Hexagonal tiling honeycomb | |
---|---|
![]() Perspective projection view within Poincaré disk model | |
Type |
Hyperbolic regular honeycomb Paracompact uniform honeycomb |
Schläfli symbols | {6,3,3} t{3,6,3} 2t{6,3,6} 2t{6,3[3]} t{3[3,3]} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{6,3}
![]() |
Faces | hexagon {6} |
Edge figure | triangle {3} |
Vertex figure |
![]() tetrahedron {3,3} |
Dual | Order-6 tetrahedral honeycomb |
Coxeter groups | , [3,3,6] , [3,6,3] , [6,3,6] , [6,3[3] , [3[3,3]] |
Properties | Regular |
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 surface in hyperbolic space that approaches a single ideal point at infinity.
The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the tetrahedron is {3,3}, the vertex figure of this honeycomb is a tetrahedron. Thus, four hexagonal tilings meet at each vertex of this honeycomb, six hexagons meet at each vertex, and four edges meet at each vertex. [1]
Viewed in perspective outside of a Poincaré disk model, the image above shows one hexagonal tiling cell within the honeycomb, and its mid-radius horosphere (the horosphere incident with edge midpoints). In this projection, the hexagons grow infinitely small towards the infinite boundary, asymptoting towards a single ideal point. It can be seen as similar to the order-3 apeirogonal tiling, {∞,3} of H2, with horocycles circumscribing vertices of apeirogonal faces.
{6,3,3} | {∞,3} |
---|---|
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One hexagonal tiling cell of the hexagonal tiling honeycomb | An order-3 apeirogonal tiling with a green apeirogon and its horocycle |
It has a total of five reflectional constructions from five related Coxeter groups all with four mirrors and only the first being regular: [6,3,3],
[3,6,3],
[6,3,6],
[6,3[3]] and [3[3,3]]
, having 1, 4, 6, 12 and 24 times
larger fundamental domains respectively. In
Coxeter notation subgroup markups, they are related as: [6,(3,3)*] (remove 3 mirrors, index 24 subgroup); [3,6,3*] or [3*,6,3] (remove 2 mirrors, index 6 subgroup); [1+,6,3,6,1+] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3[3,3]]. The ringed Coxeter diagrams are
,
,
,
and
, representing different types (colors) of hexagonal tilings in the
Wythoff construction.
The hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
11 paracompact regular honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
![]() {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} |
It is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the order-6 tetrahedral honeycomb.
[6,3,3] family honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
{6,3,3} | r{6,3,3} | t{6,3,3} | rr{6,3,3} | t0,3{6,3,3} | tr{6,3,3} | t0,1,3{6,3,3} | t0,1,2,3{6,3,3} | ||||
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{3,3,6} | r{3,3,6} | t{3,3,6} | rr{3,3,6} | 2t{3,3,6} | tr{3,3,6} | t0,1,3{3,3,6} | t0,1,2,3{3,3,6} |
It is part of a sequence of regular polychora, which include the 5-cell {3,3,3}, tesseract {4,3,3}, and 120-cell {5,3,3} of Euclidean 4-space, along with other hyperbolic honeycombs containing tetrahedral vertex figures.
