Order-6 dodecahedral honeycomb

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Order-6 dodecahedral honeycomb
H3 536 CC center.png
Perspective projection view
within Poincaré disk model
Type Hyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbol {5,3,6}
{5,3[3]}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.png
Cells {5,3} Dodecahedron.png
Faces pentagon {5}
Edge figure hexagon {6}
Vertex figure Uniform tiling 63-t2.png Uniform tiling 333-t1.png
triangular tiling
Dual Order-5 hexagonal tiling honeycomb
Coxeter group , [5,3,6]
, [5,3[3]]
Properties Regular, quasiregular

The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of faces, with all vertices as ideal points at infinity. It has Schläfli symbol {5,3,6}, with six ideal dodecahedral cells surrounding each edge of the honeycomb. Each vertex is ideal, and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure.

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.

Symmetry[]

A half symmetry construction exists as CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.png with alternately colored dodecahedral cells.

Images[]

Order-6 dodecahedral honeycomb.png
The model is cell-centered within the Poincaré disk model, with the viewpoint then placed at the origin.

The order-6 dodecahedral honeycomb is similar to the 2D hyperbolic infinite-order pentagonal tiling, {5,∞}, with pentagonal faces, and with vertices on the ideal surface.

H2 tiling 25i-4.png

Related polytopes and honeycombs[]

The order-6 dodecahedral honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.

11 paracompact regular honeycombs
H3 633 FC boundary.png
{6,3,3}
H3 634 FC boundary.png
{6,3,4}
H3 635 FC boundary.png
{6,3,5}
H3 636 FC boundary.png
{6,3,6}
H3 443 FC boundary.png
{4,4,3}
H3 444 FC boundary.png
{4,4,4}
H3 336 CC center.png
{3,3,6}
H3 436 CC center.png
{4,3,6}
H3 536 CC center.png
{5,3,6}
H3 363 FC boundary.png
{3,6,3}
H3 344 CC center.png
{3,4,4}

There are 15 uniform honeycombs in the [5,3,6] Coxeter group family, including this regular form, and its regular dual, the order-5 hexagonal tiling honeycomb.

[6,3,5] family honeycombs
{6,3,5} r{6,3,5} t{6,3,5} rr{6,3,5} t0,3{6,3,5} tr{6,3,5} t0,1,3{6,3,5} t0,1,2,3{6,3,5}
H3 635 FC boundary.png H3 635 boundary 0100.png H3 635-1100.png H3 635-1010.png H3 635-1001.png H3 635-1110.png H3 635-1101.png H3 635-1111.png
H3 536 CC center.png H3 536 CC center 0100.png H3 635-0011.png H3 635-0101.png H3 635-0110.png H3 635-0111.png H3 635-1011.png
{5,3,6} r{5,3,6} t{5,3,6} rr{5,3,6} 2t{5,3,6} tr{5,3,6} t0,1,3{5,3,6} t0,1,2,3{5,3,6}

The order-6 dodecahedral honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures:

Hyperbolic uniform honeycombs: {p,3,6}
Form Paracompact Noncompact
Name {3,3,6} {4,3,6} {5,3,6} {6,3,6} {7,3,6} {8,3,6} ... {∞,3,6}
Image H3 336 CC center.png H3 436 CC center.png H3 536 CC center.png H3 636 FC boundary.png Hyperbolic honeycomb 7-3-6 poincare.png Hyperbolic honeycomb 8-3-6 poincare.png Hyperbolic honeycomb i-3-6 poincare.png
Cells Tetrahedron.png
{3,3}
Hexahedron.png
{4,3}
Dodecahedron.png
{5,3}
Uniform tiling 63-t0.svg
{6,3}
Heptagonal tiling.svg
{7,3}
H2-8-3-dual.svg
{8,3}
H2-I-3-dual.svg
{∞,3}

It is also part of a sequence of regular polytopes and honeycombs with dodecahedral cells:

{5,3,p} polytopes
Space S3 H3
Form Finite Compact Paracompact Noncompact
Name {5,3,3} {5,3,4} {5,3,5} {5,3,6} {5,3,7} {5,3,8} ... {5,3,∞}
Image Schlegel wireframe 120-cell.png H3 534 CC center.png H3 535 CC center.png H3 536 CC center.png Hyperbolic honeycomb 5-3-7 poincare.png Hyperbolic honeycomb 5-3-8 poincare.png Hyperbolic honeycomb 5-3-i poincare.png
Vertex
figure
Tetrahedron.png
{3,3}
Octahedron.png
{3,4}
Icosahedron.png
{3,5}
Uniform tiling 63-t2.svg
{3,6}
Order-7 triangular tiling.svg
{3,7}
H2-8-3-primal.svg
{3,8}
H2 tiling 23i-4.png
{3,∞}

Rectified order-6 dodecahedral honeycomb[]

Rectified order-6 dodecahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols r{5,3,6}
t1{5,3,6}
Coxeter diagrams CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel branch.png
Cells r{5,3} Uniform polyhedron-53-t1.png
{3,6} Uniform tiling 63-t2.png
Faces triangle {3}
pentagon {5}
Vertex figure Rectified order-6 dodecahedral honeycomb verf.png
hexagonal prism
Coxeter groups , [5,3,6]
, [5,3[3]]
Properties Vertex-transitive, edge-transitive

The rectified order-6 dodecahedral honeycomb, t1{5,3,6} has icosidodecahedron and triangular tiling cells connected in a hexagonal prism vertex figure.

