Heptagonal tiling honeycomb

In the geometry of hyperbolic 3-space, the heptagonal tiling honeycomb or 7,3,3 honeycomb a regular space-filling tessellation (or honeycomb). 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.

Heptagonal tiling honeycomb
TypeRegular honeycomb
Schläfli symbol{7,3,3}
Coxeter diagram
Cells{7,3}
FacesHeptagon {7}
Vertex figuretetrahedron {3,3}
Dual{3,3,7}
Coxeter group[7,3,3]
PropertiesRegular

Geometry

The Schläfli symbol of the heptagonal tiling honeycomb is {7,3,3}, with three heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a tetrahedron, {3,3}.


Poincaré disk model
(vertex centered)
Rotating
Ideal surface

It is a part of a series of regular polytopes and honeycombs with {p,3,3} Schläfli symbol, and tetrahedral vertex figures:

{p,3,3} honeycombs
Space S3 H3
Form Finite Paracompact Noncompact
Name {3,3,3} {4,3,3} {5,3,3} {6,3,3} {7,3,3} {8,3,3} ... {∞,3,3}
Image
Coxeter diagrams
subgroups		 1
4
6
12
24
Cells
{p,3}
{3,3}
{4,3}
{5,3}
{6,3}
{7,3}
{8,3}
{∞,3}


It is a part of a series of regular honeycombs, {7,3,p}.

{7,3,3} {7,3,4} {7,3,5} {7,3,6} {7,3,7} {7,3,8} ...{7,3,∞}

It is a part of a series of regular honeycombs, with {7,p,3}.

{7,3,3} {7,4,3} {7,5,3}...

Octagonal tiling honeycomb
Octagonal tiling honeycomb
TypeRegular honeycomb
Schläfli symbol{8,3,3}
t{8,4,3}
2t{4,8,4}
t{4[3,3]}
Coxeter diagram



(all 4s)
Cells{8,3}
FacesOctagon {8}
Vertex figuretetrahedron {3,3}
Dual{3,3,8}
Coxeter group[8,3,3]
PropertiesRegular

In the geometry of hyperbolic 3-space, the octagonal tiling honeycomb or 8,3,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an octagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

The Schläfli symbol of the octagonal tiling honeycomb is {8,3,3}, with three octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an tetrahedron, {3,3}.


Poincaré disk model (vertex centered)
Direct subgroups of [8,3,3]

Apeirogonal tiling honeycomb
Apeirogonal tiling honeycomb
TypeRegular honeycomb
Schläfli symbol{∞,3,3}
t{∞,3,3}
2t{∞,∞,∞}
t{∞[3,3]}
Coxeter diagram



(all ∞)
Cells{∞,3}
FacesApeirogon {∞}
Vertex figuretetrahedron {3,3}
Dual{3,3,∞}
Coxeter group[∞,3,3]
PropertiesRegular

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

The Schläfli symbol of the apeirogonal tiling honeycomb is {∞,3,3}, with three apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an tetrahedron, {3,3}.

The "ideal surface" projection below is a plane-at-infinity, in the Poincare half-space model of H3. It shows an Apollonian gasket pattern of circles inside a largest circle.


Poincaré disk model (vertex centered)
Ideal surface

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 Archived 2016-06-10 at the Wayback Machine) 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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