Order-7 dodecahedral honeycomb

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

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

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.

File:Hyperbolic honeycomb 5-3-7 poincare cc.png Poincaré disk model Cell-centered	File:Hyperbolic honeycomb 5-3-7 poincare.png Poincaré disk model	File: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
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	File:Schlegel wireframe 120-cell.png	File:H3 534 CC center.png	File:H3 535 CC center.png	File:H3 536 CC center.png	File:Hyperbolic honeycomb 5-3-7 poincare.png	File:Hyperbolic honeycomb 5-3-8 poincare.png	File:Hyperbolic honeycomb 5-3-i poincare.png
Vertex figure	File:Tetrahedron.png {3,3}	File:Octahedron.png {3,4}	File:Icosahedron.png {3,5}	File:Uniform tiling 63-t2.svg {3,6}	File:Order-7 triangular tiling.svg {3,7}	File:H2-8-3-primal.svg {3,8}	File: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}
File:Hyperbolic honeycomb 3-3-7 poincare cc.png	File:Hyperbolic honeycomb 4-3-7 poincare cc.png	File:Hyperbolic honeycomb 5-3-7 poincare cc.png	File:Hyperbolic honeycomb 6-3-7 poincare.png	File:Hyperbolic honeycomb 7-3-7 poincare.png	File:Hyperbolic honeycomb 8-3-7 poincare.png	File:Hyperbolic honeycomb i-3-7 poincare.png

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

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.

File:Hyperbolic honeycomb 5-3-8 poincare cc.png Poincaré disk model Cell-centered	File: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, Error creating thumbnail: File:CDel split1.pngFile:CDel branch.pngFile:CDel label4.png, with alternating types or colors of dodecahedral cells.

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

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.

File:Hyperbolic honeycomb 5-3-i poincare cc.png Poincaré disk model Cell-centered	File:Hyperbolic honeycomb 5-3-i poincare.png Poincaré disk model	File: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, Error creating thumbnail: File:CDel split1.pngFile:CDel branch.pngFile:CDel labelinfin.png, with alternating types or colors of dodecahedral cells.

Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (Chapter 10, Regular Honeycombs in Hyperbolic Space Archived 2016-06-10 at the Wayback Machine) Table III
Jeffrey R. Weeks The Shape of Space, 2nd edition Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (Chapters 16–17: Geometries on Three-manifolds I, II)
George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982) [1]
Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)[2]
Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)