Order-7 dodecahedral honeycomb

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Order-7 dodecahedral honeycomb
Type Regular honeycomb
Schläfli symbols {5,3,7}
Coxeter diagrams CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
Cells {5,3} Uniform polyhedron-53-t0.png
Faces {5}
Edge figure {7}
Vertex figure {3,7}
Order-7 triangular tiling.svg
Dual {7,3,5}
Coxeter group [5,3,7]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-7 dodecahedral honeycomb is a regular space-filling tessellation (or honeycomb).

Contents

Geometry

With Schläfli symbol {5,3,7}, it has seven dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an order-7 triangular tiling vertex arrangement.

Hyperbolic honeycomb 5-3-7 poincare cc.png
Poincaré disk model
Cell-centered
Hyperbolic honeycomb 5-3-7 poincare.png
Poincaré disk model
H3 537 UHS plane at infinity.png
Ideal surface

It a part of a sequence of regular polytopes and honeycombs with dodecahedral cells, {5,3,p}.

{5,3,p} polytopes
SpaceS3 H3
FormFiniteCompactParacompactNoncompact
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,}

It a part of a sequence of honeycombs {5,p,7}.

It a part of a sequence of honeycombs {p,3,7}.

{3,3,7} {4,3,7} {5,3,7} {6,3,7} {7,3,7} {8,3,7} {∞,3,7}
Hyperbolic honeycomb 3-3-7 poincare cc.png Hyperbolic honeycomb 4-3-7 poincare cc.png Hyperbolic honeycomb 5-3-7 poincare cc.png Hyperbolic honeycomb 6-3-7 poincare.png Hyperbolic honeycomb 7-3-7 poincare.png Hyperbolic honeycomb 8-3-7 poincare.png Hyperbolic honeycomb i-3-7 poincare.png

Order-8 dodecahedral honeycomb

Order-8 dodecahedral honeycomb
Type Regular honeycomb
Schläfli symbols {5,3,8}
{5,(3,4,3)}
Coxeter diagrams CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node h0.png = CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png
Cells {5,3} Uniform polyhedron-53-t0.png
Faces {5}
Edge figure {8}
Vertex figure {3,8}, {(3,4,3)}
H2-8-3-primal.svg H2 tiling 334-4.png
Dual {8,3,5}
Coxeter group [5,3,8]
[5,((3,4,3))]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-8 dodecahedral honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {5,3,8}, it has eight dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an order-8 triangular tiling vertex arrangement.

Hyperbolic honeycomb 5-3-8 poincare cc.png
Poincaré disk model
Cell-centered
Hyperbolic honeycomb 5-3-8 poincare.png
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {5,(3,4,3)}, Coxeter diagram, CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png, with alternating types or colors of dodecahedral cells.

Infinite-order dodecahedral honeycomb

Infinite-order dodecahedral honeycomb
Type Regular honeycomb
Schläfli symbols {5,3,∞}
{5,(3,∞,3)}
Coxeter diagrams CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node h0.png = CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png
Cells {5,3} Uniform polyhedron-53-t0.png
Faces {5}
Edge figure {∞}
Vertex figure {3,∞}, {(3,∞,3)}
H2 tiling 23i-4.png H2 tiling 33i-4.png
Dual {∞,3,5}
Coxeter group [5,3,∞]
[5,((3,∞,3))]
PropertiesRegular

In the geometry of hyperbolic 3-space, the infinite-order dodecahedral honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {5,3,∞}. It has infinitely many dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.

Hyperbolic honeycomb 5-3-i poincare cc.png
Poincaré disk model
Cell-centered
Hyperbolic honeycomb 5-3-i poincare.png
Poincaré disk model
H3 53i UHS plane at infinity.png
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {5,(3,∞,3)}, Coxeter diagram, CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png, with alternating types or colors of dodecahedral cells.

See also

Related Research Articles

In the geometry of hyperbolic 3-space, the order-7 tetrahedral honeycomb is a regular space-filling tessellation with Schläfli symbol {3,3,7}. It has seven tetrahedra {3,3} around each edge. All vertices are ultra-ideal with infinitely many tetrahedra existing around each vertex in an order-7 triangular tiling vertex arrangement.

<span class="mw-page-title-main">Order-3-7 hexagonal honeycomb</span>

In the geometry of hyperbolic 3-space, the order-3-7 hexagonal honeycomb or a regular space-filling tessellation with Schläfli symbol {6,3,7}.

In the geometry of hyperbolic 3-space, the heptagonal tiling honeycomb or 7,3,3 honeycomb a regular space-filling tessellation. Each infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

In the geometry of hyperbolic 3-space, the order-7 cubic honeycomb is a regular space-filling tessellation. With Schläfli symbol {4,3,7}, it has seven cubes {4,3} around each edge. All vertices are ultra-ideal with infinitely many cubes existing around each vertex in an order-7 triangular tiling vertex arrangement.

In the geometry of hyperbolic 3-space, the order-3-4 heptagonal honeycomb or 7,3,4 honeycomb a regular space-filling tessellation. Each infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

In the geometry of hyperbolic 3-space, the order-3-5 heptagonal honeycomb a regular space-filling tessellation. Each infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

In the geometry of hyperbolic 3-space, the order-3-6 heptagonal honeycomb a regular space-filling tessellation. Each infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

In the geometry of hyperbolic 3-space, the order-3-7 heptagonal honeycomb a regular space-filling tessellation with Schläfli symbol {7,3,7}.

In the geometry of hyperbolic 3-space, the order-5 octahedral honeycomb is a regular space-filling tessellation with Schläfli symbol {3,4,5}. It has five octahedra {3,4} around each edge. All vertices are ultra-ideal with infinitely many octahedra existing around each vertex in an order-5 square tiling vertex arrangement.

In the geometry of hyperbolic 3-space, the order-4 icosahedral honeycomb is a regular space-filling tessellation with Schläfli symbol {3,5,4}.

In the geometry of hyperbolic 3-space, the order-6-4 triangular honeycomb is a regular space-filling tessellation with Schläfli symbol {3,6,4}.

In the geometry of hyperbolic 3-space, the order-4-5 square honeycomb is a regular space-filling tessellation with Schläfli symbol {4,4,5}. It has five square tiling {4,4} around each edge. All vertices are ultra-ideal with infinitely many square tiling existing around each vertex in an order-5 square tiling vertex arrangement.

In the geometry of hyperbolic 3-space, the order-5-3 square honeycomb or 4,5,3 honeycomb a regular space-filling tessellation. Each infinite cell consists of a pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

In the geometry of hyperbolic 3-space, the order-4-5 pentagonal honeycomb a regular space-filling tessellation with Schläfli symbol {5,4,5}.

In the geometry of hyperbolic 3-space, the order-5-4 square honeycomb a regular space-filling tessellation with Schläfli symbol {4,5,4}.

In the geometry of hyperbolic 3-space, the order-7-3 triangular honeycomb is a regular space-filling tessellation with Schläfli symbol {3,7,3}.

In the geometry of hyperbolic 3-space, the order-6-3 square honeycomb or 4,6,3 honeycomb is a regular space-filling tessellation. Each infinite cell consists of a hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

In the geometry of hyperbolic 3-space, the order-6-4 square honeycomb a regular space-filling tessellation with Schläfli symbol {4,6,4}.

In the geometry of hyperbolic 3-space, the order-8-3 triangular honeycomb is a regular space-filling tessellation with Schläfli symbol {3,8,3}.

In the geometry of hyperbolic 3-space, the order-infinite-3 triangular honeycomb is a regular space-filling tessellation with Schläfli symbol {3,∞,3}.

References