Order-4 hexagonal tiling honeycomb | |
---|---|
![]() Perspective projection view within Poincaré disk model | |
Type |
Hyperbolic regular honeycomb Paracompact uniform honeycomb |
Schläfli symbols | {6,3,4} {6,31,1} t0,1{(3,6)2} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{6,3}
![]() ![]() ![]() |
Faces | hexagon {6} |
Edge figure | square {4} |
Vertex figure |
![]() octahedron |
Dual | Order-6 cubic honeycomb |
Coxeter groups | , [4,3,6] , [6,31,1 , [(6,3)[2]] |
Properties | Regular, quasiregular |
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.
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 Schläfli symbol of the order-4 hexagonal tiling honeycomb is {6,3,4}. Since that of the hexagonal tiling is {6,3}, this honeycomb has four such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the octahedron is {3,4}, the vertex figure of this honeycomb is an octahedron. Thus, eight hexagonal tilings meet at each vertex of this honeycomb, and the six edges meeting at each vertex lie along three orthogonal axes. [1]
![]() Perspective projection |
![]() One cell, viewed from outside the Poincare sphere |
![]() The vertices of a t{(3,∞,3)}, ![]() ![]() ![]() ![]() |
![]() The honeycomb is analogous to the H2 order-4 apeirogonal tiling, {∞,4}, shown here with one green apeirogon outlined by its horocycle |
The order-4 hexagonal tiling honeycomb has three reflective simplex symmetry constructions.
The half-symmetry uniform construction {6,31,1} has two types (colors) of hexagonal tilings, with
Coxeter diagram ↔
. A quarter-symmetry construction also exists, with four colors of hexagonal tilings:
.
An additional two reflective symmetries exist with non-simplectic fundamental domains: [6,3*,4], which is index 6, with
Coxeter diagram ; and [6,(3,4)*], which is index 48. The latter has a
cubic fundamental domain, and an
octahedral
Coxeter diagram with three axial infinite branches:
. It can be seen as using eight colors to color the hexagonal tilings of the honeycomb.
The order-4 hexagonal tiling honeycomb contains , which tile 2-
hypercycle surfaces and are similar to the
truncated infinite-order triangular tiling,
:
The order-4 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} |
There are fifteen uniform honeycombs in the [6,3,4] Coxeter group family, including this regular form, and its dual, the order-6 cubic honeycomb.
[6,3,4] family honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
{6,3,4} | r{6,3,4} | t{6,3,4} | rr{6,3,4} | t0,3{6,3,4} | tr{6,3,4} | t0,1,3{6,3,4} | t0,1,2,3{6,3,4} | ||||
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{4,3,6} | r{4,3,6} | t{4,3,6} | rr{4,3,6} | 2t{4,3,6} | tr{4,3,6} | t0,1,3{4,3,6} | t0,1,2,3{4,3,6} |
The order-4 hexagonal tiling honeycomb has a related
alternated honeycomb, ↔
, with
triangular tiling and
octahedron cells.
It is a part of sequence of regular honeycombs of the form {6,3,p}, all of which are 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,∞} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This honeycomb is also related to the 16-cell, cubic honeycomb and order-4 dodecahedral honeycomb, all of which have octahedral vertex figures.
{p,3,4} regular honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | S3 | E3 | H3 | ||||||||
Form | Finite | Affine | Compact | Paracompact | Noncompact | ||||||
Name |
{3,3,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{4,3,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{5,3,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{6,3,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{7,3,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{8,3,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
...
