Order-8-3 triangular honeycomb

Order-8-3 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,8,3}
Coxeter diagrams
Cells {3,8}
Faces {3}
Edge figure {3}
Vertex figure {8,3}
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

edit

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.

 
Poincaré disk model
edit

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

edit
Order-8-4 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,8,4}
Coxeter diagrams        
        =      
Cells {3,8}  
Faces {3}
Edge figure {4}
Vertex figure {8,4}  
r{8,8}  
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.

 
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {3,81,1}, Coxeter diagram,      , 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

edit
Order-8-5 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,8,5}
Coxeter diagrams        
Cells {3,8}  
Faces {3}
Edge figure {5}
Vertex figure {8,5}  
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.

 
Poincaré disk model

Order-8-6 triangular honeycomb

edit
Order-8-6 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,8,6}
{3,(8,3,8)}
Coxeter diagrams        
        =      
Cells {3,8}  
Faces {3}
Edge figure {6}
Vertex figure {8,6}  
{(8,3,8)}  
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.

 
Poincaré disk model

Order-8-infinite triangular honeycomb

edit
Order-8-infinite triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,8,∞}
{3,(8,∞,8)}
Coxeter diagrams        
        =       
Cells {3,8}  
Faces {3}
Edge figure {∞}
Vertex figure {8,∞}  
{(8,∞,8)}  
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.

 
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {3,(8,∞,8)}, Coxeter diagram,         =       , 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

edit
Order-8-3 square honeycomb
Type Regular honeycomb
Schläfli symbol {4,8,3}
Coxeter diagram        
Cells {4,8}  
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}.

 
Poincaré disk model

Order-8-3 pentagonal honeycomb

edit
Order-8-3 pentagonal honeycomb
Type Regular honeycomb
Schläfli symbol {5,8,3}
Coxeter diagram        
Cells {5,8}  
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}.

 
Poincaré disk model

Order-8-3 hexagonal honeycomb

edit
Order-8-3 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbol {6,8,3}
Coxeter diagram        
Cells {6,8}  
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}.

 
Poincaré disk model

Order-8-3 apeirogonal honeycomb

edit
Order-8-3 apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbol {∞,8,3}
Coxeter diagram        
Cells {∞,8}  
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 order-8 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 {∞,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.

 
Poincaré disk model

Order-8-4 square honeycomb

edit
Order-8-4 square honeycomb
Type Regular honeycomb
Schläfli symbol {4,8,4}
Coxeter diagrams        
        =      
Cells {4,8}  
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.

 
Poincaré disk model

Order-8-5 pentagonal honeycomb

edit
Order-8-5 pentagonal honeycomb
Type Regular honeycomb
Schläfli symbol {5,8,5}
Coxeter diagrams        
Cells {5,8}  
Faces {5}
Edge figure {5}
Vertex figure {8,5}
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.

 
Poincaré disk model

Order-8-6 hexagonal honeycomb

edit
Order-8-6 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbols {6,8,6}
{6,(8,3,8)}
Coxeter diagrams        
        =      
Cells {6,8}  
Faces {6}
Edge figure {6}
Vertex figure {8,6}  
{(5,3,5)}  
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.

 
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(8,3,8)}, Coxeter diagram,      , 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

edit
Order-8-infinite apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbols {∞,8,∞}
{∞,(8,∞,8)}
Coxeter diagrams        
             
Cells {∞,8}  
Faces {∞}
Edge figure {∞}
Vertex figure   {8,∞}
  {(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 order-8 apeirogonal tiling {∞,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 infinite-order octagonal tiling vertex figure.

 
Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(8,∞,8)}, Coxeter diagram,       , with alternating types or colors of cells.

See also

edit

References

edit
  • 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)
edit