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 | [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.
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 |
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 | ||||||||
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Space | S3 | H3 | ||||||
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 ![]() | 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,∞} |
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![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
It is a part of a series of regular honeycombs, with {7,p,3}.
{7,3,3} | {7,4,3} | {7,5,3}... |
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![]() | ![]() | ![]() |
Octagonal tiling honeycomb | |
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Type | Regular honeycomb |
Schläfli symbol | {8,3,3} t{8,4,3} 2t{4,8,4} t{4[3,3]} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {8,3} ![]() |
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 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 [8,3,3] |
Apeirogonal tiling honeycomb | |
---|---|
Type | Regular honeycomb |
Schläfli symbol | {∞,3,3} t{∞,3,3} 2t{∞,∞,∞} t{∞[3,3]} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells | {∞,3} ![]() |
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 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 |
In the geometry of hyperbolic 3-space, the order-7 tetrahedral honeycomb is a regular space-filling tessellation with Schläfli symbol {3,3,7}. It has seven tetrahedra {3,3} around each edge. All vertices are ultra-ideal with infinitely many tetrahedra existing around each vertex in an order-7 triangular tiling vertex arrangement.
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 order-3-7 hexagonal honeycomb or a regular space-filling tessellation with Schläfli symbol {6,3,7}.
In the geometry of hyperbolic 3-space, the order-7 cubic honeycomb is a regular space-filling tessellation. With Schläfli symbol {4,3,7}, it has seven cubes {4,3} around each edge. All vertices are ultra-ideal with infinitely many cubes existing around each vertex in an order-7 triangular tiling vertex arrangement.
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-5 octahedral honeycomb is a regular space-filling tessellation with Schläfli symbol {3,4,5}. It has five octahedra {3,4} around each edge. All vertices are ultra-ideal with infinitely many octahedra existing around each vertex in an order-5 square tiling vertex arrangement.
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-4-5 square honeycomb is a regular space-filling tessellation with Schläfli symbol {4,4,5}. It has five square tiling {4,4} around each edge. All vertices are ultra-ideal with infinitely many square tiling existing around each vertex in an order-5 square tiling vertex arrangement.
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-5-3 square honeycomb or 4,5,3 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-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}.