{∞,3,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
Image |
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Cells |
![]() {3,3} ![]() ![]() ![]() ![]() ![]() |
![]() {4,3} ![]() ![]() ![]() ![]() ![]() |
![]() {5,3} ![]() ![]() ![]() ![]() ![]() |
![]() {6,3} ![]() ![]() ![]() ![]() ![]() |
![]() {7,3} ![]() ![]() ![]() ![]() ![]() |
![]() {8,3} ![]() ![]() ![]() ![]() ![]() |
![]() {∞,3} ![]() ![]() ![]() ![]() ![]() |
The aforementioned honeycombs are also quasiregular:
Regular and Quasiregular honeycombs: {p,3,4} and {p,31,1} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | Euclidean 4-space | Euclidean 3-space | Hyperbolic 3-space | ||||||||
Name | {3,3,4} {3,31,1} = |
{4,3,4} {4,31,1} = |
{5,3,4} {5,31,1} = |
{6,3,4} {6,31,1} = | |||||||
Coxeter diagram |
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Image |
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Cells {p,3} |
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Rectified order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbols | r{6,3,4} or t1{6,3,4} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,4}
![]() r{6,3} ![]() |
Faces |
triangle {3} hexagon {6} |
Vertex figure |
![]() square prism |
Coxeter groups | , [4,3,6] , [4,3[3] , [6,31,1 , [3[]×[]] |
Properties | Vertex-transitive, edge-transitive |
The rectified order-4 hexagonal tiling honeycomb, t1{6,3,4}, has
octahedral and
trihexagonal tiling facets, with a
square prism
vertex figure.
It is similar to the 2D hyperbolic tetraapeirogonal tiling, r{∞,4}, which alternates apeirogonal and square faces:
Truncated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t{6,3,4} or t0,1{6,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,4}
![]() t{6,3} ![]() |
Faces |
triangle {3} dodecagon {12} |
Vertex figure |
![]() square pyramid |
Coxeter groups | , [4,3,6] , [6,31,1] |
Properties | Vertex-transitive |
The truncated order-4 hexagonal tiling honeycomb, t0,1{6,3,4}, has
octahedron and
truncated hexagonal tiling facets, with a
square pyramid
vertex figure.
It is similar to the 2D hyperbolic
truncated order-4 apeirogonal tiling, t{∞,4}, with apeirogonal and square faces:
Bitruncated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | 2t{6,3,4} or t1,2{6,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
t{4,3}
![]() t{3,6} ![]() |
Faces |
square {4} hexagon {6} |
Vertex figure |
![]() digonal disphenoid |
Coxeter groups | , [4,3,6] , [4,3[3] , [6,31,1 , [3[]×[]] |
Properties | Vertex-transitive |
The bitruncated order-4 hexagonal tiling honeycomb, t1,2{6,3,4}, has
truncated octahedron and
hexagonal tiling cells, with a
digonal disphenoid
vertex figure.
Cantellated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | rr{6,3,4} or t0,2{6,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
r{3,4}
![]() {}x{4} ![]() rr{6,3} ![]() |
Faces |
triangle {3} square {4} hexagon {6} |
Vertex figure |
![]() wedge |
Coxeter groups | , [4,3,6] , [6,31,1] |
Properties | Vertex-transitive |
The cantellated order-4 hexagonal tiling honeycomb, t0,2{6,3,4}, has
cuboctahedron,
cube, and
rhombitrihexagonal tiling cells, with a
wedge
vertex figure.
Cantitruncated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | tr{6,3,4} or t0,1,2{6,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
t{3,4}
![]() {}x{4} ![]() tr{6,3} ![]() |
Faces |
square {4} hexagon {6} dodecagon {12} |
Vertex figure |
![]() mirrored sphenoid |
Coxeter groups | , [4,3,6] , [6,31,1] |
Properties | Vertex-transitive |
The cantitruncated order-4 hexagonal tiling honeycomb, t0,1,2{6,3,4}, has
truncated octahedron,
cube, and
truncated trihexagonal tiling cells, with a
mirrored sphenoid
vertex figure.
Runcinated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,3{6,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{4,3}
![]() {}x{4} ![]() {6,3} ![]() {}x{6} ![]() |
Faces |
square {4} hexagon {6} |
Vertex figure |
![]() irregular triangular antiprism |
Coxeter groups | , [4,3,6] |
Properties | Vertex-transitive |
The runcinated order-4 hexagonal tiling honeycomb, t0,3{6,3,4}, has
cube,
hexagonal tiling and
hexagonal prism cells, with an irregular
triangular antiprism
vertex figure.
It contains the 2D hyperbolic
rhombitetrahexagonal tiling, rr{4,6}, with square and hexagonal faces. The tiling also has a half symmetry construction
.
