Order-6 hexagonal tiling honeycomb | |
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Perspective projection view from center of Poincaré disk model | |
Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb |
Schläfli symbol | {6,3,6} {6,3[3]} |
Coxeter diagram | |
Cells | {6,3} |
Faces | hexagon {6} |
Edge figure | hexagon {6} |
Vertex figure | {3,6} or {3[3]} |
Dual | Self-dual |
Coxeter group | , [6,3,6] , [6,3[3]] |
Properties | Regular, quasiregular |
In the field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells with an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.
The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,6}. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has six such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the triangular tiling is {3,6}, the vertex figure of this honeycomb is a triangular tiling. Thus, infinitely many hexagonal tilings meet at each vertex of this honeycomb. [1]
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
The order-6 hexagonal tiling honeycomb is analogous to the 2D hyperbolic infinite-order apeirogonal tiling, {∞,∞}, with infinite apeirogonal faces, and with all vertices on the ideal surface.
It contains
The order-6 hexagonal tiling honeycomb has a half-symmetry construction:
It also has an index-6 subgroup, [6,3*,6], with a non-simplex fundamental domain. This subgroup corresponds to a Coxeter diagram with six order-3 branches and three infinite-order branches in the shape of a triangular prism:
The order-6 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of eleven paracompact honeycombs in 3-space.
11 paracompact regular honeycombs | |||||||||||
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{6,3,3} | {6,3,4} | {6,3,5} | {6,3,6} | {4,4,3} | {4,4,4} | ||||||
{3,3,6} | {4,3,6} | {5,3,6} | {3,6,3} | {3,4,4} |
There are nine uniform honeycombs in the [6,3,6] Coxeter group family, including this regular form.
{6,3,6} | r{6,3,6} | t{6,3,6} | rr{6,3,6} | t0,3{6,3,6} | 2t{6,3,6} | tr{6,3,6} | t0,1,3{6,3,6} | t0,1,2,3{6,3,6} |
---|---|---|---|---|---|---|---|---|
This honeycomb has a related alternated honeycomb, the triangular tiling honeycomb, but with a lower symmetry:
The order-6 hexagonal tiling honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures:
Form | Paracompact | Noncompact | |||||
---|---|---|---|---|---|---|---|
Name | {3,3,6} | {4,3,6} | {5,3,6} | {6,3,6} | {7,3,6} | {8,3,6} | ... {∞,3,6} |
Image | |||||||
Cells | {3,3} | {4,3} | {5,3} | {6,3} | {7,3} | {8,3} | {∞,3} |
It is also part of a sequence of regular polychora and honeycombs with hexagonal tiling cells:
{6,3,p} honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | H3 | ||||||||||
Form | Paracompact | Noncompact | |||||||||
Name | {6,3,3} | {6,3,4} | {6,3,5} | {6,3,6} | {6,3,7} | {6,3,8} | ... {6,3,∞} | ||||
Coxeter | |||||||||||
Image | |||||||||||
Vertex figure {3,p} | {3,3} | {3,4} | {3,5} | {3,6} | {3,7} | {3,8} | {3,∞} |
It is also part of a sequence of regular polychora and honeycombs with regular deltahedral vertex figures:
{p,3,p} regular honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | S3 | Euclidean E3 | H3 | ||||||||
Form | Finite | Affine | Compact | Paracompact | Noncompact | ||||||
Name | {3,3,3} | {4,3,4} | {5,3,5} | {6,3,6} | {7,3,7} | {8,3,8} | ...{∞,3,∞} | ||||
Image | |||||||||||
Cells | {3,3} | {4,3} | {5,3} | {6,3} | {7,3} | {8,3} | {∞,3} | ||||
Vertex figure | {3,3} | {3,4} | {3,5} | {3,6} | {3,7} | {3,8} | {3,∞} |
Rectified order-6 hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb |
Schläfli symbols | r{6,3,6} or t1{6,3,6} |
Coxeter diagrams | |
Cells | {3,6} r{6,3} |
Faces | triangle {3} hexagon {6} |
Vertex figure | hexagonal prism |
Coxeter groups | , [6,3,6] , [6,3[3]] , [3[3,3]] |
Properties | Vertex-transitive, edge-transitive |
The rectified order-6 hexagonal tiling honeycomb, t1{6,3,6},
it can also be seen as a quarter order-6 hexagonal tiling honeycomb, q{6,3,6},
It is analogous to 2D hyperbolic order-4 apeirogonal tiling, r{∞,∞} with infinite apeirogonal faces, and with all vertices on the ideal surface.
