tetrahedral-octahedral honeycomb
- 1 Vertex-transitive: gyrated tetrahedral-octahedral honeycomb

The 6 convex regular 4-polytopes are:

Edge figure | Name | Cells | Faces | Edges | Vertices | Edge figure | Vertex figure | Cells/vertex |
---|---|---|---|---|---|---|---|---|

{3,3}^{3}
| 5-cell | 5 | 10 | 10 | 5 | triangle | tetrahedron | 3 {3,3} |

{3,3}^{4}
| 16-cell | 16 | 32 | 24 | 8 | square | octahedron | 8 {3,3} |

{4,3}^{3}
| tesseract | 8 | 24 | 32 | 16 | triangle | tetrahedron | 4 {4,3} |

{3,4}^{3}
| 24-cell | 16 | 96 | 96 | 24 | triangle | cube | 6 {3,4} |

{5,3}^{3}
| 120-cell | 120 | 720 | 1200 | 600 | triangle | tetrahedron | 4 {5,3} |

{3,3}^{5}
| 600-cell | 600 | 1200 | 720 | 120 | pentagon | icosahedron | 20 {3,3} |

There are 3 semiregular polytopes. The third, the snub 24-cell, is not edge-transitive.

Edge configuration(s) | Name | Cells | Faces type | Edges | Vertices | Edge figure(s) | Vertex figure | Cells/vertex |
---|---|---|---|---|---|---|---|---|

{3,3}.{3,4}^{2}
| Rectified 5-cell | 5 {3,3} 5 {3,4} | 30 {3} | 30 | 10 | Isosceles triangle | triangular prism | 5 {3,3} 5 {3,4} |

{3,4}^{2}.{3,5}
| Rectified 600-cell | 600 {3,4} 120 {3,5} | 3600 {3} | 3600 | 720 | Isosceles triangle | pentagonal prism | 5 {3,4} 2 {3,5} |

I: {3,3}.{3,5}^{2},II: {3,3} ^{3}.{3,5}
| Snub 24-cell | 120 {3,3} 24 {3,5} | 480 {3} | 432 | 96 | Isosceles triangle, Kite | tridiminished icosahedron | 5 {3,3} 3 {3,5} |

There is one regular and two semiregular honeycombs. These are three of 28 convex uniform honeycombs that allow semiregular polyhedra cells as well as the Platonic solids.

There is only one regular honeycomb:

Edge configuration | Name | Face type | Edge figure | Vertex figure | Cells/vertex |
---|---|---|---|---|---|

{4,3}^{4}
| Cubic honeycomb | {4} | square | octahedron | 8 {4,3} |

There are two semiregular honeycombs and they contain the same edge cells, but the second is not edge-transitive as the order changes.

Edge configuration(s) | Name | Face type | Edge figure | Vertex figure | Cells/vertex |
---|---|---|---|---|---|

[{3,3}.{3.4}]^{2}
| Tetrahedral-octahedral honeycomb | {3} | rhombus | cuboctahedron | 8 {3,3} 6 {3,4} |

I: [{3,3}.{3.4}]^{2},II: {3,3} ^{2}.{3.4}^{2}
| Gyrated tetrahedral-octahedral honeycomb | {3} | rhombus, trapezoid | triangular orthobicupola | 8 {3,3} 6 {3,4} |

Vertex figures

The regular cubic honeycomb is self-dual.

The semiregular tetrahedral-octahedral honeycomb dual is called a rhombic dodecahedral honeycomb.

The semiregular gyrated tetrahedral-octahedral honeycomb dual is called a rhombo-hexagonal dodecahedron honeycomb.

Semiregular polytopes are constructed by vertex figures which are regular, semiregular or johnson polyhedra.

- If the vertex figure is a regular Platonic solid polyhedron, the polytope will be regular.
- If the vertex figure is a semiregular polyhedron, the polytope will have one type of edge configuration.
- If the vertex figure is a Johnson solid polyhedron, then the polytope will have more than one edge configuration.

Edge configurations are limited by the sum of the dihedral angles of the cells along the edge. The sum of dihedral angles must be 360 degrees or less. If it is equal to 360, the vertex figure will stay within 3D space and can be a part of an infinite tessellation.

The dihedral angle of each Platonic solid is:

Name | exact dihedral angle (in radians) | approximate dihedral angle (in degrees) |
---|---|---|

{3,3} Tetrahedron | arccos(1/3) | 70.53° |

{3,4} Octahedron | π − arccos(1/3) | 109.47° |

{4,3} Hexahedron or Cube | π/2 | 90° |

{3,5} Icosahedron | 2·arctan(φ + 1) | 138.19° |

{5,3} Dodecahedron | 2·arctan(φ) | 116.56° |

There are 17 possible edge configurations formed by the 5 platonic solids that have angle defects of zero or greater.

- Three cells/edge:
- {3,3}
^{3} - {3,3}
^{2}.{3.4} - {3,3}
^{2}.{3.5} - {3,3}.{3,4}
^{2} - {3,3}.{3.4}.{3.5}
- {3,3}.{3.5}
^{2} - {3,4}
^{3} - {3,4}
^{2}.{3,5} - {4,3}
^{3} - {5,3}
^{3} - Four cells/edge:
- {3,3}
^{4} - {3,3}
^{2}.{3.4}^{2}[Angle defect zero] - [{3,3}.{3.4}]
^{2}[Angle defect zero] - {3,3}
^{3}.{3,4} - {3,3}
^{3}.{3,5} - {4,3}
^{4}[Angle defect zero] - Five cells/edge:
- {3,3}
^{5}

As listed above, from these 17 edge configurations and a single vertex figure, there are 6 regular polytopes, and 3 semiregular polytopes, 1 regular honeycomb, and 2 semiregular honeycombs.

- Vertex/Edge/Face/Cell data
- Exploded/Unfolded cell images
- Data and Images (www.polytope.de)

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Last updated on Wednesday March 19, 2008 at 03:53:57 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday March 19, 2008 at 03:53:57 PDT (GMT -0700)

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