Order-8-3 triangular honeycomb

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Order-8-3 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,8,3}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
Cells {3,8} H2-8-3-primal.svg
Faces {3}
Edge figure {3}
Vertex figure {8,3} H2-8-3-dual.svg
Dual Self-dual
Coxeter group [3,8,3]
Properties Regular

In the geometry of hyperbolic 3-space, the order-8-3 triangular honeycomb (or 3,8,3 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,3}.

Geometry[]

It has three order-8 triangular tiling {3,8} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an octagonal tiling vertex figure.

Hyperbolic honeycomb 3-8-3 poincare.png
Poincaré disk model

Related polytopes and honeycombs[]

It is a part of a sequence of regular honeycombs with order-8 triangular tiling cells: {3,8,p}.

It is a part of a sequence of regular honeycombs with octagonal tiling vertex figures: {p,8,3}.

It is a part of a sequence of self-dual regular honeycombs: {p,8,p}.

Order-8-4 triangular honeycomb[]

Order-8-4 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,8,4}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel split1-88.pngCDel nodes.png
Cells {3,8} H2-8-3-primal.svg
Faces {3}
Edge figure {4}
Vertex figure {8,4} H2 tiling 248-1.png
r{8,8} H2 tiling 288-2.png
Dual {4,8,3}
Coxeter group [3,8,4]
Properties Regular

In the geometry of hyperbolic 3-space, the order-8-4 triangular honeycomb (or 3,8,4 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,4}.

It has four order-8 triangular tilings, {3,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 triangular tilings existing around each vertex in an order-4 hexagonal tiling vertex arrangement.

Hyperbolic honeycomb 3-8-4 poincare.png
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {3,81,1}, Coxeter diagram, CDel node 1.pngCDel 3.pngCDel node.pngCDel split1-88.pngCDel nodes.png, with alternating types or colors of order-8 triangular tiling cells. In Coxeter notation the half symmetry is [3,8,4,1+] = [3,81,1].

Order-8-5 triangular honeycomb[]

Order-8-5 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,8,5}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.pngCDel 5.pngCDel node.png
Cells {3,8} H2-8-3-primal.svg
Faces {3}
Edge figure {5}
Vertex figure {8,5} H2 tiling 258-1.png
Dual {5,8,3}
Coxeter group [3,8,5]
Properties Regular

In the geometry of hyperbolic 3-space, the order-8-3 triangular honeycomb (or 3,8,5 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,5}. It has five order-8 triangular tiling, {3,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 triangular tilings existing around each vertex in an order-5 octagonal tiling vertex figure.

Hyperbolic honeycomb 3-8-5 poincare.png
Poincaré disk model

Order-8-6 triangular honeycomb[]

Order-8-6 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,8,6}
{3,(8,3,8)}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel split1-88.pngCDel branch.png
Cells {3,8} H2-8-3-primal.svg
Faces {3}
Edge figure {6}
Vertex figure {8,6} H2 tiling 268-4.png
{(8,3,8)} H2 tiling 388-2.png
Dual {6,8,3}
Coxeter group [3,8,6]
Properties Regular

In the geometry of hyperbolic 3-space, the order-8-6 triangular honeycomb (or 3,8,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,6}. It has infinitely many order-8 triangular tiling, {3,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 triangular tilings existing around each vertex in an order-6 octagonal tiling, {8,6}, vertex figure.

Hyperbolic honeycomb 3-8-6 poincare.png
Poincaré disk model

Order-8-infinite triangular honeycomb[]

Order-8-infinite triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,8,∞}
{3,(8,∞,8)}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.pngCDel infin.pngCDel node h0.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel split1-88.pngCDel branch.pngCDel labelinfin.png
Cells {3,8} H2-8-3-primal.svg
Faces {3}
Edge figure {∞}
Vertex figure H2 tiling 28i-4.png
{(8,∞,8)} H2 tiling 88i-4.png
Dual {∞,8,3}
Coxeter group [∞,8,3]
[3,((8,∞,8))]
Properties Regular

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

Hyperbolic honeycomb 3-8-i poincare.png
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {3,(8,∞,8)}, Coxeter diagram, CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.pngCDel infin.pngCDel node h0.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel split1-88.pngCDel branch.pngCDel labelinfin.png, with alternating types or colors of order-8 triangular tiling cells. In Coxeter notation the half symmetry is [3,8,∞,1+] = [3,((8,∞,8))].

Order-8-3 square honeycomb[]

Order-8-3 square honeycomb
Type Regular honeycomb
Schläfli symbol {4,8,3}
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
Cells {4,8} H2 tiling 248-4.png
Faces {4}
Vertex figure {8,3}
Dual {3,8,4}
Coxeter group [4,8,3]
Properties Regular

In the geometry of hyperbolic 3-space, the order-8-3 square honeycomb (or 4,8,3 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 order-8-3 square honeycomb is {4,8,3}, with three order-4 octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, {8,3}.

