Square tiling honeycomb

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Square tiling honeycomb
H3 443 FC boundary.png
Type Hyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbols {4,4,3}
r{4,4,4}
{41,1,1}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel split1-44.pngCDel nodes.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.png
CDel nodes 11.pngCDel 2a2b-cross.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node 1.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel branchu 11.pngCDel 2.pngCDel branchu 11.pngCDel 2.pngCDel branchu 11.pngCDel 2.pngCDel branchu 11.pngCDel node 1.pngCDel 4.pngCDel node g.pngCDel 4sg.pngCDel node g.pngCDel 3g.pngCDel node g.png
Cells {4,4} Square tiling uniform coloring 1.png Square tiling uniform coloring 9.png Square tiling uniform coloring 7.png
Faces Square {4}
Edge figure Triangle {3}
Vertex figure Square tiling honeycomb verf.png
cube, {4,3}
Dual Order-4 octahedral honeycomb
Coxeter groups [4,4,3]
[41,1,1] ↔ [4,4,3*]
Properties Regular

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}, has three square tilings, {4,4} around each edge, and 6 square tilings around each vertex in an cubic {4,3} vertex figure.[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.

Rectified order-4 square tiling[edit]

It is also seen as a rectified order-4 square tiling honeycomb, r{4,4,4}:

{4,4,4} r{4,4,4} = {4,4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
H3 444 FC boundary.png H3 444 boundary 0100.png

Symmetry[edit]

It has three reflective symmetry constructions, CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png as a regular honeycomb, a half symmetry CDel node 1.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel nodes 11.pngCDel 2a2b-cross.pngCDel nodes.pngCDel split2.pngCDel node.png and lastly CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel split1-44.pngCDel nodes.png with 3 types (colors) of checkered square tilings. [4,4,3*] ↔ [41,1,1], index 6, and a final radial subgroup [4,(4,3)*], index 48, with a right dihedral angled octahedral fundamental domain, and 4 pairs of ultraparallel mirrors: CDel branchu 11.pngCDel 2.pngCDel branchu 11.pngCDel 2.pngCDel branchu 11.pngCDel 2.pngCDel branchu 11.png.

This honeycomb contains CDel node.pngCDel 3.pngCDel node.pngCDel ultra.pngCDel node 1.png that tile 2-hypercycle surfaces, similar to this paracompact tiling, CDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node 1.png:

H2 tiling 23i-1.png

Related polytopes and honeycombs[edit]

It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures.

There are fifteen uniform honeycombs in the [4,4,3] Coxeter group family, including this regular form, and its dual, the order-4 octahedral honeycomb, {3,4,4}.

[4,4,3] family honeycombs
{4,4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
r{4,4,3}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
t{4,4,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
rr{4,4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,3{4,4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
tr{4,4,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,1,3{4,4,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,2,3{4,4,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
H3 443 FC boundary.png H3 443 boundary 0100.png H3 443-1100.png H3 443-1010.png H3 443-1001.png H3 443-1110.png H3 443-1101.png H3 443-1111.png
H3 344 CC center.png H3 344 CC center 0100.png H3 443-0011.png H3 443-0101.png H3 443-0110.png H3 443-0111.png H3 443-1011.png
{3,4,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
r{3,4,4}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
t{3,4,4}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
rr{3,4,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
2t{3,4,4}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
tr{3,4,4}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
t0,1,3{3,4,4}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
t0,1,2,3{3,4,4}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png

This honeycomb is related to the 24-cell, {3,4,3}, with a cubic vertex figure.

It is a part of a sequence of honeycombs with square tiling cells:

Rectified square tiling honeycomb[edit]

Rectified square tiling honeycomb
Type Paracompact uniform honeycomb
Semiregular honeycomb
Schläfli symbols r{4,4,3} or t1{4,4,3}
2r{3,41,1}
r{41,1,1}
Coxeter diagrams CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel nodes 11.pngCDel split2-44.pngCDel node.pngCDel 3.pngCDel node.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel nodes 11.pngCDel split2-44.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.png
CDel node 1.pngCDel split1-uu.pngCDel nodes 11.pngCDel 2a2b-cross.pngCDel nodes 11.pngCDel split2-uu.pngCDel node 1.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.png
Cells {4,3} Uniform polyhedron-43-t0.png
r{4,4}Uniform tiling 44-t1.png
Faces square {4}
Vertex figure Rectified square tiling honeycomb verf.png
Coxeter groups [4,4,3]
[41,1,1] ↔ [4,4,3*]
Properties Vertex-transitive, edge-transitive

The rectified square tiling honeycomb, t1{4,4,3}, CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png has cube and square tiling facets, with a triangular prism vertex figure.

