Order-7 tetrahedral honeycomb

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Order-7 tetrahedral honeycomb
Type Hyperbolic regular honeycomb
Schläfli symbols {3,3,7}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
Cells {3,3} Uniform polyhedron-33-t0.png
Faces {3}
Edge figure {7}
Vertex figure {3,7} H2 tiling 237-4.png
Dual {7,3,3}
Coxeter group [7,3,3]
Properties Regular

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[edit]

Hyperbolic honeycomb 3-3-7 poincare cc.png
Poincaré disk model (cell-centered)
H3 337 UHS plane at infinity.png
Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model

Related polytopes and honeycombs[edit]

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

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}
Hyperbolic honeycomb 3-3-7 poincare cc.png Hyperbolic honeycomb 4-3-7 poincare cc.png Hyperbolic honeycomb 5-3-7 poincare cc.png Hyperbolic honeycomb 6-3-7 poincare.png Hyperbolic honeycomb 7-3-7 poincare.png Hyperbolic honeycomb 8-3-7 poincare.png Hyperbolic honeycomb i-3-7 poincare.png

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

Order-8 tetrahedral honeycomb[edit]

Order-8 tetrahedral honeycomb
Type Hyperbolic regular honeycomb
Schläfli symbols {3,3,8}
{3,(3,4,3)}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node h0.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png
Cells {3,3} Uniform polyhedron-33-t0.png
Faces {3}
Edge figure {8}
Vertex figure {3,8} H2 tiling 238-4.png
{(3,4,3)} Uniform tiling 433-t2.png
Dual {8,3,3}
Coxeter group [3,3,8]
[3,((3,4,3))]
Properties Regular

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.

Hyperbolic honeycomb 3-3-8 poincare cc.png
Poincaré disk model (cell-centered)
H3 338 UHS plane at infinity.png
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, CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png, 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[edit]

Infinite-order tetrahedral honeycomb
Type Hyperbolic regular honeycomb
Schläfli symbols {3,3,∞}
{3,(3,∞,3)}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node h0.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png
Cells {3,3} Uniform polyhedron-33-t0.png
Faces {3}
Edge figure {∞}
Vertex figure {3,∞} H2 tiling 23i-4.png
{(3,∞,3)} H2 tiling 33i-4.png
Dual {∞,3,3}
Coxeter group [∞,3,3]
[3,((3,∞,3))]
Properties Regular

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.

Hyperbolic honeycomb 3-3-i poincare cc.png
Poincaré disk model (cell-centered)
H3 33i UHS plane at infinity.png
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, CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node h0.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png, with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [3,3,∞,1+] = [3,((3,∞,3))].

See also[edit]

References[edit]

External links[edit]