Cubic-octahedral honeycomb

In the geometry of hyperbolic 3-space, the cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from cube, octahedron, and cuboctahedron cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

Cube-octahedron honeycomb
TypeCompact uniform honeycomb
Schläfli symbol{(3,4,3,4)} or {(4,3,4,3)}
Coxeter diagrams
Cells{4,3}
{3,4}
r{4,3}
Facestriangle {3}
square {4}
Vertex figure
rhombicuboctahedron
Coxeter group[(4,3)[2]]
PropertiesVertex-transitive, edge-transitive

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

Images

Wide-angle perspective views:

It contains a subgroup H2 tiling, the alternated order-4 hexagonal tiling, , with vertex figure (3.4)4.

Symmetry

A lower symmetry form, index 6, of this honeycomb can be constructed with [(4,3,4,3*)] symmetry, represented by a trigonal trapezohedron fundamental domain, and Coxeter diagram . This lower symmetry can be extended by restoring one mirror as .

Cells

=

=

=

There are 5 related uniform honeycombs generated within the same family, generated with 2 or more rings of the Coxeter group : , , , , .

Rectified cubic-octahedral honeycomb

Rectified cubic-octahedral honeycomb
TypeCompact uniform honeycomb
Schläfli symbolr{(4,3,4,3)}
Coxeter diagrams
Cellsr{4,3}
rr{3,4}
Facestriangle {3}
square {4}
Vertex figure
cuboid
Coxeter group[[(4,3)[2]]],
PropertiesVertex-transitive, edge-transitive

The rectified cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from cuboctahedron and rhombicuboctahedron cells, in a cuboid vertex figure. It has a Coxeter diagram .

Perspective view from center of rhombicuboctahedron

Cyclotruncated cubic-octahedral honeycomb

Cyclotruncated cubic-octahedral honeycomb
TypeCompact uniform honeycomb
Schläfli symbolct{(4,3,4,3)}
Coxeter diagrams
Cellst{4,3}
{3,4}
Facestriangle {3}
octagon {8}
Vertex figure
square antiprism
Coxeter group[[(4,3)[2]]],
PropertiesVertex-transitive, edge-transitive

The cyclotruncated cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from truncated cube and octahedron cells, in a square antiprism vertex figure. It has a Coxeter diagram .

Perspective view from center of octahedron

It can be seen as somewhat analogous to the trioctagonal tiling, which has truncated square and triangle facets:

Cyclotruncated octahedral-cubic honeycomb

Cyclotruncated octahedral-cubic honeycomb
TypeCompact uniform honeycomb
Schläfli symbolct{(3,4,3,4)}
Coxeter diagrams
Cells{4,3}
t{3,4}
Facessquare {4}
hexagon {6}
Vertex figure
triangular antiprism
Coxeter group[[(4,3)[2]]],
PropertiesVertex-transitive, edge-transitive

The cyclotruncated octahedral-cubic honeycomb is a compact uniform honeycomb, constructed from cube and truncated octahedron cells, in a triangular antiprism vertex figure. It has a Coxeter diagram .

Perspective view from center of cube

It contains an H2 subgroup tetrahexagonal tiling alternating square and hexagonal faces, with Coxeter diagram or half symmetry :

Symmetry

Fundamental domains

Trigonal trapezohedron

Half domain

H2 subgroup, rhombic *3232

A radial subgroup symmetry, index 6, of this honeycomb can be constructed with [(4,3,4,3*)], , represented by a trigonal trapezohedron fundamental domain, and Coxeter diagram . This lower symmetry can be extended by restoring one mirror as .


Cells

=

=

Truncated cubic-octahedral honeycomb

Truncated cubic-octahedral honeycomb
TypeCompact uniform honeycomb
Schläfli symbolt{(4,3,4,3)}
Coxeter diagrams
Cellst{3,4}
t{4,3}
rr{3,4}
tr{4,3}
Facestriangle {3}
square {4}
hexagon {6}
octagon {8}
Vertex figure
rectangular pyramid
Coxeter group[(4,3)[2]]
PropertiesVertex-transitive

The truncated cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from truncated octahedron, truncated cube, rhombicuboctahedron, and truncated cuboctahedron cells, in a rectangular pyramid vertex figure. It has a Coxeter diagram .

Perspective view from center of rhombicuboctahedron

Omnitruncated cubic-octahedral honeycomb

Omnitruncated cubic-octahedral honeycomb
TypeCompact uniform honeycomb
Schläfli symboltr{(4,3,4,3)}
Coxeter diagrams
Cellstr{3,4}
Facessquare {4}
hexagon {6}
octagon {8}
Vertex figure
Rhombic disphenoid
Coxeter group[2[(4,3)[2]]] or [(2,2)+[(4,3)[2]]],
PropertiesVertex-transitive, edge-transitive, cell-transitive

The omnitruncated cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from truncated cuboctahedron cells, in a rhombic disphenoid vertex figure. It has a Coxeter diagram with [2,2]+ (order 4) extended symmetry in its rhombic disphenoid vertex figure.

Perspective view from center of truncated cuboctahedron

See also

References

  • Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
  • Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
  • Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I,II)
  • Norman Johnson Uniform Polytopes, Manuscript
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
    • N.W. Johnson: Geometries and Transformations, (2018) Chapter 13: Hyperbolic Coxeter groups
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