H3 443 boundary 0100.png

It is similar to the 2D hyperbolic uniform triapeirogonal tiling, r{∞,3}, with triangle and apeirogonal faces.

H2 tiling 23i-2.png

Truncated square tiling honeycomb[edit]

Truncated square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t{4,4,3} or t0,1{4,4,3}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1-44.pngCDel nodes 11.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node h0.png
CDel nodes 11.pngCDel split2-44.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.png
Cells {4,3} Uniform polyhedron-43-t0.png
t{4,4}Uniform tiling 44-t01.png
Faces square {4}
octagon {8}
Vertex figure Truncated square tiling honeycomb verf.png
triangular pyramid
Coxeter groups [4,4,3]
Properties Vertex-transitive

The truncated square tiling honeycomb, t{4,4,3}, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png has cube and truncated square tiling facets, with a triangular pyramid vertex figure. It is the same as the cantitruncated order-4 square tiling honeycomb, tr{4,4,4}, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png.

H3 443-1100.png

Bitruncated square tiling honeycomb[edit]

Bitruncated square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols 2t{4,4,3} or t1,2{4,4,3}
Coxeter diagram CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells t{4,3} Uniform polyhedron-43-t01.png
t{4,4}Uniform tiling 44-t01.png
Faces triangular {3}
square {4}
octagon {8}
Vertex figure Bitruncated square tiling honeycomb verf.png
digonal disphenoid
Coxeter groups [4,4,3]
Properties Vertex-transitive

The bitruncated square tiling honeycomb, 2t{4,4,3}, CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png has truncated cube and truncated square tiling facets, with a digonal disphenoid vertex figure.

H3 443-0110.png

Cantellated square tiling honeycomb[edit]

Cantellated square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols rr{4,4,3} or t0,2{4,4,3}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel nodes 11.pngCDel 2a2b-cross.pngCDel nodes 11.pngCDel split2.pngCDel node.pngCDel node 1.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells r{4,3} Uniform polyhedron-43-t1.png
rr{4,4}Uniform tiling 44-t02.png
Faces triangular {3}
square {4}
Vertex figure Cantellated square tiling honeycomb verf.png
triangular prism
Coxeter groups [4,4,3]
Properties Vertex-transitive

The cantellated square tiling honeycomb, rr{4,4,3}, CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png has cube and truncated square tiling facets, with a triangular prism vertex figure.

H3 443-1010.png

Cantitruncated square tiling honeycomb[edit]

Cantitruncated square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols tr{4,4,3} or t0,1,2{4,4,3}
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells t{4,3} Uniform polyhedron-43-t01.png
tr{4,4}Uniform tiling 44-t012.png
{}x{3} Triangular prism.png
Faces triangular {3}
square {4}
octagon {8}
Vertex figure Cantitruncated square tiling honeycomb verf.png
tetrahedron
Coxeter groups [4,4,3]
Properties Vertex-transitive

The cantitruncated square tiling honeycomb, tr{4,4,3}, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png has truncated cube and truncated square tiling facets, with a tetrahedron vertex figure.

H3 443-1110.png

Runcinated square tiling honeycomb[edit]

Runcinated square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,3{4,4,3}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel nodes 11.pngCDel 2a2b-cross.pngCDel nodes.pngCDel split2.pngCDel node 1.pngCDel node 1.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells {3,4} Uniform polyhedron-43-t2.png
{4,4}Uniform tiling 44-t0.svg
{}x{4} Hexahedron.png
{}x{3} Triangular prism.png
Faces triangle {3}
square {4}
Vertex figure Runcinated square tiling honeycomb verf.png
triangular antiprism
Coxeter groups [4,4,3]
Properties Vertex-transitive

The runcinated square tiling honeycomb, t0,3{4,4,3}, CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png has octahedron, triangular prism, cube, and square tiling facets, with a triangular antiprism vertex figure.

