Misplaced Pages

Heptagonal tiling honeycomb

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
(Redirected from Octagonal tiling honeycomb)
Heptagonal tiling honeycomb
Type Regular honeycomb
Schläfli symbol {7,3,3}
Coxeter diagram
Cells {7,3}
Faces Heptagon {7}
Vertex figure tetrahedron {3,3}
Dual {3,3,7}
Coxeter group
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

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 a tetrahedron, {3,3}.


Poincaré disk model
(vertex centered)

Rotating

Ideal surface

Related polytopes and honeycombs

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

{p,3,3} honeycombs
Space S H
Form Finite Paracompact Noncompact
Name {3,3,3} {4,3,3} {5,3,3} {6,3,3} {7,3,3} {8,3,3} ... {∞,3,3}
Image
Coxeter diagrams
subgroups
1
4
6
12
24
Cells
{p,3}

{3,3}

{4,3}



{5,3}

{6,3}



{7,3}

{8,3}



{∞,3}


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,∞}

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

{7,3,3} {7,4,3} {7,5,3}...

Octagonal tiling honeycomb

Octagonal tiling honeycomb
Type Regular honeycomb
Schläfli symbol {8,3,3}
t{8,4,3}
2t{4,8,4}
t{4}
Coxeter diagram



(all 4s)
Cells {8,3}
Faces Octagon {8}
Vertex figure tetrahedron {3,3}
Dual {3,3,8}
Coxeter group
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 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 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}.


Poincaré disk model (vertex centered)

Direct subgroups of

Apeirogonal tiling honeycomb

Apeirogonal tiling honeycomb
Type Regular honeycomb
Schläfli symbol {∞,3,3}
t{∞,3,3}
2t{∞,∞,∞}
t{∞}
Coxeter diagram



(all ∞)
Cells {∞,3}
Faces Apeirogon {∞}
Vertex figure tetrahedron {3,3}
Dual {3,3,∞}
Coxeter group
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 an 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 an Apollonian gasket pattern of circles inside a largest circle.


Poincaré disk model (vertex centered)

Ideal surface

See also

References

External links

Categories:
Ad.

Before you begin

Get Life Coaching Tips
Or continue to this article
X