Semiregular 4-polytope

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.

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.

Summary

Polychora
* 6 regular polychora
* 2 Vertex-transitive AND edge-transitive: rectified 5-cell, Rectified 600-cell
* 1 Vertex-transitive: snub 24-cellHoneycombs
* 1 Regular honeycomb: cubic honeycomb
* 1 Vertex-transitive AND edge-transitive: tetrahedral-octahedral honeycomb
* 1 Vertex-transitive: gyrated tetrahedral-octahedral honeycomb

Regular polytopes

The 6 convex regular 4-polytopes are:

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

Vertex figures

Dual honeycombs

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.

Existence enumeration by edge configurations

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°

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

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.

ee also

*Convex regular 4-polytope
*Uniform polychora
*Semiregular polyhedron
*List of regular polytopes

External links

* Vertex/Edge/Face/Cell data
** [http://members.aol.com/Polycell/section1.html Dispentachoron] [2]
** [http://members.aol.com/Polycell/section4.html Icosahedral hexacosihecatonicosachoron] [34]
** [http://members.aol.com/Polycell/section3.html Snub icositetrachoron] [31]
* Exploded/Unfolded cell images
** [http://members.aol.com/Polycell/nets.html Snub icositetrachoron]
** [http://members.aol.com/Polycell/nets.html Icosahedral hexacosihecatonicosachoron]
* Data and Images (www.polytope.de)
** [http://www.polytope.de/nr01.html 5-cell]
** [http://www.polytope.de/nr06.html 8-cell]
** [http://www.polytope.de/nr02.html 16-cell]
** [http://www.polytope.de/nr07.html 24-cell]
** [http://www.polytope.de/nr49.html 120-cell]
** [http://www.polytope.de/nr24.html 600-cell]
** [http://www.polytope.de/nr03.html Rectified 5-cell]
** [http://www.polytope.de/nr45.html Rectified 600-cell]
** [http://www.polytope.de/nr22.html Snub icositetrachoron]