{p,3,3} honeycombs | ||||||||
---|---|---|---|---|---|---|---|---|
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 |
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Coxeter diagrams![]() |
1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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4 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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6 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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12 | ![]() ![]() ![]() ![]() ![]() |
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24 | ![]() ![]() ![]() |
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Cells {p,3} ![]() ![]() ![]() ![]() ![]() |
![]() {3,3} ![]() ![]() ![]() ![]() ![]() |
![]() {4,3} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {5,3} ![]() ![]() ![]() ![]() ![]() |
![]() {6,3} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {7,3} ![]() ![]() ![]() ![]() ![]() |
![]() {8,3} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {∞,3} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
It is also part of a sequence of regular honeycombs of the form {6,3,p}, which are each composed of 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![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Image |
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Vertex figure {3,p} ![]() ![]() ![]() ![]() ![]() |
![]() {3,3} ![]() ![]() ![]() ![]() ![]() |
![]() {3,4} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {3,5} ![]() ![]() ![]() ![]() ![]() |
![]() {3,6} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {3,7} ![]() ![]() ![]() ![]() ![]() |
![]() {3,8} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {3,∞} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Rectified hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbols | r{6,3,3} or t1{6,3,3} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,3}
![]() r{6,3} ![]() ![]() |
Faces |
triangle {3} hexagon {6} |
Vertex figure |
![]() triangular prism |
Coxeter groups | , [3,3,6] , [3,3[3]] |
Properties | Vertex-transitive, edge-transitive |
The rectified hexagonal tiling honeycomb, t1{6,3,3}, has
tetrahedral and
trihexagonal tiling facets, with a
triangular prism
vertex figure. The
half-symmetry construction alternates two types of tetrahedra.
Hexagonal tiling honeycomb![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Rectified hexagonal tiling honeycomb![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Related H2 tilings | |
Order-3 apeirogonal tiling![]() ![]() ![]() ![]() ![]() |
Triapeirogonal tiling![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Truncated hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t{6,3,3} or t0,1{6,3,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,3}
![]() t{6,3} ![]() |
Faces |
triangle {3} dodecagon {12} |
Vertex figure |
![]() triangular pyramid |
Coxeter groups | , [3,3,6] |
Properties | Vertex-transitive |
The truncated hexagonal tiling honeycomb, t0,1{6,3,3}, has
tetrahedral and
truncated hexagonal tiling facets, with a
triangular pyramid
vertex figure.
It is similar to the 2D hyperbolic truncated order-3 apeirogonal tiling, t{∞,3} with apeirogonal and triangle faces:
Bitruncated hexagonal tiling honeycomb Bitruncated order-6 tetrahedral honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | 2t{6,3,3} or t1,2{6,3,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
t{3,3}
![]() t{3,6} ![]() |
Faces |
triangle {3} hexagon {6} |
Vertex figure |
![]() digonal disphenoid |
Coxeter groups | , [3,3,6] , [3,3[3]] |
Properties | Vertex-transitive |
The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t1,2{6,3,3}, has
truncated tetrahedron and
hexagonal tiling cells, with a
digonal disphenoid
vertex figure.
Cantellated hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | rr{6,3,3} or t0,2{6,3,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
r{3,3}
![]() rr{6,3} ![]() {}×{3} ![]() |
Faces |
triangle {3} square {4} hexagon {6} |
Vertex figure |
![]() wedge |
Coxeter groups | , [3,3,6] |
Properties | Vertex-transitive |
The cantellated hexagonal tiling honeycomb, t0,2{6,3,3}, has
octahedron,
rhombitrihexagonal tiling, and
triangular prism cells, with a
wedge
vertex figure.
Cantitruncated hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | tr{6,3,3} or t0,1,2{6,3,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
t{3,3}
![]() tr{6,3} ![]() {}×{3} ![]() |
Faces |
triangle {3} square {4} hexagon {6} dodecagon {12} |
Vertex figure |
![]() mirrored sphenoid |
Coxeter groups | , [3,3,6] |
Properties | Vertex-transitive |
The cantitruncated hexagonal tiling honeycomb, t0,1,2{6,3,3}, has
truncated tetrahedron,
truncated trihexagonal tiling, and
triangular prism cells, with a
mirrored sphenoid
vertex figure.
Runcinated hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,3{6,3,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,3}
![]() {6,3} ![]() {}×{6} ![]() {}×{3} ![]() |
Faces |
triangle {3} square {4} hexagon {6} |
Vertex figure |
![]() irregular triangular antiprism |
Coxeter groups | , [3,3,6] |
Properties | Vertex-transitive |
The runcinated hexagonal tiling honeycomb, t0,3{6,3,3}, has
tetrahedron,
hexagonal tiling,
hexagonal prism, and
triangular prism cells, with an irregular
triangular antiprism
vertex figure.