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---|
Runcitruncated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,1,3{6,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
rr{3,4}
![]() {}x{4} ![]() {}x{12} ![]() t{6,3} ![]() |
Faces |
triangle {3} square {4} dodecagon {12} |
Vertex figure |
![]() isosceles-trapezoidal pyramid |
Coxeter groups | , [4,3,6] |
Properties | Vertex-transitive |
The runcitruncated order-4 hexagonal tiling honeycomb, t0,1,3{6,3,4}, has
rhombicuboctahedron,
cube,
dodecagonal prism, and
truncated hexagonal tiling cells, with an
isosceles-trapezoidal
pyramid
vertex figure.
The runcicantellated order-4 hexagonal tiling honeycomb is the same as the runcitruncated order-6 cubic honeycomb.
Omnitruncated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,1,2,3{6,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
tr{4,3}
![]() tr{6,3} ![]() {}x{12} ![]() {}x{8} ![]() |
Faces |
square {4} hexagon {6} octagon {8} dodecagon {12} |
Vertex figure |
![]() irregular tetrahedron |
Coxeter groups | , [4,3,6] |
Properties | Vertex-transitive |
The omnitruncated order-4 hexagonal tiling honeycomb, t0,1,2,3{6,3,4}, has
truncated cuboctahedron,
truncated trihexagonal tiling,
dodecagonal prism, and
octagonal prism cells, with an irregular
tetrahedron
vertex figure.
Alternated order-4 hexagonal tiling honeycomb | |
---|---|
Type |
Paracompact uniform honeycomb Semiregular honeycomb |
Schläfli symbols | h{6,3,4} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3[3]}
![]() {3,4} ![]() |
Faces | triangle {3} |
Vertex figure |
![]() truncated octahedron |
Coxeter groups | , [4,3[3]] |
Properties | Vertex-transitive, edge-transitive, quasiregular |
The alternated order-4 hexagonal tiling honeycomb, ↔
, is composed of
triangular tiling and
octahedron cells, in a
truncated octahedron
vertex figure.
Cantic order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbols | h2{6,3,4} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
h2{6,3}
![]() t{3,4} ![]() r{3,4} ![]() |
Faces |
triangle {3} square {4} hexagon {6} |
Vertex figure |
![]() wedge |
Coxeter groups | , [4,3[3]] |
Properties | Vertex-transitive |
The cantic order-4 hexagonal tiling honeycomb, ↔
, is composed of
trihexagonal tiling,
truncated octahedron, and
cuboctahedron cells, with a
wedge
vertex figure.
Runcic order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbols | h3{6,3,4} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3[3]}
![]() rr{3,4} ![]() {4,3} ![]() {}x{3} ![]() |
Faces |
triangle {3} square {4} |
Vertex figure |
![]() triangular cupola |
Coxeter groups | , [4,3[3]] |
Properties | Vertex-transitive |
The runcic order-4 hexagonal tiling honeycomb, ↔
, is composed of
triangular tiling,
rhombicuboctahedron,
cube, and
triangular prism cells, with a
triangular cupola
vertex figure.
Runcicantic order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbols | h2,3{6,3,4} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
h2{6,3}
![]() tr{3,4} ![]() t{4,3} ![]() {}x{3} ![]() |
Faces |
triangle {3} square {4} hexagon {6} octagon {8} |
Vertex figure |
![]() rectangular pyramid |
Coxeter groups | , [4,3[3]] |
Properties | Vertex-transitive |
The runcicantic order-4 hexagonal tiling honeycomb, ↔
, is composed of
trihexagonal tiling,
truncated cuboctahedron,
truncated cube, and
triangular prism cells, with a
rectangular
pyramid
vertex figure.
Quarter order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | q{6,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3[3]}
![]() {3,3} ![]() t{3,3} ![]() h2{6,3} ![]() |
Faces |
triangle {3} hexagon {6} |
Vertex figure |
![]() triangular cupola |
Coxeter groups | , [3[]x[]] |
Properties | Vertex-transitive |
The quarter order-4 hexagonal tiling honeycomb, q{6,3,4}, or
, is composed of
triangular tiling,
trihexagonal tiling,
tetrahedron, and
truncated tetrahedron cells, with a
triangular cupola
vertex figure.
