Order-4 dodecahedral honeycomb

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Order-4 dodecahedral honeycomb
H3 534 CC center.png
Type Hyperbolic regular honeycomb
Schläfli symbol {5,3,4}
{5,31,1}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png
Cells {5,3} Uniform polyhedron-53-t0.png
Faces pentagon {5}
Edge figure square {4}
Vertex figure Order-4 dodecahedral honeycomb verf.png
octahedron
Dual Order-5 cubic honeycomb
Coxeter group BH3, [5,3,4]
DH3, [5,31,1]
Properties Regular, Quasiregular honeycomb

In the geometry of hyperbolic 3-space, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs). With Schläfli symbol {5,3,4}, it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb.

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.

Description[edit]

The dihedral angle of a dodecahedron is ~116.6°, so it is impossible to fit 4 of them on an edge in Euclidean 3-space. However in hyperbolic space a properly scaled dodecahedron can be scaled so that its dihedral angles are reduced to 90 degrees, and then four fit exactly on every edge.

Symmetry[edit]

It a half symmetry construction, {5,31,1}, with two types (colors) of hexagonal tilings in the Wythoff construction. CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png.

Images[edit]

It contains 2D hyperbolic order-4 pentagonal tiling, {5,4}

Hyperbolic orthogonal dodecahedral honeycomb.png
Beltrami-Klein model

Related polytopes and honeycomb[edit]

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

Four regular compact honeycombs in H3
H3 534 CC center.png
{5,3,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
H3 435 CC center.png
{4,3,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
H3 353 CC center.png
{3,5,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
H3 535 CC center.png
{5,3,5}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png

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

[5,3,4] family honeycombs
{5,3,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t1{5,3,4}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t0,1{5,3,4}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t0,2{5,3,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t0,3{5,3,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
t0,1,2{5,3,4}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t0,1,3{5,3,4}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
t0,1,2,3{5,3,4}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
H3 534 CC center.png H3 534 CC center 0100.png H3 534-0011 center ultrawide.png H3 534-1010 center ultrawide.png H3 534-1001 center ultrawide.png H3 534-1110 center ultrawide.png H3 534-1101 center ultrawide.png H3 534-1111 center ultrawide.png
H3 435 CC center.png H3 435 CC center 0100.png H3 435-0011 center ultrawide.png H3 534-0101 center ultrawide.png H3 534-0110 center ultrawide.png H3 534-0111 center ultrawide.png H3 534-1011 center ultrawide.png
{4,3,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
t1{4,3,5}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
t0,1{4,3,5}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
t0,2{4,3,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
t1,2{4,3,5}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
t0,1,2{4,3,5}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
t0,1,3{4,3,5}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
t0,1,2,3{4,3,5}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png

There are eleven uniform honeycombs in the bifurcating [5,31,1] Coxeter group family, including this honeycomb in its alternated form. This construction can be represented by alternation (checkerboard) with two colors of dodecahedral cells.

This honeycomb is also related to the 16-cell, cubic honeycomb, and order-5 dodecahedral honeycomb all which have octahedral vertex figures:

{p,3,4}
Space S3 E3 H3
Form Finite Affine Compact Paracompact Noncompact
Name {3,3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
{4,3,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel labelinfin.pngCDel branch 10.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.png
{5,3,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png
{6,3,4}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node.pngCDel ultra.pngCDel node 1.pngCDel split1.pngCDel branch 11.pngCDel uaub.pngCDel nodes.png
{7,3,4}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 7.pngCDel node.pngCDel split1.pngCDel nodes.png
{8,3,4}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel split1.pngCDel nodes.png
... {∞,3,4}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel split1.pngCDel nodes.png
Image Stereographic polytope 16cell.png Cubic honeycomb.png H3 534 CC center.png H3 634 FC boundary.png
Cells Tetrahedron.png
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Hexahedron.png
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Dodecahedron.png
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 63-t0.png
{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 237-1.png
{7,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 238-1.png
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 23i-1.png
{∞,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png

This honeycomb is a part of a sequence of polychora and honeycombs with dodecahedral cells:

{5,3,p}
Space S3 H3
Form Finite Compact Paracompact Noncompact
Name {5,3,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{5,3,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png
{5,3,5}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{5,3,6}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.png
{5,3,7}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
{5,3,8}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png
... {5,3,∞}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png
Image Schlegel wireframe 120-cell.png H3 534 CC center.png H3 535 CC center.png H3 536 CC center.png H3 53i UHS plane at infinity.png
Vertex
figure
CDel node 1.pngCDel 3.pngCDel node.pngCDel p.pngCDel node.png
Tetrahedron.png
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Octahedron.png
{3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Icosahedron.png
{3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 63-t2.png
{3,6}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
H2 tiling 237-4.png
{3,7}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
H2 tiling 238-4.png
{3,8}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 23i-4.png
{3,∞}
CDel node 1.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png