Runcitruncated hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,1,3{6,3,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
rr{3,3}
![]() {}x{3} ![]() {}x{12} ![]() t{6,3} ![]() |
Faces |
triangle {3} square {4} dodecagon {12} |
Vertex figure |
![]() isosceles-trapezoidal pyramid |
Coxeter groups | , [3,3,6] |
Properties | Vertex-transitive |
The runcitruncated hexagonal tiling honeycomb, t0,1,3{6,3,3}, has
cuboctahedron,
triangular prism,
dodecagonal prism, and
truncated hexagonal tiling cells, with an
isosceles-trapezoidal
pyramid
vertex figure.
Runcicantellated hexagonal tiling honeycomb runcitruncated order-6 tetrahedral honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,2,3{6,3,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
t{3,3}
![]() {}x{6} ![]() rr{6,3} ![]() |
Faces |
triangle {3} square {4} hexagon {6} |
Vertex figure |
![]() isosceles-trapezoidal pyramid |
Coxeter groups | , [3,3,6] |
Properties | Vertex-transitive |
The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t0,2,3{6,3,3}, has
truncated tetrahedron,
hexagonal prism, and
rhombitrihexagonal tiling cells, with an
isosceles-trapezoidal
pyramid
vertex figure.
Omnitruncated hexagonal tiling honeycomb Omnitruncated order-6 tetrahedral honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,1,2,3{6,3,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
tr{3,3}
![]() {}x{6} ![]() {}x{12} ![]() tr{6,3} ![]() |
Faces |
square {4} hexagon {6} dodecagon {12} |
Vertex figure |
![]() irregular tetrahedron |
Coxeter groups | , [3,3,6] |
Properties | Vertex-transitive |
The omnitruncated hexagonal tiling honeycomb or omnitruncated order-6 tetrahedral honeycomb, t0,1,2,3{6,3,3}, has
truncated octahedron,
hexagonal prism,
dodecagonal prism, and
truncated trihexagonal tiling cells, with an irregular
tetrahedron
vertex figure.
Hexagonal tiling honeycomb | |
---|---|
![]() Perspective projection view within Poincaré disk model | |
Type |
Hyperbolic regular honeycomb Paracompact uniform honeycomb |
Schläfli symbols | {6,3,3} t{3,6,3} 2t{6,3,6} 2t{6,3[3]} t{3[3,3]} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{6,3}
![]() |
Faces | hexagon {6} |
Edge figure | triangle {3} |
Vertex figure |
![]() tetrahedron {3,3} |
Dual | Order-6 tetrahedral honeycomb |
Coxeter groups | , [3,3,6] , [3,6,3] , [6,3,6] , [6,3[3] , [3[3,3]] |
Properties | Regular |
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 surface in hyperbolic space that approaches a single ideal point at infinity.
The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the tetrahedron is {3,3}, the vertex figure of this honeycomb is a tetrahedron. Thus, four hexagonal tilings meet at each vertex of this honeycomb, six hexagons meet at each vertex, and four edges meet at each vertex. [1]
Viewed in perspective outside of a Poincaré disk model, the image above shows one hexagonal tiling cell within the honeycomb, and its mid-radius horosphere (the horosphere incident with edge midpoints). In this projection, the hexagons grow infinitely small towards the infinite boundary, asymptoting towards a single ideal point. It can be seen as similar to the order-3 apeirogonal tiling, {∞,3} of H2, with horocycles circumscribing vertices of apeirogonal faces.
{6,3,3} | {∞,3} |
---|---|
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One hexagonal tiling cell of the hexagonal tiling honeycomb | An order-3 apeirogonal tiling with a green apeirogon and its horocycle |
It has a total of five reflectional constructions from five related Coxeter groups all with four mirrors and only the first being regular: [6,3,3],
[3,6,3],
[6,3,6],
[6,3[3]] and [3[3,3]]
, having 1, 4, 6, 12 and 24 times
larger fundamental domains respectively. In
Coxeter notation subgroup markups, they are related as: [6,(3,3)*] (remove 3 mirrors, index 24 subgroup); [3,6,3*] or [3*,6,3] (remove 2 mirrors, index 6 subgroup); [1+,6,3,6,1+] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3[3,3]]. The ringed Coxeter diagrams are
,
,
,
and
, representing different types (colors) of hexagonal tilings in the
Wythoff construction.
The hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
11 paracompact regular honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
![]() {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} |
It is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the order-6 tetrahedral honeycomb.
[6,3,3] family honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
{6,3,3} | r{6,3,3} | t{6,3,3} | rr{6,3,3} | t0,3{6,3,3} | tr{6,3,3} | t0,1,3{6,3,3} | t0,1,2,3{6,3,3} | ||||
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{3,3,6} | r{3,3,6} | t{3,3,6} | rr{3,3,6} | 2t{3,3,6} | tr{3,3,6} | t0,1,3{3,3,6} | t0,1,2,3{3,3,6} |
It is part of a sequence of regular polychora, which include the 5-cell {3,3,3}, tesseract {4,3,3}, and 120-cell {5,3,3} of Euclidean 4-space, along with other hyperbolic honeycombs containing tetrahedral vertex figures.
{p,3,3} honeycombs | ||||||||
---|---|---|---|---|---|---|---|---|
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 |
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Coxeter diagrams![]() |
1 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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4 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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6 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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12 | ![]() ![]() ![]() ![]() ![]() |
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24 | ![]() ![]() ![]() |
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Cells {p,3} ![]() ![]() ![]() ![]() ![]() |
![]() {3,3} ![]() ![]() ![]() ![]() ![]() |
![]() {4,3} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {5,3} ![]() ![]() ![]() ![]() ![]() |
![]() {6,3} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {7,3} ![]() ![]() ![]() ![]() ![]() |
![]() {8,3} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {∞,3} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
It is also part of a sequence of regular honeycombs of the form {6,3,p}, which are each composed of 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![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Image |
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Vertex figure {3,p} ![]() ![]() ![]() ![]() ![]() |
![]() {3,3} ![]() ![]() ![]() ![]() ![]() |
![]() {3,4} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {3,5} ![]() ![]() ![]() ![]() ![]() |
![]() {3,6} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {3,7} ![]() ![]() ![]() ![]() ![]() |
![]() {3,8} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() {3,∞} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Rectified hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb |
Schläfli symbols | r{6,3,3} or t1{6,3,3} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,3}
![]() r{6,3} ![]() ![]() |
Faces |
triangle {3} hexagon {6} |
Vertex figure |
![]() triangular prism |
Coxeter groups | , [3,3,6] , [3,3[3]] |
Properties | Vertex-transitive, edge-transitive |
The rectified hexagonal tiling honeycomb, t1{6,3,3}, has
tetrahedral and
trihexagonal tiling facets, with a
triangular prism
vertex figure. The
half-symmetry construction alternates two types of tetrahedra.
Hexagonal tiling honeycomb![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Rectified hexagonal tiling honeycomb![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Related H2 tilings | |
Order-3 apeirogonal tiling![]() ![]() ![]() ![]() ![]() |
Triapeirogonal tiling![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() ![]() |
Truncated hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t{6,3,3} or t0,1{6,3,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,3}
![]() t{6,3} ![]() |
Faces |
triangle {3} dodecagon {12} |
Vertex figure |
![]() triangular pyramid |
Coxeter groups | , [3,3,6] |
Properties | Vertex-transitive |
The truncated hexagonal tiling honeycomb, t0,1{6,3,3}, has
tetrahedral and
truncated hexagonal tiling facets, with a
triangular pyramid
vertex figure.
It is similar to the 2D hyperbolic truncated order-3 apeirogonal tiling, t{∞,3} with apeirogonal and triangle faces:
Bitruncated hexagonal tiling honeycomb Bitruncated order-6 tetrahedral honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | 2t{6,3,3} or t1,2{6,3,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
t{3,3}
![]() t{3,6} ![]() |
Faces |
triangle {3} hexagon {6} |
Vertex figure |
![]() digonal disphenoid |
Coxeter groups | , [3,3,6] , [3,3[3]] |
Properties | Vertex-transitive |
The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t1,2{6,3,3}, has
truncated tetrahedron and
hexagonal tiling cells, with a
digonal disphenoid
vertex figure.
