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词条 Order-5 cubic honeycomb
释义

  1. Description

  2. Symmetry

  3. Related polytopes and honeycombs

      Compact regular honeycombs   543 honeycombs   Polytopes with icosahedral vertex figures   Related polytopes and honeycombs with cubic cells   Rectified order-5 cubic honeycomb    Related honeycomb   Truncated order-5 cubic honeycomb    Related honeycombs    Bitruncated order-5 cubic honeycomb    Cantellated order-5 cubic honeycomb    Related honeycombs   Cantitruncated order-5 cubic honeycomb    Related honeycombs   Runcinated order-5 cubic honeycomb    Related honeycombs   Runcitruncated order-5 cubic honeycomb    Related honeycombs   Omnitruncated order-5 cubic honeycomb    Related honeycombs   Alternated order-5 cubic honeycomb    Related honeycombs  Cantic order-5 cubic honeycomb  Runcic order-5 cubic honeycomb  Runcicantic order-5 cubic honeycomb 

  4. See also

  5. References

Order-5 cubic honeycomb

Poincaré disk models
TypeHyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Schläfli symbol {4,3,5}
Coxeter diagramnode_1|4|node|3|node|5|node}}
Cells{4,3}
Facessquare {4}
Edge figurepentagon {5}
Vertex figure
icosahedron
Coxeter groupBH}}3, [5,3,4]
DualOrder-4 dodecahedral honeycomb
PropertiesRegular

The order-5 cubic honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.

{{Honeycomb}}

Description


One cell, centered in Poincare ball model
Main cells
Cells with extended edges to ideal boundary

Symmetry

It a radial subgroup symmetry construction with dodecahedral fundamental domains: Coxeter notation: [4,(3,5)*], index 120.

Related polytopes and honeycombs

It has a related alternation honeycomb, represented by {{CDD|node_h1|4|node|3|node|5|node}} ↔ {{CDD|nodes_10ru|split2|node|5|node}}, having icosahedron and tetrahedron cells.

Compact regular honeycombs

There are four regular compact honeycombs in 3D hyperbolic space:

{{Regular compact H3 honeycombs}}

543 honeycombs

There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including this regular form:

{{534 family}}

Polytopes with icosahedral vertex figures

It is in a sequence of polychora and honeycomb with icosahedron vertex figures:

{{Icosahedral vertex figure tessellations}}

Related polytopes and honeycombs with cubic cells

It in a sequence of regular polychora and honeycombs with cubic cells. The first polytope in the sequence is the tesseract, and the second is the Euclidean cubic honeycomb.

{{Cubic cell tessellations}}

Rectified order-5 cubic honeycomb

Rectified order-5 cubic honeycomb

TypeUniform honeycombs in hyperbolic space
Schläfli symbolr{4,3,5} or 2r{5,3,4}
2r{5,31,1}
Coxeter diagramnode|5|node|3|node_1|4|node}}
{{CDD|node|5|node|3|node_1|4|node_h0}} ↔ {{CDD|node|5|node|split1|nodes_11}}
Cellsr{4,3}
{3,5}
Facestriangle {3}
square {4}
Vertex figure
pentagonal prism
Coxeter groupBH}}3, [5,3,4]
{{overline|DH}}3, [5,31,1]
PropertiesVertex-transitive, edge-transitive

The rectified order-5 cubic honeycomb, {{CDD|node|5|node|3|node_1|4|node}}, has alternating icosahedron and cuboctahedron cells, with a pentagonal prism vertex figure.

Related honeycomb

There are four rectified compact regular honeycombs:

{{Rectified compact H3 honeycombs}}{{Pentagonal prism vertex figure tessellations}}{{-}}

Truncated order-5 cubic honeycomb

Truncated order-5 cubic honeycomb

TypeUniform honeycombs in hyperbolic space
Schläfli symbolt{4,3,5}
Coxeter diagramnode|5|node|3|node_1|4|node_1}}
Cellst{4,3}
{3,5}
Facestriangle {3}
square {4}
pentagon {5}
Vertex figure
pentagonal pyramid
Coxeter groupBH}}3, [5,3,4]
PropertiesVertex-transitive

The truncated order-5 cubic honeycomb, {{CDD|node|5|node|3|node_1|4|node_1}}, has truncated cube and icosahedron cells, with a pentagonal pyramid vertex figure.