Rectified order-4 dodecahedral honeycomb[edit]

Rectified order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol r{5,3,4}
r{5,31,1}
Coxeter diagram CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel nodes.png
Cells r{5,3} Uniform polyhedron-53-t1.png
{3,4} Uniform polyhedron-43-t2.png
Faces triangle {3}
pentagon {5}
Vertex figure Rectified order-4 dodecahedral honeycomb verf.png
cube
Coxeter group BH3, [5,3,4]
DH3, [5,31,1]
Properties Vertex-transitive, edge-transitive

The rectified order-4 dodecahedral honeycomb, CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png, has alternating octahedron and icosidodecahedron cells, with a cube vertex figure.

H3 534 CC center 0100.pngRectified order 4 dodecahedral honeycomb.png
It can be seen as analogous to the 2D hyperbolic tetrapentagonal tiling, r{5,4}

Related honeycombs[edit]

There are four rectified compact regular honeycombs:

Four rectified regular compact honeycombs in H3
Image H3 534 CC center 0100.png H3 435 CC center 0100.png H3 353 CC center 0100.png H3 535 CC center 0100.png
Symbols r{5,3,4}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
r{4,3,5}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
r{3,5,3}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
r{5,3,5}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Vertex
figure
Rectified order-4 dodecahedral honeycomb verf.png Rectified order-5 cubic honeycomb verf.png Rectified icosahedral honeycomb verf.png Rectified order-5 dodecahedral honeycomb verf.png


Truncated order-4 dodecahedral honeycomb[edit]

Truncated order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t{5,3,4}
t{5,31,1}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel nodes.png
Cells t{5,3} Uniform polyhedron-53-t01.png
{3,4} Uniform polyhedron-43-t2.png
Faces triangle {3}
decagon {10}
Vertex figure Truncated order-4 dodecahedral honeycomb verf.png
Square pyramid
Coxeter group BH3, [5,3,4]
DH3, [5,31,1]
Properties Vertex-transitive

The truncated order-4 dodecahedral honeycomb, CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png, has octahedron and truncated dodecahedron cells, with a cube vertex figure.

H3 435-0011 center ultrawide.png

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

H2 tiling 245-3.png

Related honeycombs[edit]

Four truncated regular compact honeycombs in H3
Image H3 435-0011 center ultrawide.png H3 534-0011 center ultrawide.png H3 353-0011 center ultrawide.png H3 535-0011 center ultrawide.png
Symbols t{5,3,4}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t{4,3,5}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
t{3,5,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
t{5,3,5}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Vertex
figure
Truncated order-4 dodecahedral honeycomb verf.png Truncated order-5 cubic honeycomb verf.png Truncated icosahedral honeycomb verf.png Truncated order-5 dodecahedral honeycomb verf.png


Bitruncated order-4 dodecahedral honeycomb[edit]

Bitruncated order-4 dodecahedral honeycomb
Bitruncated order-5 cubic honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol 2t{5,3,4}
2t{5,31,1}
Coxeter diagram CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h0.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png
Cells t{3,5} Uniform polyhedron-53-t12.png
t{3,4} Uniform polyhedron-43-t12.png
Faces triangle {3}
square {4}
hexagon {6}
Vertex figure Bitruncated order-4 dodecahedral honeycomb verf.png
tetrahedron
Coxeter group BH3, [5,3,4]
DH3, [5,31,1]
Properties Vertex-transitive

The bitruncated order-4 dodecahedral honeycomb, or bitruncated order-5 cubic honeycomb, CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png, has truncated octahedron and truncated icosahedron cells, with a tetrahedron vertex figure.

