Order-5-3 square honeycomb

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Order-5-3 square honeycomb
Type Regular honeycomb
Schläfli symbol {4,5,3}
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Cells {4,5} Uniform tiling 45-t0.png
Faces {4}
Vertex figure {5,3}
Dual {3,5,4}
Coxeter group [4,5,3]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-5-3 square honeycomb or 4,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

Contents

Geometry

The Schläfli symbol of the order-5-3 square honeycomb is {4,5,3}, with three order-4 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.

Hyperbolic honeycomb 4-5-3 poincare vc.png
Poincaré disk model
(Vertex centered)
H3 453 UHS plane at infinity.png
Ideal surface

It is a part of a series of regular polytopes and honeycombs with {p,5,3} Schläfli symbol, and dodecahedral vertex figures:

Order-5-3 pentagonal honeycomb

Order-5-3 pentagonal honeycomb
Type Regular honeycomb
Schläfli symbol {5,5,3}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Cells {5,5} Uniform tiling 55-t0.png
Faces {5}
Vertex figure {5,3}
Dual {3,5,5}
Coxeter group [5,5,3]
PropertiesRegular

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

Hyperbolic honeycomb 5-5-3 poincare vc.png
Poincaré disk model
(Vertex centered)
H3 553 UHS plane at infinity.png
Ideal surface

Order-5-3 hexagonal honeycomb

Order-5-3 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbol {6,5,3}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Cells {6,5} Uniform tiling 65-t0.png
Faces {6}
Vertex figure {5,3}
Dual {3,5,6}
Coxeter group [6,5,3]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-5-3 hexagonal honeycomb or 6,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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-5-3 hexagonal honeycomb is {6,5,3}, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.

Hyperbolic honeycomb 6-5-3 poincare vc.png
Poincaré disk model
(Vertex centered)
H3 653 UHS plane at infinity.png
Ideal surface

Order-5-3 heptagonal honeycomb

Order-5-3 heptagonal honeycomb
Type Regular honeycomb
Schläfli symbol {7,5,3}
Coxeter diagram CDel node 1.pngCDel 7.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Cells {7,5} Uniform tiling 75-t0.png
Faces {7}
Vertex figure {5,3}
Dual {3,5,7}
Coxeter group [7,5,3]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-5-3 heptagonal honeycomb or 7,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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-5-3 heptagonal honeycomb is {7,5,3}, with three order-5 heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.

Hyperbolic honeycomb 7-5-3 poincare vc.png
Poincaré disk model
(Vertex centered)
H3 753 UHS plane at infinity.png
Ideal surface

Order-5-3 octagonal honeycomb

Order-5-3 octagonal honeycomb
Type Regular honeycomb
Schläfli symbol {8,5,3}
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Cells {8,5} Uniform tiling 85-t0.png
Faces {8}
Vertex figure {5,3}
Dual {3,5,8}
Coxeter group [8,5,3]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-5-3 octagonal honeycomb or 8,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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-5-3 octagonal honeycomb is {8,5,3}, with three order-5 octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.

Hyperbolic honeycomb 8-5-3 poincare vc.png
Poincaré disk model
(Vertex centered)

Order-5-3 apeirogonal honeycomb

Order-5-3 apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbol {∞,5,3}
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Cells {∞,5} H2 tiling 25i-1.png
Faces Apeirogon {∞}
Vertex figure {5,3}
Dual {3,5,∞}
Coxeter group [∞,5,3]
PropertiesRegular

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

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.

Hyperbolic honeycomb i-5-3 poincare vc.png
Poincaré disk model
(Vertex centered)
H3 i53 UHS plane at infinity.png
Ideal surface

See also

Related Research Articles

<span class="mw-page-title-main">Order-6 cubic honeycomb</span>

The order-6 cubic honeycomb is a paracompact regular space-filling tessellation in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of facets, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,6}, the honeycomb has six ideal cubes meeting along each edge. Its vertex figure is an infinite triangular tiling. Its dual is the order-4 hexagonal tiling honeycomb.

<span class="mw-page-title-main">Order-6 dodecahedral honeycomb</span> Regular geometrical object in hyperbolic space

The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of faces, with all vertices as ideal points at infinity. It has Schläfli symbol {5,3,6}, with six ideal dodecahedral cells surrounding each edge of the honeycomb. Each vertex is ideal, and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure.

<span class="mw-page-title-main">Order-5 hexagonal tiling honeycomb</span>

In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere, a flat plane in hyperbolic space that approaches a single ideal point at infinity.

In the geometry of hyperbolic 3-space, the order-7 dodecahedral honeycomb is a regular space-filling tessellation.

In the geometry of hyperbolic 3-space, the heptagonal tiling honeycomb or 7,3,3 honeycomb a regular space-filling tessellation. 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.

In the geometry of hyperbolic 3-space, the order-3-4 heptagonal honeycomb or 7,3,4 honeycomb a regular space-filling tessellation. 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.

In the geometry of hyperbolic 3-space, the order-3-5 heptagonal honeycomb a regular space-filling tessellation. 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.

In the geometry of hyperbolic 3-space, the order-3-6 heptagonal honeycomb a regular space-filling tessellation. 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.

In the geometry of hyperbolic 3-space, the order-3-7 heptagonal honeycomb a regular space-filling tessellation with Schläfli symbol {7,3,7}.

In the geometry of hyperbolic 3-space, the order-4 icosahedral honeycomb is a regular space-filling tessellation with Schläfli symbol {3,5,4}.

In the geometry of hyperbolic 3-space, the order-4-3 pentagonal honeycomb or 5,4,3 honeycomb is a regular space-filling tessellation. Each infinite cell is an order-4 pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

In the geometry of hyperbolic 3-space, the order-4-4 pentagonal honeycomb a regular space-filling tessellation. Each infinite cell consists of a pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

In the geometry of hyperbolic 3-space, the order-4-5 pentagonal honeycomb a regular space-filling tessellation with Schläfli symbol {5,4,5}.

In the geometry of hyperbolic 3-space, the order-5-4 square honeycomb a regular space-filling tessellation with Schläfli symbol {4,5,4}.

In the geometry of hyperbolic 3-space, the order-7-3 triangular honeycomb is a regular space-filling tessellation with Schläfli symbol {3,7,3}.

In the geometry of hyperbolic 3-space, the order-6-3 square honeycomb or 4,6,3 honeycomb is a regular space-filling tessellation. Each infinite cell consists of a hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

In the geometry of hyperbolic 3-space, the order-6-4 square honeycomb a regular space-filling tessellation with Schläfli symbol {4,6,4}.

In the geometry of hyperbolic 3-space, the order-8-3 triangular honeycomb is a regular space-filling tessellation with Schläfli symbol {3,8,3}.

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

References