Hyperbolic honeycomb 4-8-3 poincare.png
Poincaré disk model

Order-8-3 pentagonal honeycomb[]

Order-8-3 pentagonal honeycomb
Type Regular honeycomb
Schläfli symbol {5,8,3}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
Cells {5,8} H2 tiling 258-4.png
Faces {5}
Vertex figure {8,3}
Dual {3,8,5}
Coxeter group [5,8,3]
Properties Regular

In the geometry of hyperbolic 3-space, the order-8-3 pentagonal honeycomb (or 5,8,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-8 pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

The Schläfli symbol of the order-6-3 pentagonal honeycomb is {5,8,3}, with three order-8 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, {8,3}.

Hyperbolic honeycomb 5-8-3 poincare.png
Poincaré disk model

Order-8-3 hexagonal honeycomb[]

Order-8-3 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbol {6,8,3}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
Cells {6,8} H2 tiling 268-4.png
Faces {6}
Vertex figure {8,3}
Dual {3,8,6}
Coxeter group [6,8,3]
Properties Regular

In the geometry of hyperbolic 3-space, the order-8-3 hexagonal honeycomb (or 6,8,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

The Schläfli symbol of the order-8-3 hexagonal honeycomb is {6,8,3}, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, {8,3}.

Hyperbolic honeycomb 6-8-3 poincare.png
Poincaré disk model

Order-8-3 apeirogonal honeycomb[]

Order-8-3 apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbol {∞,8,3}
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
Cells H2 tiling 28i-1.png
Faces Apeirogon {∞}
Vertex figure {8,3}
Dual {3,8,∞}
Coxeter group [∞,8,3]
Properties Regular

In the geometry of hyperbolic 3-space, the order-8-3 apeirogonal honeycomb (or ∞,8,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an 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 {∞,8,3}, with three order-8 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, {8,3}.

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

Hyperbolic honeycomb i-8-3 poincare.png
Poincaré disk model

Order-8-4 square honeycomb[]

Order-8-4 square honeycomb
Type Regular honeycomb
Schläfli symbol {4,8,4}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node h0.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel split1-88.pngCDel nodes.png
Cells {4,8} H2 tiling 248-4.png
Faces {4}
Edge figure {4}
Vertex figure {8,4}
Dual self-dual
Coxeter group [4,8,4]
Properties Regular

In the geometry of hyperbolic 3-space, the order-8-4 square honeycomb (or 4,8,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,8,4}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-5 square tilings existing around each edge and with an order-4 octagonal tiling vertex figure.

Hyperbolic honeycomb 4-8-4 poincare.png
Poincaré disk model

Order-8-5 pentagonal honeycomb[]

Order-8-5 pentagonal honeycomb
Type Regular honeycomb
Schläfli symbol {5,8,5}
Coxeter diagrams CDel node 1.pngCDel 5.pngCDel node.pngCDel 8.pngCDel node.pngCDel 5.pngCDel node.png
Cells {5,8} H2 tiling 258-1.png
Faces {5}
Edge figure {5}
Vertex figure
Dual self-dual
Coxeter group [5,8,5]
Properties Regular

In the geometry of hyperbolic 3-space, the order-8-5 pentagonal honeycomb (or 5,8,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,8,5}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-8 pentagonal tilings existing around each edge and with an order-5 pentagonal tiling vertex figure.

Hyperbolic honeycomb 5-8-5 poincare.png
Poincaré disk model

Order-8-6 hexagonal honeycomb[]

Order-8-6 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbols {6,8,6}
{6,(8,3,8)}
Coxeter diagrams CDel node 1.pngCDel 6.pngCDel node.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel 6.pngCDel node.pngCDel split1-88.pngCDel branch.png
Cells H2 tiling 258-4.png
Faces {6}
Edge figure {6}
Vertex figure {8,6} H2 tiling 258-4.png
{(5,3,5)} H2 tiling 358-1.png
Dual self-dual
Coxeter group [6,8,6]
[6,((8,3,8))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-8-6 hexagonal honeycomb (or 6,8,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,8,6}. It has six order-8 hexagonal tilings, {6,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an order-6 octagonal tiling vertex arrangement.

Hyperbolic honeycomb 6-8-6 poincare.png
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(8,3,8)}, Coxeter diagram, CDel node 1.pngCDel 6.pngCDel node.pngCDel split1-88.pngCDel branch.png, with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,8,6,1+] = [6,((8,3,8))].

Order-8-infinite apeirogonal honeycomb[]

Order-8-infinite apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbols {∞,8,∞}
{∞,(8,∞,8)}
Coxeter diagrams CDel node 1.pngCDel infin.pngCDel node.pngCDel 8.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel 8.pngCDel node.pngCDel infin.pngCDel node h0.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel split1-88.pngCDel branch.pngCDel labelinfin.png
Cells {∞,8} H2 tiling 28i-1.png
Faces {∞}
Edge figure {∞}
Vertex figure H2 tiling 28i-4.png
H2 tiling 88i-4.png {(8,∞,8)}
Dual self-dual
Coxeter group [∞,8,∞]
[∞,((8,∞,8))]
Properties Regular

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

Hyperbolic honeycomb i-8-i poincare.png
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(8,∞,8)}, Coxeter diagram, CDel node 1.pngCDel infin.pngCDel node.pngCDel split1-88.pngCDel branch.pngCDel labelinfin.png, with alternating types or colors of cells.

See also[]

  • Convex uniform honeycombs in hyperbolic space
  • List of regular polytopes

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) [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[]

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