Heptagonal tiling honeycomb

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Heptagonal tiling honeycomb
Type Regular honeycomb
Schläfli symbol {7,3,3}
Coxeter diagram CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Cells {7,3} Uniform tiling 73-t0.png
Faces Heptagon {7}
Vertex figure tetrahedron {3,3}
Dual {3,3,7}
Coxeter group [7,3,3]
Properties Regular

In the geometry of hyperbolic 3-space, the heptagonal tiling honeycomb or 7,3,3 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.

Geometry[edit]

The Schläfli symbol of the heptagonal tiling honeycomb is {7,3,3}, with three heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an tetrahedron, {3,3}.

Hyperbolic honeycomb 7-3-3 poincare vc.png
Poincaré disk model
(vertex centered)
H3 733 UHS plane at infinity.png
Ideal surface

Related polytopes and honeycombs[edit]

It is a part of a series of regular polytopes and honeycombs with {p,3,3} Schläfli symbol, and tetrahedral vertex figures:


It is a part of a series of regular honeycombs, {7,3,p}.

{7,3,3} {7,3,4} {7,3,5} {7,3,6} {7,3,7} {7,3,8} ...{7,3,∞}
Hyperbolic honeycomb 7-3-3 poincare vc.png Hyperbolic honeycomb 7-3-4 poincare vc.png Hyperbolic honeycomb 7-3-5 poincare vc.png Hyperbolic honeycomb 7-3-6 poincare.png Hyperbolic honeycomb 7-3-7 poincare.png Hyperbolic honeycomb 7-3-8 poincare.png Hyperbolic honeycomb 7-3-i poincare.png

It is a part of a series of regular honeycombs,with {7,p,3}.

{7,3,3} {7,4,3} {7,5,3}...
Hyperbolic honeycomb 7-3-3 poincare vc.png Hyperbolic honeycomb 7-4-3 poincare vc.png Hyperbolic honeycomb 7-5-3 poincare vc.png

Octagonal tiling honeycomb[edit]

Octagonal tiling honeycomb
Type Regular honeycomb
Schläfli symbol {8,3,3}
t{8,4,3}
2t{4,8,4}
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel branch 11.pngCDel split2-44.pngCDel node 1.pngCDel 8.pngCDel node.png
Cells {8,3} Uniform tiling 83-t0.png
Faces Octagon {8}
Vertex figure tetrahedron {3,3}
Dual {3,3,8}
Coxeter group [8,3,3]
Properties Regular

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

Hyperbolic honeycomb 8-3-3 poincare vc.png
Poincaré disk model
(vertex centered)

Apeirogonal tiling honeycomb[edit]

Apeirogonal tiling honeycomb
Type Regular honeycomb
Schläfli symbol {∞,3,3}
t{∞,3,3}
2t{∞,∞,∞}
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel labelinfin.pngCDel branch 11.pngCDel split2-ii.pngCDel node 1.pngCDel infin.pngCDel node.png
Cells {∞,3} H2 tiling 23i-1.png
Faces Apeirogon {∞}
Vertex figure tetrahedron {3,3}
Dual {3,3,∞}
Coxeter group [∞,3,3]
Properties Regular

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

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

Hyperbolic honeycomb i-3-3 poincare vc.png
Poincaré disk model
(vertex centered)
H3 i33 UHS plane at infinity.png
Ideal surface

See also[edit]

References[edit]

External links[edit]