Infinite-order dodecahedral honeycomb

Infinite-order dodecahedral honeycomb
Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model
Type	Hyperbolic regular honeycomb
Schläfli symbols	{5,3,∞} {5,(3,∞,3)}
Coxeter diagrams	=
Cells	{5,3}
Faces	{5}
Edge figure	{∞}
Vertex figure	{3,∞}, {(3,∞,3)}
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.

Symmetry constructions

It has a second construction as a uniform honeycomb, Schläfli symbol {5,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of dodecahedral cells.

Related polytopes and honeycombs

It a part of a sequence of regular polychora and honeycombs with dodecahedral cells.

{5,3,p} polytopes
Space	S^3	H^3
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
Vertex figure	{3,3}	{3,4}	{3,5}	{3,6}	{3,7}	{3,8}	{3,∞}

See also

• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes
• Infinite-order hexagonal tiling honeycomb

References

• The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space)
• George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982)
• Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)