Semiregular 4-polytope

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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 semiregular 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.

Contents 1 Summary 2 Regular polytopes 3 Semiregular polytopes 4 Infinite tessellations (Honeycombs) 5 Dual honeycombs 6 Existence enumeration by edge configurations 7 See also 8 External links

[edit] Summary

Polychora

• 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

[edit] Regular polytopes

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}

[edit] Semiregular polytopes

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}

[edit] Infinite tessellations (Honeycombs)

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

[edit] 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.

[edit] Existence enumeration by edge configurations

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

1. If the vertex figure is a regular Platonic solid polyhedron, the polytope will be regular.
2. If the vertex figure is a semiregular polyhedron, the polytope will have one type of edge configuration.
3. 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:

1. {3,3}³
2. {3,3}².{3.4}
3. {3,3}².{3.5}
4. {3,3}.{3,4}²
5. {3,3}.{3.4}.{3.5}
6. {3,3}.{3.5}²
7. {3,4}³
8. {3,4}².{3,5}
9. {4,3}³
10. {5,3}³

• Four cells/edge:

1. {3,3}⁴
2. {3,3}².{3.4}² [Angle defect zero]
3. [{3,3}.{3.4}]² [Angle defect zero]
4. {3,3}³.{3,4}
5. {3,3}³.{3,5}
6. {4,3}⁴ [Angle defect zero]

• Five cells/edge:

1. {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.

[edit] See also

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

[edit] External links

• Vertex/Edge/Face/Cell data
  • Dispentachoron [2]
  • Icosahedral hexacosihecatonicosachoron [34]
  • Snub icositetrachoron [31]
• Exploded/Unfolded cell images
  • Snub icositetrachoron
  • Icosahedral hexacosihecatonicosachoron
• Data and Images (www.polytope.de)
  • 5-cell
  • 8-cell
  • 16-cell
  • 24-cell
  • 120-cell
  • 600-cell
  • Rectified 5-cell
  • Rectified 600-cell
  • Snub icositetrachoron