Definitions

# 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.

Polychora

Honeycombs

## 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}

## 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}

## 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

## 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.

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/290°
{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.

## External links

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

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