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.
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}.
Related polytopes and honeycombs 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
File:Stereographic polytope 5cell.png File:Stereographic polytope 8cell.png File:Stereographic polytope 120cell faces.png File:H3 633 FC boundary.png File:Hyperbolic honeycomb 7-3-3 poincare.png File:Hyperbolic honeycomb 8-3-3 poincare.png File:Hyperbolic honeycomb i-3-3 poincare.png
Coxeter diagrams subgroups 1
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 6.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 7.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel infin.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
4
File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png
File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 6.png File:CDel node.png File:CDel 3.png File:CDel node.png
File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel infin.png File:CDel node 1.png File:CDel infin.png File:CDel node.png File:CDel 3.png File:CDel node.png
6
File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:CDel 4.png File:CDel node.png
File:CDel node.png File:CDel 6.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 6.png File:CDel node.png
File:CDel node.png File:CDel 8.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel node.png File:CDel infin.png File:CDel node 1.png File:CDel infin.png File:CDel node 1.png File:CDel infin.png File:CDel node.png
12
File:CDel nodes 11.png File:CDel 2.png File:CDel node 1.png File:CDel 4.png File:CDel node.png
File:CDel branch 11.png File:CDel split2.png File:CDel node 1.png File:CDel 6.png File:CDel node.png
File:CDel label4.png File:CDel branch 11.png File:CDel split2-44.png File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel labelinfin.png File:CDel branch 11.png File:CDel split2-ii.png File:CDel node 1.png File:CDel infin.png File:CDel node.png
24
File:CDel nodes 11.png File:CDel 2.png File:CDel nodes 11.png
File:CDel branch 11.png File:CDel splitcross.png File:CDel branch 11.png
File:Cdel tet4 1111.png File:Cdel tetinfin 1111.png
CellsFile:CDel node 1.png File:CDel p.png File:CDel node.png File:CDel 3.png File:CDel node.png
File:Tetrahedron.png {3,3} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:Hexahedron.png {4,3} File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:CDel nodes 11.png File:CDel 2.png File:CDel node 1.png File:Dodecahedron.png {5,3} File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:Uniform tiling 63-t0.svg {6,3} File:CDel node 1.png File:CDel 6.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel 6.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel branch 11.png File:CDel split2.png File:CDel node 1.png File:Heptagonal tiling.svg {7,3} File:CDel node 1.png File:CDel 7.png File:CDel node.png File:CDel 3.png File:CDel node.png File:H2-8-3-dual.svg {8,3} File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel 8.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel label4.png File:CDel branch 11.png File:CDel split2-44.png File:CDel node 1.png File:H2-I-3-dual.svg {∞,3} File:CDel node 1.png File:CDel infin.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel infin.png File:CDel node 1.png File:CDel infin.png File:CDel node 1.png File:CDel labelinfin.png File:CDel branch 11.png File:CDel split2-ii.png File:CDel node 1.png
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,∞}
File:Hyperbolic honeycomb 7-3-3 poincare vc.png File:Hyperbolic honeycomb 7-3-4 poincare vc.png File:Hyperbolic honeycomb 7-3-5 poincare vc.png File:Hyperbolic honeycomb 7-3-6 poincare.png File:Hyperbolic honeycomb 7-3-7 poincare.png File:Hyperbolic honeycomb 7-3-8 poincare.png File:Hyperbolic honeycomb 7-3-i poincare.png
It is a part of a series of regular honeycombs, with {7,p ,3}.
Octagonal tiling honeycomb
Octagonal tiling honeycomb
Type
Regular honeycomb
Schläfli symbol {8,3,3}[3,3] }
Coxeter diagram File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel 8.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel branch 11.png File:CDel split2-44.png File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel label4.png File:CDel branch 11.png File:CDel splitcross.png File:CDel branch 11.png File:CDel label4.png
Cells
{8,3} File:H2-8-3-dual.svg
Faces
Octagon {8}
Vertex figure tetrahedron {3,3}
Dual
{3,3,8}
Coxeter group [8,3,3]
Properties
Regular
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}.
Apeirogonal tiling honeycomb
Apeirogonal tiling honeycomb
Type
Regular honeycomb
Schläfli symbol {∞,3,3}[3,3] }
Coxeter diagram File:CDel node 1.png File:CDel infin.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel infin.png File:CDel node 1.png File:CDel infin.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel infin.png File:CDel node 1.png File:CDel infin.png File:CDel node 1.png File:CDel infin.png File:CDel node.png File:CDel labelinfin.png File:CDel branch 11.png File:CDel split2-ii.png File:CDel node 1.png File:CDel infin.png File:CDel node.png File:CDel labelinfin.png File:CDel branch 11.png File:CDel splitcross.png File:CDel branch 11.png File:CDel labelinfin.png
Cells
{∞,3} File:H2-I-3-dual.svg
Faces
Apeirogon {∞}
Vertex figure tetrahedron {3,3}
Dual
{3,3,∞}
Coxeter group [∞,3,3]
Properties
Regular
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.
See also References
Coxeter , Regular Polytopes 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 IIIJeffrey 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) [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)External links