The order-6 hexagonal tiling honeycomb is part of a series of honeycombs with hexagonal prism vertex figures:
Space | H3 | ||||||
---|---|---|---|---|---|---|---|
Form | Paracompact | Noncompact | |||||
Name | r{3,3,6} | r{4,3,6} | r{5,3,6} | r{6,3,6} | r{7,3,6} | ... r{∞,3,6} | |
Image | |||||||
Cells {3,6} | r{3,3} | r{4,3} | r{5,3} | r{6,3} | r{7,3} | r{∞,3} |
It is also part of a matrix of 3-dimensional quarter honeycombs: q{2p,4,2q}
Euclidean/hyperbolic(paracompact/noncompact) quarter honeycombs q{p,3,q} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
p \ q | 4 | 6 | 8 | ... ∞ | |||||||
4 | q{4,3,4} | q{4,3,6} | q{4,3,8} | q{4,3,∞} | |||||||
6 | q{6,3,4} | q{6,3,6} | q{6,3,8} | q{6,3,∞} | |||||||
8 | q{8,3,4} | q{8,3,6} | q{8,3,8} | q{8,3,∞} | |||||||
... ∞ | q{∞,3,4} | q{∞,3,6} | q{∞,3,8} | q{∞,3,∞} |
Truncated order-6 hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb |
Schläfli symbol | t{6,3,6} or t0,1{6,3,6} |
Coxeter diagram | |
Cells | {3,6} t{6,3} |
Faces | triangle {3} dodecagon {12} |
Vertex figure | hexagonal pyramid |
Coxeter groups | , [6,3,6] , [6,3[3]] |
Properties | Vertex-transitive |
The truncated order-6 hexagonal tiling honeycomb, t0,1{6,3,6},
Bitruncated order-6 hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb |
Schläfli symbol | bt{6,3,6} or t1,2{6,3,6} |
Coxeter diagram | |
Cells | t{3,6} |
Faces | hexagon {6} |
Vertex figure | tetrahedron |
Coxeter groups | , [[6,3,6]] , [6,3[3]] , [3,3,6] |
Properties | Regular |
The bitruncated order-6 hexagonal tiling honeycomb is a lower symmetry construction of the regular hexagonal tiling honeycomb,
Cantellated order-6 hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb |
Schläfli symbol | rr{6,3,6} or t0,2{6,3,6} |
Coxeter diagram | |
Cells | r{3,6} rr{6,3} {}x{6} |
Faces | triangle {3} square {4} hexagon {6} |
Vertex figure | wedge |
Coxeter groups | , [6,3,6] , [6,3[3]] |
Properties | Vertex-transitive |
The cantellated order-6 hexagonal tiling honeycomb, t0,2{6,3,6},
Cantitruncated order-6 hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb |
Schläfli symbol | tr{6,3,6} or t0,1,2{6,3,6} |
Coxeter diagram | |
Cells | tr{3,6} t{3,6} {}x{6} |
Faces | triangle {3} square {4} hexagon {6} dodecagon {12} |
Vertex figure | mirrored sphenoid |
Coxeter groups | , [6,3,6] , [6,3[3]] |
Properties | Vertex-transitive |
The cantitruncated order-6 hexagonal tiling honeycomb, t0,1,2{6,3,6},
Runcinated order-6 hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,3{6,3,6} |
Coxeter diagram | |
Cells | {6,3} {}×{6} |
Faces | triangle {3} square {4} hexagon {6} |
Vertex figure | triangular antiprism |
Coxeter groups | , [[6,3,6]] |
Properties | Vertex-transitive, edge-transitive |
The runcinated order-6 hexagonal tiling honeycomb, t0,3{6,3,6},
It is analogous to the 2D hyperbolic rhombihexahexagonal tiling, rr{6,6},
Runcitruncated order-6 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,1,3{6,3,6} |
Coxeter diagram | |
Cells | t{6,3} rr{6,3} {}x{6} {}x{12} |
Faces | triangle {3} square {4} hexagon {6} dodecagon {12} |
Vertex figure | isosceles-trapezoidal pyramid |
Coxeter groups | , [6,3,6] |
Properties | Vertex-transitive |
The runcitruncated order-6 hexagonal tiling honeycomb, t0,1,3{6,3,6},
Omnitruncated order-6 hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,1,2,3{6,3,6} |
Coxeter diagram | |
Cells | tr{6,3} {}x{12} |
Faces | square {4} hexagon {6} dodecagon {12} |
Vertex figure | phyllic disphenoid |
Coxeter groups | , [[6,3,6]] |
Properties | Vertex-transitive |
The omnitruncated order-6 hexagonal tiling honeycomb, t0,1,2,3{6,3,6},
Alternated order-6 hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb |
Schläfli symbols | h{6,3,6} |
Coxeter diagrams | |
Cells | {3,6} {3[3]} |
Faces | triangle {3} |
Vertex figure | hexagonal tiling |
Coxeter groups | , [6,3[3]] |
Properties | Regular, quasiregular |
The alternated order-6 hexagonal tiling honeycomb is a lower-symmetry construction of the regular triangular tiling honeycomb,
Cantic order-6 hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb |
Schläfli symbols | h2{6,3,6} |
Coxeter diagrams | |
Cells | t{3,6} r{6,3} h2{6,3} |