Order-4 hexagonal tiling honeycomb | |
---|---|
![]() Perspective projection view within Poincaré disk model | |
Type |
Hyperbolic regular honeycomb Paracompact uniform honeycomb |
Schläfli symbols | {6,3,4} {6,31,1} t0,1{(3,6)2} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{6,3}
![]() ![]() ![]() |
Faces | hexagon {6} |
Edge figure | square {4} |
Vertex figure |
![]() octahedron |
Dual | Order-6 cubic honeycomb |
Coxeter groups | , [4,3,6] , [6,31,1 , [(6,3)[2]] |
Properties | Regular, quasiregular |
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.
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 Schläfli symbol of the order-4 hexagonal tiling honeycomb is {6,3,4}. Since that of the hexagonal tiling is {6,3}, this honeycomb has four such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the octahedron is {3,4}, the vertex figure of this honeycomb is an octahedron. Thus, eight hexagonal tilings meet at each vertex of this honeycomb, and the six edges meeting at each vertex lie along three orthogonal axes. [1]
![]() Perspective projection |
![]() One cell, viewed from outside the Poincare sphere |
![]() The vertices of a t{(3,∞,3)}, ![]() ![]() ![]() ![]() |
![]() The honeycomb is analogous to the H2 order-4 apeirogonal tiling, {∞,4}, shown here with one green apeirogon outlined by its horocycle |
The order-4 hexagonal tiling honeycomb has three reflective simplex symmetry constructions.
The half-symmetry uniform construction {6,31,1} has two types (colors) of hexagonal tilings, with
Coxeter diagram ↔
. A quarter-symmetry construction also exists, with four colors of hexagonal tilings:
.
An additional two reflective symmetries exist with non-simplectic fundamental domains: [6,3*,4], which is index 6, with
Coxeter diagram ; and [6,(3,4)*], which is index 48. The latter has a
cubic fundamental domain, and an
octahedral
Coxeter diagram with three axial infinite branches:
. It can be seen as using eight colors to color the hexagonal tilings of the honeycomb.
The order-4 hexagonal tiling honeycomb contains , which tile 2-
hypercycle surfaces and are similar to the
truncated infinite-order triangular tiling,
:
The order-4 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} |
There are fifteen uniform honeycombs in the [6,3,4] Coxeter group family, including this regular form, and its dual, the order-6 cubic honeycomb.
[6,3,4] family honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
{6,3,4} | r{6,3,4} | t{6,3,4} | rr{6,3,4} | t0,3{6,3,4} | tr{6,3,4} | t0,1,3{6,3,4} | t0,1,2,3{6,3,4} | ||||
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{4,3,6} | r{4,3,6} | t{4,3,6} | rr{4,3,6} | 2t{4,3,6} | tr{4,3,6} | t0,1,3{4,3,6} | t0,1,2,3{4,3,6} |
The order-4 hexagonal tiling honeycomb has a related
alternated honeycomb, ↔
, with
triangular tiling and
octahedron cells.
It is a part of sequence of regular honeycombs of the form {6,3,p}, all of which are 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,∞} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This honeycomb is also related to the 16-cell, cubic honeycomb and order-4 dodecahedral honeycomb, all of which have octahedral vertex figures.
{p,3,4} regular honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | S3 | E3 | H3 | ||||||||
Form | Finite | Affine | Compact | Paracompact | Noncompact | ||||||
Name |
{3,3,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{4,3,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{5,3,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{6,3,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{7,3,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
{8,3,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
...
