Order-4 hexagonal tiling honeycomb

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Order-4 hexagonal tiling honeycomb

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.

Order-4 hexagonal tiling honeycomb

Perspective projection view
within Poincaré disk model
TypeHyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbols{6,3,4}
{6,31,1}
t0,1{(3,6)2}
Coxeter diagrams



Cells{6,3}
Faceshexagon {6}
Edge figuresquare {4}
Vertex figure
octahedron
DualOrder-6 cubic honeycomb
Coxeter groups, [4,3,6]
, [6,31,1]
, [(6,3)[2]]
PropertiesRegular, quasiregular

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]

Images


Perspective projection

One cell, viewed from outside the Poincare sphere

The vertices of a t{(3,,3)}, tiling exist as a 2-hypercycle within this honeycomb

The honeycomb is analogous to the H2 order-4 apeirogonal tiling, {,4}, shown here with one green apeirogon outlined by its horocycle

Symmetry

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Subgroup relations

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, :

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Summarize
Perspective

The order-4 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.

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.

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:

More information Space, H3 ...
{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 Thumb Thumb Thumb Thumb
Vertex
figure
{3,p}

{3,3}

{3,4}


{3,5}

{3,6}


{3,7}

{3,8}


{3,}

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This honeycomb is also related to the 16-cell, cubic honeycomb and order-4 dodecahedral honeycomb, all of which have octahedral vertex figures.

More information {p,3,4} regular honeycombs, Space ...
{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 Thumb Thumb
Cells
{3,3}

{4,3}

{5,3}

{6,3}

{7,3}

{8,3}

{,3}
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The aforementioned honeycombs are also quasiregular:

More information = ...
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
= = = =
Image Thumb Thumb Thumb Thumb
Cells
{p,3}
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Thumb
Thumb
Thumb
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Rectified order-4 hexagonal tiling honeycomb

More information , ...
Rectified order-4 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolsr{6,3,4} or t1{6,3,4}
Coxeter diagrams


Cells{3,4}
r{6,3}
Facestriangle {3}
hexagon {6}
Vertex figureThumb
square prism
Coxeter groups, [4,3,6]
, [4,3[3]]
, [6,31,1]
, [3[]×[]]
PropertiesVertex-transitive, edge-transitive
Close

The rectified order-4 hexagonal tiling honeycomb, t1{6,3,4}, has octahedral and trihexagonal tiling facets, with a square prism vertex figure.

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It is similar to the 2D hyperbolic tetraapeirogonal tiling, r{,4}, which alternates apeirogonal and square faces:

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Truncated order-4 hexagonal tiling honeycomb

More information , ...
Truncated order-4 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt{6,3,4} or t0,1{6,3,4}
Coxeter diagram
Cells{3,4}
t{6,3}
Facestriangle {3}
dodecagon {12}
Vertex figureThumb
square pyramid
Coxeter groups, [4,3,6]
, [6,31,1]
PropertiesVertex-transitive
Close

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.

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It is similar to the 2D hyperbolic truncated order-4 apeirogonal tiling, t{,4}, with apeirogonal and square faces:

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Bitruncated order-4 hexagonal tiling honeycomb

More information , ...
Bitruncated order-4 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbol2t{6,3,4} or t1,2{6,3,4}
Coxeter diagram


Cellst{4,3}
t{3,6}
Facessquare {4}
hexagon {6}
Vertex figureThumb
digonal disphenoid
Coxeter groups, [4,3,6]
, [4,3[3]]
, [6,31,1]
, [3[]×[]]
PropertiesVertex-transitive
Close

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.

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Cantellated order-4 hexagonal tiling honeycomb

More information , ...
Cantellated order-4 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolrr{6,3,4} or t0,2{6,3,4}
Coxeter diagram
Cellsr{3,4}
{}x{4}
rr{6,3}
Facestriangle {3}
square {4}
hexagon {6}
Vertex figureThumb
wedge
Coxeter groups, [4,3,6]
, [6,31,1]
PropertiesVertex-transitive
Close

The cantellated order-4 hexagonal tiling honeycomb, t0,2{6,3,4}, has cuboctahedron, cube, and rhombitrihexagonal tiling cells, with a wedge vertex figure.