H3 534-0110 center ultrawide.png

Related honeycombs[edit]

Three bitruncated regular compact honeycombs in H3
Image H3 534-0110 center ultrawide.png H3 353-0110 center ultrawide.png H3 535-0110 center ultrawide.png
Symbols 2t{4,3,5}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
2t{3,5,3}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
2t{5,3,5}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
Vertex
figure
Bitruncated order-5 cubic honeycomb verf.png Bitruncated icosahedral honeycomb verf.png Bitruncated order-5 dodecahedral honeycomb verf.png


Cantellated order-4 dodecahedral honeycomb[edit]

Cantellated order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol rr{5,3,4}
rr{5,31,1}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h0.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes 11.png
Cells rr{3,5} Uniform polyhedron-53-t02.png
r{3,4} Uniform polyhedron-43-t2.png
{}x{4} cube Tetragonal prism.png
Faces triangle {3}
square {4}
pentagon {5}
Vertex figure Cantellated order-4 dodecahedral honeycomb verf.png
Triangular prism
Coxeter group BH3, [5,3,4]
DH3, [5,31,1]
Properties Vertex-transitive

The cantellated order-4 dodecahedral honeycomb,CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png, has rhombicosidodecahedron and cuboctahedron, and cube cells, with a triangular prism vertex figure.

H3 534-1010 center ultrawide.png

Related honeycombs[edit]

Four cantellated regular compact honeycombs in H3
Image H3 534-1010 center ultrawide.png H3 534-0101 center ultrawide.png H3 353-1010 center ultrawide.png H3 535-1010 center ultrawide.png
Symbols rr{5,3,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
rr{4,3,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
rr{3,5,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
rr{5,3,5}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
Vertex
figure
Cantellated order-4 dodecahedral honeycomb verf.png Cantellated order-5 cubic honeycomb verf.png Cantellated icosahedral honeycomb verf.png Cantellated order-5 dodecahedral honeycomb verf.png


Cantitruncated order-4 dodecahedral honeycomb[edit]

Cantitruncated order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol tr{5,3,4}
tr{5,31,1}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h0.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png
Cells tr{3,5} Uniform polyhedron-53-t012.png
t{3,4} Uniform polyhedron-43-t12.png
{}x{4} cube Tetragonal prism.png
Faces square {4}
hexagon {6}
decagon {10}
Vertex figure Cantitruncated order-4 dodecahedral honeycomb verf.png
mirrored sphenoid
Coxeter group BH3, [5,3,4]
DH3, [5,31,1]
Properties Vertex-transitive

The cantitruncated order-4 dodecahedral honeycomb, is a uniform honeycomb constructed with a CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png coxeter diagram, and mirrored sphenoid vertex figure.

H3 534-1110 center ultrawide.png

Related honeycombs[edit]

Four cantitruncated regular compact honeycombs in H3
Image H3 534-1110 center ultrawide.png H3 534-0111 center ultrawide.png H3 353-1110 center ultrawide.png H3 535-1110 center ultrawide.png
Symbols tr{5,3,4}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
tr{4,3,5}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
tr{3,5,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
tr{5,3,5}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
Vertex
figure
Cantitruncated order-4 dodecahedral honeycomb verf.png Cantitruncated order-5 cubic honeycomb verf.png Cantitruncated icosahedral honeycomb verf.png Cantitruncated order-5 dodecahedral honeycomb verf.png


Runcitruncated order-4 dodecahedral honeycomb[edit]

Runcitruncated order-4 dodecahedral honeycomb
Type Uniform honeycombs in hyperbolic space
Schläfli symbol t0,1,3{5,3,4}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Cells t{5,3} Uniform polyhedron-53-t01.png
rr{3,4} Uniform polyhedron-43-t02.png
{}x{10} Decagonal prism.png
{}x{4} Tetragonal prism.png
Faces triangle {3}
square {4}
decagon {10}
Vertex figure Runcitruncated order-4 dodecahedral honeycomb verf.png
quad pyramid
Coxeter group BH3, [5,3,4]
Properties Vertex-transitive

The runcititruncated order-4 dodecahedral honeycomb, is a uniform honeycomb constructed with a CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png coxeter diagram, and a quadrilateral pyramid vertex figure.

H3 534-1101 center ultrawide.png

Related honeycombs[edit]

Four runcitruncated regular compact honeycombs in H3
Image H3 534-1101 center ultrawide.png H3 534-1011 center ultrawide.png H3 353-1101 center ultrawide.png H3 535-1101 center ultrawide.png
Symbols t0,1,3{5,3,4}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
t0,1,3{4,3,5}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
t0,1,3{3,5,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,3{5,3,5}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.png
Vertex
figure
Runcitruncated order-4 dodecahedral honeycomb verf.png Runcitruncated order-5 cubic honeycomb verf.png Runcitruncated icosahedral honeycomb verf.png Runcitruncated order-5 dodecahedral honeycomb verf.png

See also[edit]

References[edit]

  • 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, Manuscript, (2011) Chapter 13: Hyperbolic Coxeter groups