Cantellated hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | rr{6,3,3} or t0,2{6,3,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
r{3,3}
![]() rr{6,3} ![]() {}×{3} ![]() |
Faces |
triangle {3} square {4} hexagon {6} |
Vertex figure |
![]() wedge |
Coxeter groups | , [3,3,6] |
Properties | Vertex-transitive |
The cantellated hexagonal tiling honeycomb, t0,2{6,3,3}, has
octahedron,
rhombitrihexagonal tiling, and
triangular prism cells, with a
wedge
vertex figure.
Cantitruncated hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | tr{6,3,3} or t0,1,2{6,3,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
t{3,3}
![]() tr{6,3} ![]() {}×{3} ![]() |
Faces |
triangle {3} square {4} hexagon {6} dodecagon {12} |
Vertex figure |
![]() mirrored sphenoid |
Coxeter groups | , [3,3,6] |
Properties | Vertex-transitive |
The cantitruncated hexagonal tiling honeycomb, t0,1,2{6,3,3}, has
truncated tetrahedron,
truncated trihexagonal tiling, and
triangular prism cells, with a
mirrored sphenoid
vertex figure.
Runcinated hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,3{6,3,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,3}
![]() {6,3} ![]() {}×{6} ![]() {}×{3} ![]() |
Faces |
triangle {3} square {4} hexagon {6} |
Vertex figure |
![]() irregular triangular antiprism |
Coxeter groups | , [3,3,6] |
Properties | Vertex-transitive |
The runcinated hexagonal tiling honeycomb, t0,3{6,3,3}, has
tetrahedron,
hexagonal tiling,
hexagonal prism, and
triangular prism cells, with an irregular
triangular antiprism
vertex figure.
Runcitruncated hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,1,3{6,3,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
rr{3,3}
![]() {}x{3} ![]() {}x{12} ![]() t{6,3} ![]() |
Faces |
triangle {3} square {4} dodecagon {12} |
Vertex figure |
![]() isosceles-trapezoidal pyramid |
Coxeter groups | , [3,3,6] |
Properties | Vertex-transitive |
The runcitruncated hexagonal tiling honeycomb, t0,1,3{6,3,3}, has
cuboctahedron,
triangular prism,
dodecagonal prism, and
truncated hexagonal tiling cells, with an
isosceles-trapezoidal
pyramid
vertex figure.
Runcicantellated hexagonal tiling honeycomb runcitruncated order-6 tetrahedral honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,2,3{6,3,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
t{3,3}
![]() {}x{6} ![]() rr{6,3} ![]() |
Faces |
triangle {3} square {4} hexagon {6} |
Vertex figure |
![]() isosceles-trapezoidal pyramid |
Coxeter groups | , [3,3,6] |
Properties | Vertex-transitive |
The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t0,2,3{6,3,3}, has
truncated tetrahedron,
hexagonal prism, and
rhombitrihexagonal tiling cells, with an
isosceles-trapezoidal
pyramid
vertex figure.
Omnitruncated hexagonal tiling honeycomb Omnitruncated order-6 tetrahedral honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,1,2,3{6,3,3} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
tr{3,3}
![]() {}x{6} ![]() {}x{12} ![]() tr{6,3} ![]() |
Faces |
square {4} hexagon {6} dodecagon {12} |
Vertex figure |
![]() irregular tetrahedron |
Coxeter groups | , [3,3,6] |
Properties | Vertex-transitive |
The omnitruncated hexagonal tiling honeycomb or omnitruncated order-6 tetrahedral honeycomb, t0,1,2,3{6,3,3}, has
truncated octahedron,
hexagonal prism,
dodecagonal prism, and
truncated trihexagonal tiling cells, with an irregular
tetrahedron
vertex figure.