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Cantitruncated order-4 hexagonal tiling honeycomb

More information , ...
Cantitruncated order-4 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symboltr{6,3,4} or t0,1,2{6,3,4}
Coxeter diagram
Cellst{3,4}
{}x{4}
tr{6,3}
Facessquare {4}
hexagon {6}
dodecagon {12}
Vertex figureThumb
mirrored sphenoid
Coxeter groups, [4,3,6]
, [6,31,1]
PropertiesVertex-transitive
Close

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.

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Runcinated order-4 hexagonal tiling honeycomb

More information ...
Runcinated order-4 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt0,3{6,3,4}
Coxeter diagram
Cells{4,3}
{}x{4}
{6,3}
{}x{6}
Facessquare {4}
hexagon {6}
Vertex figureThumb
irregular triangular antiprism
Coxeter groups, [4,3,6]
PropertiesVertex-transitive
Close

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.

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It contains the 2D hyperbolic rhombitetrahexagonal tiling, rr{4,6}, with square and hexagonal faces. The tiling also has a half symmetry construction .

More information = ...
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=
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Runcitruncated order-4 hexagonal tiling honeycomb

More information ...
Runcitruncated order-4 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt0,1,3{6,3,4}
Coxeter diagram
Cellsrr{3,4}
{}x{4}
{}x{12}
t{6,3}
Facestriangle {3}
square {4}
dodecagon {12}
Vertex figureThumb
isosceles-trapezoidal pyramid
Coxeter groups, [4,3,6]
PropertiesVertex-transitive
Close

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.

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Runcicantellated order-4 hexagonal tiling honeycomb

The runcicantellated order-4 hexagonal tiling honeycomb is the same as the runcitruncated order-6 cubic honeycomb.

Omnitruncated order-4 hexagonal tiling honeycomb

More information ...
Omnitruncated order-4 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt0,1,2,3{6,3,4}
Coxeter diagram
Cellstr{4,3}
tr{6,3}
{}x{12}
{}x{8}
Facessquare {4}
hexagon {6}
octagon {8}
dodecagon {12}
Vertex figureThumb
irregular tetrahedron
Coxeter groups, [4,3,6]
PropertiesVertex-transitive
Close

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.

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Alternated order-4 hexagonal tiling honeycomb

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Alternated order-4 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Semiregular honeycomb
Schläfli symbolsh{6,3,4}
Coxeter diagrams
Cells{3[3]}
{3,4}
Facestriangle {3}
Vertex figure
truncated octahedron
Coxeter groups, [4,3[3]]
PropertiesVertex-transitive, edge-transitive, quasiregular
Close

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

More information ...
Cantic order-4 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolsh2{6,3,4}
Coxeter diagrams
Cellsh2{6,3}
t{3,4}
r{3,4}
Facestriangle {3}
square {4}
hexagon {6}
Vertex figureThumb
wedge
Coxeter groups, [4,3[3]]
PropertiesVertex-transitive
Close

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

More information ...
Runcic order-4 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolsh3{6,3,4}
Coxeter diagrams
Cells{3[3]}
rr{3,4}
{4,3}
{}x{3}
Facestriangle {3}
square {4}
Vertex figureThumb
triangular cupola
Coxeter groups, [4,3[3]]
PropertiesVertex-transitive
Close

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

More information ...
Runcicantic order-4 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolsh2,3{6,3,4}
Coxeter diagrams
Cellsh2{6,3}
tr{3,4}
t{4,3}
{}x{3}
Facestriangle {3}
square {4}
hexagon {6}
octagon {8}
Vertex figureThumb
rectangular pyramid
Coxeter groups, [4,3[3]]
PropertiesVertex-transitive
Close

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

More information ...
Quarter order-4 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolq{6,3,4}
Coxeter diagram
Cells{3[3]}
{3,3}
t{3,3}
h2{6,3}
Facestriangle {3}
hexagon {6}
Vertex figureThumb
triangular cupola
Coxeter groups, [3[]x[]]
PropertiesVertex-transitive
Close

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.

See also

References

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