{∞,3,4}![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||
Image |
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Cells |
![]() {3,3} ![]() ![]() ![]() ![]() ![]() |
![]() {4,3} ![]() ![]() ![]() ![]() ![]() |
![]() {5,3} ![]() ![]() ![]() ![]() ![]() |
![]() {6,3} ![]() ![]() ![]() ![]() ![]() |
![]() {7,3} ![]() ![]() ![]() ![]() ![]() |
![]() {8,3} ![]() ![]() ![]() ![]() ![]() |
![]() {∞,3} ![]() ![]() ![]() ![]() ![]() |
The aforementioned honeycombs are also quasiregular:
Regular and Quasiregular honeycombs: {p,3,4} and {p,31,1} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | Euclidean 4-space | Euclidean 3-space | Hyperbolic 3-space | ||||||||
Name | {3,3,4} {3,31,1} = |
{4,3,4} {4,31,1} = |
{5,3,4} {5,31,1} = |
{6,3,4} {6,31,1} = | |||||||
Coxeter diagram |
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Image |
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Cells {p,3} |
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Rectified order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbols | r{6,3,4} or t1{6,3,4} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,4}
![]() r{6,3} ![]() |
Faces |
triangle {3} hexagon {6} |
Vertex figure |
![]() square prism |
Coxeter groups | , [4,3,6] , [4,3[3] , [6,31,1 , [3[]×[]] |
Properties | Vertex-transitive, edge-transitive |
The rectified order-4 hexagonal tiling honeycomb, t1{6,3,4}, has
octahedral and
trihexagonal tiling facets, with a
square prism
vertex figure.
It is similar to the 2D hyperbolic tetraapeirogonal tiling, r{∞,4}, which alternates apeirogonal and square faces:
Truncated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t{6,3,4} or t0,1{6,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3,4}
![]() t{6,3} ![]() |
Faces |
triangle {3} dodecagon {12} |
Vertex figure |
![]() square pyramid |
Coxeter groups | , [4,3,6] , [6,31,1] |
Properties | Vertex-transitive |
The truncated order-4 hexagonal tiling honeycomb, t0,1{6,3,4}, has
octahedron and
truncated hexagonal tiling facets, with a
square pyramid
vertex figure.
It is similar to the 2D hyperbolic
truncated order-4 apeirogonal tiling, t{∞,4}, with apeirogonal and square faces:
Bitruncated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | 2t{6,3,4} or t1,2{6,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
t{4,3}
![]() t{3,6} ![]() |
Faces |
square {4} hexagon {6} |
Vertex figure |
![]() digonal disphenoid |
Coxeter groups | , [4,3,6] , [4,3[3] , [6,31,1 , [3[]×[]] |
Properties | Vertex-transitive |
The bitruncated order-4 hexagonal tiling honeycomb, t1,2{6,3,4}, has
truncated octahedron and
hexagonal tiling cells, with a
digonal disphenoid
vertex figure.
Cantellated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | rr{6,3,4} or t0,2{6,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
r{3,4}
![]() {}x{4} ![]() rr{6,3} ![]() |
Faces |
triangle {3} square {4} hexagon {6} |
Vertex figure |
![]() wedge |
Coxeter groups | , [4,3,6] , [6,31,1] |
Properties | Vertex-transitive |
The cantellated order-4 hexagonal tiling honeycomb, t0,2{6,3,4}, has
cuboctahedron,
cube, and
rhombitrihexagonal tiling cells, with a
wedge
vertex figure.
Cantitruncated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | tr{6,3,4} or t0,1,2{6,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
t{3,4}
![]() {}x{4} ![]() tr{6,3} ![]() |
Faces |
square {4} hexagon {6} dodecagon {12} |
Vertex figure |
![]() mirrored sphenoid |
Coxeter groups | , [4,3,6] , [6,31,1] |
Properties | Vertex-transitive |
The cantitruncated order-4 hexagonal tiling honeycomb, t0,1,2{6,3,4}, has
truncated octahedron,
cube, and
truncated trihexagonal tiling cells, with a
mirrored sphenoid
vertex figure.
Runcinated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,3{6,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{4,3}
![]() {}x{4} ![]() {6,3} ![]() {}x{6} ![]() |
Faces |
square {4} hexagon {6} |
Vertex figure |
![]() irregular triangular antiprism |
Coxeter groups | , [4,3,6] |
Properties | Vertex-transitive |
The runcinated order-4 hexagonal tiling honeycomb, t0,3{6,3,4}, has
cube,
hexagonal tiling and
hexagonal prism cells, with an irregular
triangular antiprism
vertex figure.
It contains the 2D hyperbolic
rhombitetrahexagonal tiling, rr{4,6}, with square and hexagonal faces. The tiling also has a half symmetry construction
.
