Order-7 dodecahedral honeycomb
TypeRegular honeycomb
Schläfli symbols{5,3,7}
Coxeter diagrams
Cells{5,3}
Faces{5}
Edge figure{7}
Vertex figure{3,7}
Dual{7,3,5}
Coxeter group[5,3,7]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-7 dodecahedral honeycomb 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.


Poincaré disk model
Cell-centered
Poincaré disk model
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
Vertex
figure
{3,3}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{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}

Order-8 dodecahedral honeycomb

Order-8 dodecahedral honeycomb
TypeRegular honeycomb
Schläfli symbols{5,3,8}
{5,(3,4,3)}
Coxeter diagrams
=
Cells{5,3}
Faces{5}
Edge figure{8}
Vertex figure{3,8}, {(3,4,3)}
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.


Poincaré disk model
Cell-centered
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {5,(3,4,3)}, Coxeter diagram, , with alternating types or colors of dodecahedral cells.

Infinite-order dodecahedral honeycomb

Infinite-order dodecahedral honeycomb
TypeRegular 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))]
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.


Poincaré disk model
Cell-centered
Poincaré disk model
Ideal surface

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

See also

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 (Chapters 16–17: Geometries on Three-manifolds I,II)
    • 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)
    • Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
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