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Order-infinite-3 triangular honeycomb

(Redirected from Order-infinite-3 pentagonal honeycomb)
Order-infinite-3 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,∞,3}
Coxeter diagrams
Cells {3,∞}
Faces {3}
Edge figure {3}
Vertex figure {∞,3}
Dual Self-dual
Coxeter group [3,∞,3]
Properties Regular

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

Geometry

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

 
Poincaré disk model 		 
Ideal surface
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It is a part of a sequence of regular honeycombs with Infinite-order triangular tiling cells: {3,∞,p}.

It is a part of a sequence of regular honeycombs with order-3 apeirogonal tiling vertex figures: {p,∞,3}.

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

Order-infinite-4 triangular honeycomb

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Order-infinite-4 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,∞,4}
{3,∞1,1}
Coxeter diagrams        
        =      
Cells {3,∞}  
Faces {3}
Edge figure {4}
Vertex figure {∞,4}  
r{∞,∞}  
Dual {4,∞,3}
Coxeter group [3,∞,4]
[3,∞1,1]
Properties Regular

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

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

 
Poincaré disk model 		 
Ideal surface

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

Order-infinite-5 triangular honeycomb

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Order-infinite-5 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,∞,5}
Coxeter diagrams        
Cells {3,∞}  
Faces {3}
Edge figure {5}
Vertex figure {∞,5}  
Dual {5,∞,3}
Coxeter group [3,∞,5]
Properties Regular

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

 
Poincaré disk model 		 
Ideal surface

Order-infinite-6 triangular honeycomb

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Order-infinite-6 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,∞,6}
{3,(∞,3,∞)}
Coxeter diagrams        
        =      
Cells {3,∞}  
Faces {3}
Edge figure {6}
Vertex figure {∞,6}  
{(∞,3,∞)}  
Dual {6,∞,3}
Coxeter group [3,∞,6]
Properties Regular

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

 
Poincaré disk model 		 
Ideal surface

Order-infinite-7 triangular honeycomb

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Order-infinite-7 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,∞,7}
Coxeter diagrams        
Cells {3,∞}  
Faces {3}
Edge figure {7}
Vertex figure {∞,7}  
Dual {7,∞,3}
Coxeter group [3,∞,7]
Properties Regular

In the geometry of hyperbolic 3-space, the order-infinite-7 triangular honeycomb (or 3,∞,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,∞,7}. It has infinitely many infinite-order triangular tiling, {3,∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an order-7 apeirogonal tiling, {∞,7}, vertex figure.

 
Ideal surface

Order-infinite-infinite triangular honeycomb

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Order-infinite-infinite triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,∞,∞}
{3,(∞,∞,∞)}
Coxeter diagrams        
        =       
Cells {3,∞}  
Faces {3}
Edge figure {∞}
Vertex figure {∞,∞}  
{(∞,∞,∞)}  
Dual {∞,∞,3}
Coxeter group [∞,∞,3]
[3,((∞,∞,∞))]
Properties Regular

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

 
Poincaré disk model 		 
Ideal surface

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

Order-infinite-3 square honeycomb

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Order-infinite-3 square honeycomb
Type Regular honeycomb
Schläfli symbol {4,∞,3}
Coxeter diagram        
Cells {4,∞}  
Faces {4}
Vertex figure {∞,3}
Dual {3,∞,4}
Coxeter group [4,∞,3]
Properties Regular

In the geometry of hyperbolic 3-space, the order-infinite-3 square honeycomb (or 4,∞,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.

The Schläfli symbol of the order-infinite-3 square honeycomb is {4,∞,3}, with three infinite-order square tilings meeting at each edge. The vertex figure of this honeycomb is an order-3 apeirogonal tiling, {∞,3}.

 
Poincaré disk model 		 
Ideal surface

Order-infinite-3 pentagonal honeycomb

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Order-infinite-3 pentagonal honeycomb
Type Regular honeycomb
Schläfli symbol {5,∞,3}
Coxeter diagram        
Cells {5,∞}  
Faces {5}
Vertex figure {∞,3}
Dual {3,∞,5}
Coxeter group [5,∞,3]
Properties Regular

In the geometry of hyperbolic 3-space, the order-infinite-3 pentagonal honeycomb (or 5,∞,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an infinite-order 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,∞,3}, with three infinite-order pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a heptagonal tiling, {∞,3}.