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---|
Runcitruncated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,1,3{6,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
rr{3,4}
![]() {}x{4} ![]() {}x{12} ![]() t{6,3} ![]() |
Faces |
triangle {3} square {4} dodecagon {12} |
Vertex figure |
![]() isosceles-trapezoidal pyramid |
Coxeter groups | , [4,3,6] |
Properties | Vertex-transitive |
The runcitruncated order-4 hexagonal tiling honeycomb, t0,1,3{6,3,4}, has
rhombicuboctahedron,
cube,
dodecagonal prism, and
truncated hexagonal tiling cells, with an
isosceles-trapezoidal
pyramid
vertex figure.
The runcicantellated order-4 hexagonal tiling honeycomb is the same as the runcitruncated order-6 cubic honeycomb.
Omnitruncated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,1,2,3{6,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
tr{4,3}
![]() tr{6,3} ![]() {}x{12} ![]() {}x{8} ![]() |
Faces |
square {4} hexagon {6} octagon {8} dodecagon {12} |
Vertex figure |
![]() irregular tetrahedron |
Coxeter groups | , [4,3,6] |
Properties | Vertex-transitive |
The omnitruncated order-4 hexagonal tiling honeycomb, t0,1,2,3{6,3,4}, has
truncated cuboctahedron,
truncated trihexagonal tiling,
dodecagonal prism, and
octagonal prism cells, with an irregular
tetrahedron
vertex figure.
Alternated order-4 hexagonal tiling honeycomb | |
---|---|
Type |
Paracompact uniform honeycomb Semiregular honeycomb |
Schläfli symbols | h{6,3,4} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3[3]}
![]() {3,4} ![]() |
Faces | triangle {3} |
Vertex figure |
![]() truncated octahedron |
Coxeter groups | , [4,3[3]] |
Properties | Vertex-transitive, edge-transitive, quasiregular |
The alternated order-4 hexagonal tiling honeycomb, ↔
, is composed of
triangular tiling and
octahedron cells, in a
truncated octahedron
vertex figure.
Cantic order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbols | h2{6,3,4} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
h2{6,3}
![]() t{3,4} ![]() r{3,4} ![]() |
Faces |
triangle {3} square {4} hexagon {6} |
Vertex figure |
![]() wedge |
Coxeter groups | , [4,3[3]] |
Properties | Vertex-transitive |
The cantic order-4 hexagonal tiling honeycomb, ↔
, is composed of
trihexagonal tiling,
truncated octahedron, and
cuboctahedron cells, with a
wedge
vertex figure.
Runcic order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbols | h3{6,3,4} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3[3]}
![]() rr{3,4} ![]() {4,3} ![]() {}x{3} ![]() |
Faces |
triangle {3} square {4} |
Vertex figure |
![]() triangular cupola |
Coxeter groups | , [4,3[3]] |
Properties | Vertex-transitive |
The runcic order-4 hexagonal tiling honeycomb, ↔
, is composed of
triangular tiling,
rhombicuboctahedron,
cube, and
triangular prism cells, with a
triangular cupola
vertex figure.
Runcicantic order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbols | h2,3{6,3,4} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
h2{6,3}
![]() tr{3,4} ![]() t{4,3} ![]() {}x{3} ![]() |
Faces |
triangle {3} square {4} hexagon {6} octagon {8} |
Vertex figure |
![]() rectangular pyramid |
Coxeter groups | , [4,3[3]] |
Properties | Vertex-transitive |
The runcicantic order-4 hexagonal tiling honeycomb, ↔
, is composed of
trihexagonal tiling,
truncated cuboctahedron,
truncated cube, and
triangular prism cells, with a
rectangular
pyramid
vertex figure.
Quarter order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | q{6,3,4} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Cells |
{3[3]}
![]() {3,3} ![]() t{3,3} ![]() h2{6,3} ![]() |
Faces |
triangle {3} hexagon {6} |
Vertex figure |
![]() triangular cupola |
Coxeter groups | , [3[]x[]] |
Properties | Vertex-transitive |
The quarter order-4 hexagonal tiling honeycomb, q{6,3,4}, or
, is composed of
triangular tiling,
trihexagonal tiling,
tetrahedron, and
truncated tetrahedron cells, with a
triangular cupola
vertex figure.