Order-7 tetrahedral honeycomb
TypeHyperbolic regular honeycomb
Schläfli symbols{3,3,7}
Coxeter diagrams
Cells{3,3}
Faces{3}
Edge figure{7}
Vertex figure{3,7}
Dual{7,3,3}
Coxeter group[7,3,3]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-7 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,7}. It has seven tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an order-7 triangular tiling vertex arrangement.

Images


Poincaré disk model (cell-centered)
Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model

It is a part of a sequence of regular polychora and honeycombs with tetrahedral cells, {3,3,p}.

{3,3,p} polytopes
Space S3 H3
Form Finite Paracompact Noncompact
Name {3,3,3}
 {3,3,4}

 {3,3,5}
 {3,3,6}

 {3,3,7}
 {3,3,8}

 ... {3,3,∞}

Image
Vertex
figure
{3,3}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{3,∞}

It is a part of a sequence of hyperbolic honeycombs with order-7 triangular tiling vertex figures, {p,3,7}.

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

It is a part of a sequence of hyperbolic honeycombs, {3,p,7}.

Order-8 tetrahedral honeycomb

Order-8 tetrahedral honeycomb
TypeHyperbolic regular honeycomb
Schläfli symbols{3,3,8}
{3,(3,4,3)}
Coxeter diagrams
=
Cells{3,3}
Faces{3}
Edge figure{8}
Vertex figure{3,8}
{(3,4,3)}
Dual{8,3,3}
Coxeter group[3,3,8]
[3,((3,4,3))]
PropertiesRegular

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


Poincaré disk model (cell-centered)
Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model

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

Infinite-order tetrahedral honeycomb

Infinite-order tetrahedral honeycomb
TypeHyperbolic regular honeycomb
Schläfli symbols{3,3,∞}
{3,(3,∞,3)}
Coxeter diagrams
=
Cells{3,3}
Faces{3}
Edge figure{∞}
Vertex figure{3,∞}
{(3,∞,3)}
Dual{∞,3,3}
Coxeter group[∞,3,3]
[3,((3,∞,3))]
PropertiesRegular

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


Poincaré disk model (cell-centered)
Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model

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

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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