It can be seen as analogous to the 2D hyperbolic truncated order-5 square tiling, t{4,5} with truncated square and pentagonal faces:

It is similar to the Euclidean (order-4) truncated cubic honeycomb, t{4,3,4}, with octahedral cells at the truncated vertices.

Related honeycombs

{{Truncated compact H3 honeycombs}}{{-}}

Bitruncated order-5 cubic honeycomb

Same as Bitruncated order-4 dodecahedral honeycomb

Cantellated order-5 cubic honeycomb

Cantellated order-5 cubic honeycomb

TypeUniform honeycombs in hyperbolic space
Schläfli symbolrr{4,3,5}
Coxeter diagramnode|5|node_1|3|node|4|node_1}}
Cellsrr{4,3}
r{3,5}
{}x{5}
Facestriangle {3}
square {4}
pentagon {5}
Vertex figure
wedge
Coxeter groupBH}}3, [5,3,4]
PropertiesVertex-transitive

The cantellated order-5 cubic honeycomb, {{CDD|node|5|node_1|3|node|4|node_1}}, has rhombicuboctahedron and icosidodecahedron cells, with a wedge vertex figure.

Related honeycombs

It is similar to the Euclidean (order-4) cantellated cubic honeycomb, rr{4,3,4}:

{{Cantellated compact H3 honeycombs}}{{-}}

Cantitruncated order-5 cubic honeycomb

Cantitruncated order-5 cubic honeycomb

TypeUniform honeycombs in hyperbolic space
Schläfli symboltr{4,3,5}
Coxeter diagramnode|5|node_1|3|node_1|4|node_1}}
Cellstr{4,3}
t{3,5}
Facessquare {4}
pentagon {5}
hexagon {6}
octahedron {8}
Vertex figure
Mirrored sphenoid
Coxeter groupBH}}3, [5,3,4]
PropertiesVertex-transitive

The cantitruncated order-5 cubic honeycomb, {{CDD|node|5|node_1|3|node|4|node_1}}, has rhombicuboctahedron and icosidodecahedron cells, with a mirrored sphenoid vertex figure.

Related honeycombs

It is similar to the Euclidean (order-4) cantitruncated cubic honeycomb, tr{4,3,4}:

{{Cantitruncated compact H3 honeycombs}}{{-}}

Runcinated order-5 cubic honeycomb

Runcinated order-5 cubic honeycomb

TypeUniform honeycombs in hyperbolic space
Semiregular honeycomb
Schläfli symbolt0,3{4,3,5}
Coxeter diagramnode_1|5|node|3|node|4|node_1}}
Cells{4,3}
{5,3}
{}x{5}
FacesSquare {4}
Pentagon {5}
Vertex figure
octahedron
Coxeter groupBH}}3, [5,3,4]
PropertiesVertex-transitive

The runcinated order-5 cubic honeycomb or runcinated order-4 dodecahedral honeycomb {{CDD|node_1|5|node|3|node|4|node_1}}, has cube, dodecahedron, and pentagonal prism cells, with an octahedron vertex figure.

It is analogous to the 2D hyperbolic rhombitetrapentagonal tiling, rr{4,5}, {{CDD|node_1|5|node|4|node_1}} with square and pentagonal faces:

Related honeycombs

It is similar to the Euclidean (order-4) runcinated cubic honeycomb, t0,3{4,3,4}:

{{Runcinated compact H3 honeycombs}}{{-}}

Runcitruncated order-5 cubic honeycomb

Runctruncated order-5 cubic honeycomb
Runcicantellated order-4 dodecahedral honeycomb

TypeUniform honeycombs in hyperbolic space
Schläfli symbolt0,1,3{4,3,5}
Coxeter diagramnode_1|5|node|3|node_1|4|node_1}}
Cellst{4,3}
rr{5,3}
{}x{5}
{}x{8}
FacesTriangle {3}
Square {4}
Pentagon {5}
Octagon {8}
Vertex figure
quad-pyramid
Coxeter groupBH}}3, [5,3,4]
PropertiesVertex-transitive

The runcitruncated order-5 cubic honeycomb or runcicantellated order-4 dodecahedral honeycomb {{CDD|node_1|5|node|3|node_1|4|node_1}}, has cube, dodecahedron, and pentagonal prism cells, with a quad-pyramid vertex figure.