 
Poincaré disk model 		 
Ideal surface

Order-infinite-3 hexagonal honeycomb

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Order-infinite-3 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbol {6,∞,3}
Coxeter diagram        
Cells {6,∞}  
Faces {6}
Vertex figure {∞,3}
Dual {3,∞,6}
Coxeter group [6,∞,3]
Properties Regular

In the geometry of hyperbolic 3-space, the order-infinite-3 hexagonal honeycomb (or 6,∞,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-3 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 order-infinite-3 hexagonal honeycomb is {6,∞,3}, with three infinite-order hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is an order-3 apeirogonal tiling, {∞,3}.

 
Poincaré disk model 		 
Ideal surface

Order-infinite-3 heptagonal honeycomb

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Order-infinite-3 heptagonal honeycomb
Type Regular honeycomb
Schläfli symbol {7,∞,3}
Coxeter diagram        
Cells {7,∞}  
Faces {7}
Vertex figure {∞,3}
Dual {3,∞,7}
Coxeter group [7,∞,3]
Properties Regular

In the geometry of hyperbolic 3-space, the order-infinite-3 heptagonal honeycomb (or 7,∞,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an infinite-order heptagonal 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-infinite-3 heptagonal honeycomb is {7,∞,3}, with three infinite-order heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an order-3 apeirogonal tiling, {∞,3}.

 
Ideal surface

Order-infinite-3 apeirogonal honeycomb

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Order-infinite-3 apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbol {∞,∞,3}
Coxeter diagram        
Cells {∞,∞}  
Faces Apeirogon {∞}
Vertex figure {∞,3}
Dual {3,∞,∞}
Coxeter group [∞,∞,3]
Properties Regular

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

 
Poincaré disk model 		 
Ideal surface

Order-infinite-4 square honeycomb

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Order-infinite-4 square honeycomb
Type Regular honeycomb
Schläfli symbol {4,∞,4}
Coxeter diagrams        
        =      
Cells {4,∞}  
Faces {4}
Edge figure {4}
Vertex figure {∞,4}
{∞,∞}
Dual self-dual
Coxeter group [4,∞,4]
Properties Regular

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

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

 
Poincaré disk model 		 
Ideal surface

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

Order-infinite-5 pentagonal honeycomb

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Order-infinite-5 pentagonal honeycomb
Type Regular honeycomb
Schläfli symbol {5,∞,5}
Coxeter diagrams        
Cells {5,∞}  
Faces {5}
Edge figure {5}
Vertex figure {∞,5}
Dual self-dual
Coxeter group [5,∞,5]
Properties Regular

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

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

 
Poincaré disk model 		 
Ideal surface

Order-infinite-6 hexagonal honeycomb

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Order-infinite-6 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbols {6,∞,6}
{6,(∞,3,∞)}
Coxeter diagrams        
        =      
Cells {6,∞}  
Faces {6}
Edge figure {6}
Vertex figure {∞,6}  
{(5,3,5)}  
Dual self-dual
Coxeter group [6,∞,6]
[6,((∞,3,∞))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-infinite-6 hexagonal honeycomb (or 6,∞,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,∞,6}. It has six infinite-order hexagonal tilings, {6,∞}, 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 apeirogonal tiling vertex figure.

 
Poincaré disk model 		 
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(∞,3,∞)}, Coxeter diagram,      , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,∞,6,1+] = [6,((∞,3,∞))].

Order-infinite-7 heptagonal honeycomb

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Order-infinite-7 heptagonal honeycomb
Type Regular honeycomb
Schläfli symbols {7,∞,7}
Coxeter diagrams        
Cells {7,∞}  
Faces {7}
Edge figure {7}
Vertex figure {∞,7}  
Dual self-dual
Coxeter group [7,∞,7]
Properties Regular

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

 
Ideal surface

Order-infinite-infinite apeirogonal honeycomb

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Order-infinite-infinite apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbols {∞,∞,∞}
{∞,(∞,∞,∞)}
Coxeter diagrams        
        ↔       
Cells {∞,∞}  
Faces {∞}
Edge figure {∞}
Vertex figure   {∞,∞}
  {(∞,∞,∞)}
Dual self-dual
Coxeter group [∞,∞,∞]
[∞,((∞,∞,∞))]
Properties Regular

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

 
Poincaré disk model 		 
Ideal surface

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

See also

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References

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  • 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)
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