Faces | triangle {3} hexagon {6} |
Vertex figure | triangular prism |
Coxeter groups | , [6,3[3]] |
Properties | Vertex-transitive, edge-transitive |
The cantic order-6 hexagonal tiling honeycomb is a lower-symmetry construction of the rectified triangular tiling honeycomb,
Runcic order-6 hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb |
Schläfli symbols | h3{6,3,6} |
Coxeter diagrams | |
Cells | rr{3,6} {6,3} {3[3]} {3}x{} |
Faces | triangle {3} square {4} hexagon {6} |
Vertex figure | triangular cupola |
Coxeter groups | , [6,3[3]] |
Properties | Vertex-transitive |
The runcic hexagonal tiling honeycomb, h3{6,3,6},
Runcicantic order-6 hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb |
Schläfli symbols | h2,3{6,3,6} |
Coxeter diagrams | |
Cells | tr{6,3} t{6,3} h2{6,3} {}x{3} |
Faces | triangle {3} square {4} hexagon {6} dodecagon {12} |
Vertex figure | rectangular pyramid |
Coxeter groups | , [6,3[3]] |
Properties | Vertex-transitive |
The runcicantic order-6 hexagonal tiling honeycomb, h2,3{6,3,6},
The icosahedral honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {3,5,3}, there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure.
The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed entirely of triangular prisms.
In geometry, a quasiregular polyhedron is a uniform polyhedron which has exactly two kinds of regular faces, which alternate around each vertex. They are edge-transitive, and hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive. Their dual figures are also sometimes considered quasiregular, except that they are edge-transitive, are face-transitive, and alternate between two regular vertex figures.
In the field of hyperbolic geometry, the hexagonal tiling honeycomb is 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 is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.
In hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation. It is paracompact because it has vertex figures composed of an infinite number of faces, and has all vertices as ideal points at infinity. With Schläfli symbol {3,3,6}, the order-6 tetrahedral honeycomb has six ideal tetrahedra around each edge. All vertices are ideal, with infinitely many tetrahedra existing around each vertex in a triangular tiling vertex figure.
In the field of hyperbolic geometry, the order-4 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 is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.
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.
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.
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.
The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations in hyperbolic 3-space. It is called paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. It has Schläfli symbol {3,6,3}, being composed of triangular tiling cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Its vertex figure is a hexagonal tiling.
In the geometry of hyperbolic 3-space, the square tiling honeycomb is one of 11 paracompact regular honeycombs. It is called paracompact because it has infinite cells, whose vertices exist on horospheres and converge to a single ideal point at infinity. Given by Schläfli symbol {4,4,3}, it has three square tilings, {4,4}, around each edge, and six square tilings around each vertex, in a cubic {4,3} vertex figure.
In the geometry of hyperbolic 3-space, the order-4 square tiling honeycomb is one of 11 paracompact regular honeycombs. It is paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {4,4,4}, it has four square tilings around each edge, and infinite square tilings around each vertex in a square tiling vertex figure.
The order-4 octahedral honeycomb is a regular paracompact honeycomb in hyperbolic 3-space. It is paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {3,4,4}, it has four ideal octahedra around each edge, and infinite octahedra around each vertex in a square tiling vertex figure.
In three-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h{6,3,3},
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-4-5 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-6-4 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}.