Related honeycombs

It is similar to the Euclidean (order-4) runcitruncated cubic honeycomb, t0,1,3{4,3,4}:

{{Runcitruncated compact H3 honeycombs}}{{-}}

Omnitruncated order-5 cubic honeycomb

Omnitruncated order-5 cubic honeycomb

TypeUniform honeycombs in hyperbolic space
Semiregular honeycomb
Schläfli symbolt0,1,2,3{4,3,5}
Coxeter diagramnode_1|5|node_1|3|node_1|4|node_1}}
Cellstr{5,3}
tr{4,3}
{10}x{}
{8}x{}
FacesSquare {4}
Hexagon {6}
Octagon {8}
Decagon {10}
Vertex figure
tetrahedron
Coxeter groupBH}}3, [5,3,4]
PropertiesVertex-transitive

The omnitruncated order-5 cubic honeycomb or omnitruncated order-4 dodecahedral honeycomb has Coxeter diagram {{CDD|node_1|5|node_1|3|node_1|4|node_1}}.

Related honeycombs

It is similar to the Euclidean (order-4) omnitruncated cubic honeycomb, t0,1,2,3{4,3,4}:

{{Omnitruncated compact H3 honeycombs}}{{-}}

Alternated order-5 cubic honeycomb

Alternated order-5 cubic honeycomb

TypeUniform honeycombs in hyperbolic space
Schläfli symbolh{4,3,5}
Coxeter diagramnode_h1|4|node|3|node|5|node}} ↔ {{CDD|nodes_10ru|split2|node|5|node}}
Cells{3,3}
{3,5}
Facestriangle {3}
pentagon {5}
Vertex figure
icosidodecahedron
Coxeter groupDH}}3, [5,31,1]
Propertiesquasiregular

In 3-dimensional hyperbolic geometry, the alternated order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). With Schläfli symbol h{4,3,5}, it can be considered a quasiregular honeycomb, alternating icosahedra and tetrahedra around each vertex in an icosidodecahedron vertex figure.

{{-}}

Related honeycombs

It has 3 related forms: the cantic order-5 cubic honeycomb, {{CDD|node_h1|4|node|3|node_1|5|node}}, the runcic order-5 cubic honeycomb, {{CDD|node_h1|4|node|3|node|5|node_1}}, and the runcicantic order-5 cubic honeycomb, {{CDD|node_h1|4|node|3|node_1|5|node_1}}.

Cantic order-5 cubic honeycomb

Cantic order-5 cubic honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolh2{4,3,5}
Coxeter diagramnode_h1|4|node|3|node_1|5|node}} ↔ {{CDD|nodes_10ru|split2|node_1|5|node}}
Cellsr{5,3}
t{3,5}
t{3,3}
FacesTriangle {3}
Pentagon {5}
Hexagon {6}
Vertex figure
Rectangular pyramid
Coxeter groupDH}}3, [5,31,1]
PropertiesVertex-transitive

The cantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). It has Schläfli symbol h2{4,3,5} and a rectangular pyramid vertex figure.

{{-}}

Runcic order-5 cubic honeycomb

Runcic order-5 cubic honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolh3{4,3,5}
Coxeter diagramnode_h1|4|node|3|node|5|node_1}} ↔ {{CDD|nodes_10ru|split2|node|5|node_1}}
Cells{5,3}
rr{5,3}
{3,3}
FacesTriangle {3}
square {4}
pentagon {5}
Vertex figure
triangular prism
Coxeter groupDH}}3, [5,31,1]
PropertiesVertex-transitive

The runcic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). It has Schläfli symbol h3{4,3,5} and a triangular prism vertex figure.

{{-}}

Runcicantic order-5 cubic honeycomb

Runcicantic order-5 cubic honeycomb
TypeUniform honeycombs in hyperbolic space
Schläfli symbolh2,3{4,3,5}
Coxeter diagramnode_h1|4|node|3|node_1|5|node_1}} ↔ {{CDD|nodes_10ru|split2|node_1|5|node_1}}
Cellst{5,3}
tr{5,3}
t{3,3}
FacesTriangle {3}
square {4}
hexagon {6}
dodecagon {10}
Vertex figure
mirrored sphenoid
Coxeter groupDH}}3, [5,31,1]
PropertiesVertex-transitive

The runcicantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). It has Schläfli symbol h2,3{4,3,5} and a mirrored sphenoid vertex figure.

{{-}}

See also

  • Convex uniform honeycombs in hyperbolic space

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)
  • 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, (2015) Chapter 13: Hyperbolic Coxeter groups

1 : Honeycombs (geometry)
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