Order-3-7 heptagonal honeycomb

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

Order-3-7 heptagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{7,3,7}
Coxeter diagrams
Cells{7,3}
Faces{7}
Edge figure{7}
Vertex figure{3,7}
Dualself-dual
Coxeter group[7,3,7]
PropertiesRegular

Geometry

All vertices are ultra-ideal (existing beyond the ideal boundary) with seven heptagonal tilings existing around each edge and with an order-7 triangular tiling vertex figure.


Poincaré disk model

Ideal surface

It a part of a sequence of regular polychora and honeycombs {p,3,p}:

{p,3,p} regular honeycombs
Space S3 Euclidean E3 H3
Form Finite Affine Compact Paracompact Noncompact
Name {3,3,3} {4,3,4} {5,3,5} {6,3,6} {7,3,7} {8,3,8} ...{,3,}
Image
Cells
{3,3}

{4,3}

{5,3}

{6,3}

{7,3}

{8,3}

{,3}
Vertex
figure

{3,3}

{3,4}

{3,5}

{3,6}

{3,7}

{3,8}

{3,}

Order-3-8 octagonal honeycomb

Order-3-8 octagonal honeycomb
TypeRegular honeycomb
Schläfli symbols{8,3,8}
{8,(3,4,3)}
Coxeter diagrams
=
Cells{8,3}
Faces{8}
Edge figure{8}
Vertex figure{3,8}
{(3,8,3)}
Dualself-dual
Coxeter group[8,3,8]
[8,((3,4,3))]
PropertiesRegular

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


Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {8,(3,4,3)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [8,3,8,1+] = [8,((3,4,3))].

Order-3-infinite apeirogonal honeycomb

Order-3-infinite apeirogonal honeycomb
TypeRegular honeycomb
Schläfli symbols{∞,3,∞}
{∞,(3,∞,3)}
Coxeter diagrams
Cells{∞,3}
Faces{∞}
Edge figure{∞}
Vertex figure {3,∞}
{(3,∞,3)}
Dualself-dual
Coxeter group[∞,3,∞]
[∞,((3,∞,3))]
PropertiesRegular

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


Poincaré disk model

Ideal surface

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