H3 443-1001.png

Runcitruncated square tiling honeycomb[edit]

Runcitruncated square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t0,1,3{4,4,3}
s2,3{3,4,4}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
Cells rr{4,3} Uniform polyhedron-43-t02.png
t{4,4}Uniform tiling 44-t01.png
{}x{3} Triangular prism.png
{}x{8} Octagonal prism.png
Faces triangle {3}
square {4}
Vertex figure Runcitruncated square tiling honeycomb verf.png
trapezoidal pyramid
Coxeter groups [4,4,3]
Properties Vertex-transitive

The runcitruncated square tiling honeycomb, t0,1,3{4,4,3}, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png has rhombicuboctahedron, octagonal prism, triangular prism and truncated square tiling facets, with a trapezoidal pyramid vertex figure.

H3 443-1101.png

Omnitruncated square tiling honeycomb[edit]

Omnitruncated square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,1,2,3{4,4,3}
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells {4,4} Uniform tiling 44-t012.png
{}x{6} Hexagonal prism.png
{}x{8} Octagonal prism.png
tr{4,3} Uniform polyhedron-43-t012.png
Faces Square {4}
Hexagon {6}
Octagon {8}
Vertex figure Omnitruncated square tiling honeycomb verf.png
tetrahedron
Coxeter groups [4,4,3]
Properties Vertex-transitive

The omnitruncated square tiling honeycomb, t0,1,2,3{4,4,3}, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png has truncated square tiling, truncated cuboctahedron, hexagonal prism, octagonal prism facets, with a tetrahedron vertex figure.

H3 443-1111.png

Omnisnub square tiling honeycomb[edit]

Omnisnub square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol h(t0,1,2,3{4,4,3})
Coxeter diagram CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
Cells sr{4,4} Uniform tiling 44-snub.png
sr{2,3} Octahedron.png
sr{2,4} Square antiprism.png
sr{4,3} Uniform polyhedron-43-s012.png
Faces Triangular {3}
Square {4}
Vertex figure Irr. tetrahedron
Coxeter group [4,4,3]+
Properties Nonuniform vertex-transitive

The alternated omnitruncated square tiling honeycomb (or omnisnub square tiling honeycomb), h(t0,1,2,3{4,4,3}), CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png has snub square tiling, snub cube, triangular antiprism, square antiprism, and tetrahedron cells, with an irregular vertex figure.

Alternated square tiling honeycomb[edit]

Alternated square tiling honeycomb
Type Paracompact uniform honeycomb
Semiregular honeycomb
Schläfli symbol h{4,4,3}
hr{4,4,4}
{(4,3,3,4)}
h{41,1,1}
Coxeter diagrams CDel nodes 10ru.pngCDel split2-44.pngCDel node.pngCDel 3.pngCDel node.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel nodes 10.pngCDel 2a2b-cross.pngCDel nodes 10ru.pngCDel split2-44.pngCDel node.png
CDel node 1.pngCDel split1-44.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel split1-43.pngCDel nodes 10lu.png
CDel node h.pngCDel split1-44.pngCDel nodes.pngCDel split2-44.pngCDel node h.pngCDel node h0.pngCDel 4.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.png
CDel nodes.pngCDel split2-44.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel node.pngCDel 4.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node 1.pngCDel split1-uu.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes 11.pngCDel split2-uu.pngCDel node.png
Cells Uniform tiling 44-t0.svg (4.4.4.4)
Uniform polyhedron-43-t0.png (4.4.4)
Faces Square {4}
Vertex figure Uniform polyhedron-43-t1.png (4.3.4.3)
Coxeter groups [1+,4,4,3] ↔ [3,41,1]
[4,1+,4,4] ↔ [∞,4,4,∞]
[(4,4,3,3)]
[1+,41,1,1] ↔ [∞[6]]
Properties vertex-transitive, edge-transitive, quasiregular

The alternated square tiling honeycomb is a paracompact uniform honeycomb in hyperbolic 3-space, composed of cube, and square tiling facets in a cuboctahedron vertex figure.

Alternated rectified square tiling honeycomb[edit]

Alternated rectified square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol hr{4,4,3}
Coxeter diagrams CDel node.pngCDel 4.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel nodes 10.pngCDel 2a2b-cross.pngCDel nodes 10ru.pngCDel split2.pngCDel node.png
Cells
Faces
Vertex figure Triangular prism
Coxeter groups [4,1+,4,3] = [∞,3,3,∞]
Properties Nonsimplectic, vertex-transitive

The alternated rectified square tiling honeycomb is a paracompact uniform honeycomb in hyperbolic 3-space.

See also[edit]

References[edit]

  1. ^ Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III