H3 536 CC center 0100.png
Perspective projection view within Poincaré disk model

It is similar to the 2D hyperbolic pentaapeirogonal tiling, r{5,∞} with pentagon and apeirogonal faces.

H2 tiling 25i-2.png
r{p,3,6}
Space H3
Form Paracompact Noncompact
Name r{3,3,6}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
r{4,3,6}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
r{5,3,6}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
r{6,3,6}
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
r{7,3,6}
CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
... r{∞,3,6}
CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
Image H3 336 CC center 0100.png H3 436 CC center 0100.png H3 536 CC center 0100.png H3 636 boundary 0100.png
Cells
Uniform tiling 63-t2.svg
{3,6}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
Uniform polyhedron-33-t1.png
r{3,3}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cuboctahedron.png
r{4,3}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Icosidodecahedron.png
r{5,3}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 63-t1.svg
r{6,3}
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Triheptagonal tiling.svg
r{7,3}
CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png
H2 tiling 23i-2.png
r{∞,3}
CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png

Truncated order-6 dodecahedral honeycomb[]

Truncated order-6 dodecahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t{5,3,6}
t0,1{5,3,6}
Coxeter diagrams CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel branch.png
Cells t{5,3} Uniform polyhedron-53-t01.png
{3,6} Uniform tiling 63-t2.png
Faces triangle {3}
decagon {10}
Vertex figure Truncated order-6 dodecahedral honeycomb verf.png
hexagonal pyramid
Coxeter groups , [5,3,6]
, [5,3[3]]
Properties Vertex-transitive

The truncated order-6 dodecahedral honeycomb, t0,1{5,3,6} has truncated dodecahedron and triangular tiling cells connected in a hexagonal pyramid vertex figure.

H3 635-0011.png

Bitruncated order-6 dodecahedral honeycomb[]

The bitruncated order-6 dodecahedral honeycomb is the same as the bitruncated order-5 hexagonal tiling honeycomb.

Cantellated order-6 dodecahedral honeycomb[]

Cantellated order-6 dodecahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols rr{5,3,6}
t0,2{5,3,6}
Coxeter diagrams CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node h0.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch 11.png
Cells rr{5,3} Uniform polyhedron-53-t02.png
rr{6,3} Uniform tiling 63-t1.png
{}x{6} Hexagonal prism.png
Faces triangle {3}
square {4}
pentagon {5}
hexagon {6}
Vertex figure Cantellated order-6 dodecahedral honeycomb verf.png
wedge
Coxeter groups , [5,3,6]
, [5,3[3]]
Properties Vertex-transitive

The cantellated order-6 dodecahedral honeycomb, t0,2{5,3,6}, has rhombicosidodecahedron, trihexagonal tiling, and hexagonal prism cells, with a wedge vertex figure.

H3 635-0101.png

Cantitruncated order-6 dodecahedral honeycomb[]

Cantitruncated order-6 dodecahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols tr{5,3,6}
t0,1,2{5,3,6}
Coxeter diagrams CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node h0.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel branch 11.png
Cells tr{5,3} Uniform polyhedron-53-t012.png
t{3,6} Uniform tiling 63-t12.png
{}x{6} Hexagonal prism.png
Faces square {4}
hexagon {6}
decagon {10}
Vertex figure Cantitruncated order-6 dodecahedral honeycomb verf.png
mirrored sphenoid
Coxeter groups , [5,3,6]
, [5,3[3]]
Properties Vertex-transitive

The cantitruncated order-6 dodecahedral honeycomb, t0,1,2{5,3,6} has truncated icosidodecahedron, hexagonal tiling, and hexagonal prism facets, with a mirrored sphenoid vertex figure.

H3 635-0111.png

Runcinated order-6 dodecahedral honeycomb[]

The runcinated order-6 dodecahedral honeycomb is the same as the runcinated order-5 hexagonal tiling honeycomb.

Runcitruncated order-6 dodecahedral honeycomb[]

Runcitruncated order-6 dodecahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t0,1,3{5,3,6}
Coxeter diagrams CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.png
Cells t{5,3} Uniform polyhedron-53-t01.png
rr{6,3} Uniform tiling 63-t02.png
{}x{10} Decagonal prism.png
{}x{6} Hexagonal prism.png
Faces square {4}
hexagon {6}
decagon {10}
Vertex figure Runcitruncated order-6 dodecahedral honeycomb verf.png
isosceles-trapezoidal pyramid
Coxeter groups , [5,3,6]
Properties Vertex-transitive

The runcitruncated order-6 dodecahedral honeycomb, t0,1,3{5,3,6} has truncated dodecahedron, rhombitrihexagonal tiling, decagonal prism, and hexagonal prism facets, with an isosceles-trapezoidal pyramid vertex figure.

H3 635-1011.png

Runcicantellated order-6 dodecahedral honeycomb[]

The runcicantellated order-6 dodecahedral honeycomb is the same as the runcitruncated order-5 hexagonal tiling honeycomb.

Omnitruncated order-6 dodecahedral honeycomb[]

The omnitruncated order-6 dodecahedral honeycomb is the same as the omnitruncated order-5 hexagonal tiling honeycomb.

See also[]

  • Convex uniform honeycombs in hyperbolic space
  • Regular tessellations of hyperbolic 3-space
  • Paracompact uniform honeycombs

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)
  • The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
  • 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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