Order-4 square tiling honeycomb

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Order-4 square tiling honeycomb
H3 444 FC boundary.png
Type Hyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbols {4,4,4}
h{4,4,4} ↔ {4,41,1}
{4[4]}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel split1-44.pngCDel nodes.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes 10ru.pngCDel split2-44.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node.pngCDel split1-44.pngCDel nodes 10luru.pngCDel split2-44.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel split1-44.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes.png
CDel node 1.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel nodes 11.pngCDel 2a2b-cross.pngCDel nodes.pngCDel split2-44.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node 1.pngCDel split1-uu.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes.pngCDel split2-uu.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node g.pngCDel 4sg.pngCDel node g.pngCDel branchu.pngCDel split2-44.pngCDel node 1.pngCDel split1-44.pngCDel branchu.png
CDel node 1.pngCDel 4.pngCDel node g.pngCDel 4sg.pngCDel node g.pngCDel 4.pngCDel node.pngCDel branchu 10.pngCDel 2.pngCDel branchu 10.pngCDel 2.pngCDel branchu 10.pngCDel 2.pngCDel branchu 10.png
Cells {4,4}
Square tiling uniform coloring 1.png 40px 40px Square tiling uniform coloring 9.png
Faces square {4}
Edge figure square {4}
Vertex figure square tiling, {4,4}
Square tiling uniform coloring 1.png 40px 40px Square tiling uniform coloring 9.png
Dual Self-dual
Coxeter groups [math]\displaystyle{ \overline{N}_3 }[/math], [4,4,4]
[math]\displaystyle{ \overline{M}_3 }[/math], [41,1,1]
[math]\displaystyle{ \widehat{RR}_3 }[/math], [4[4]]
Properties Regular, quasiregular

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.[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.

Symmetry

The order-4 square tiling honeycomb has many reflective symmetry constructions: CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png as a regular honeycomb, CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel split1-44.pngCDel nodes.png with alternating types (colors) of square tilings, and CDel node 1.pngCDel split1-44.pngCDel nodes.pngCDel split2-44.pngCDel node.png with 3 types (colors) of square tilings in a ratio of 2:1:1.

Two more half symmetry constructions with pyramidal domains have [4,4,1+,4] symmetry: CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel split1-44.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes.png, and CDel node 1.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel nodes 11.pngCDel 2a2b-cross.pngCDel nodes.pngCDel split2-44.pngCDel node.png.

There are two high-index subgroups, both index 8: [4,4,4*] ↔ [(4,4,4,4,1+)], with a pyramidal fundamental domain: [((4,∞,4)),((4,∞,4))] or CDel branchu.pngCDel split2-44.pngCDel node 1.pngCDel split1-44.pngCDel branchu.png; and [4,4*,4], with 4 orthogonal sets of ultra-parallel mirrors in an octahedral fundamental domain: CDel branchu 10.pngCDel 2.pngCDel branchu 10.pngCDel 2.pngCDel branchu 10.pngCDel 2.pngCDel branchu 10.png.

Images

The order-4 square tiling honeycomb is analogous to the 2D hyperbolic infinite-order apeirogonal tiling, {∞,∞}, with infinite apeirogonal faces, and with all vertices on the ideal surface.

H2 tiling 2ii-4.png

It contains CDel node.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node 1.png and CDel node 1.pngCDel 4.pngCDel node.pngCDel ultra.pngCDel node.png that tile 2-hypercycle surfaces, which are similar to these paracompact order-4 apeirogonal tilings CDel node.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node 1.png:

H2 tiling 24i-1.png H2 tiling 24i-4.png

Related polytopes and honeycombs

The order-4 square tiling honeycomb is a regular hyperbolic honeycomb in 3-space. It is one of eleven regular paracompact honeycombs.

There are nine uniform honeycombs in the [4,4,4] Coxeter group family, including this regular form.

It is part of a sequence of honeycombs with a square tiling vertex figure:

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

It is part of a sequence of quasiregular polychora and honeycombs:

Rectified order-4 square tiling honeycomb

Rectified order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols r{4,4,4} or t1{4,4,4}
Coxeter diagrams CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel split1-44.pngCDel nodes.png
Cells {4,4} Uniform tiling 44-t0.svg
r{4,4} Uniform tiling 44-t1.svg
Faces square {4}
Vertex figure Rectified order-4 square tiling honeycomb verf.png
cube
Coxeter groups [math]\displaystyle{ \overline{N}_3 }[/math], [4,4,4]
[math]\displaystyle{ \overline{M}_3 }[/math], [41,1,1]
Properties Quasiregular or regular, depending on symmetry

The rectified order-4 hexagonal tiling honeycomb, t1{4,4,4}, CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png has square tiling facets, with a cubic vertex figure. It is the same as the regular square tiling honeycomb, {4,4,3}, CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png.

H3 443 FC boundary.png

Truncated order-4 square tiling honeycomb

Truncated order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t{4,4,4} or t0,1{4,4,4}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1-44.pngCDel nodes.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel split1-44.pngCDel nodes 11.pngCDel 2a2b-cross.pngCDel nodes.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node g.pngCDel 4sg.pngCDel node g.pngCDel branchu 11.pngCDel split2-44.pngCDel node 1.pngCDel split1-44.pngCDel branchu 11.png
Cells {4,4} Uniform tiling 44-t0.svg
t{4,4} Uniform tiling 44-t01.png
Faces square {4}
octagon {8}
Vertex figure Truncated order-4 square tiling honeycomb verf.png
square pyramid
Coxeter groups [math]\displaystyle{ \overline{N}_3 }[/math], [4,4,4]
[math]\displaystyle{ \overline{M}_3 }[/math], [41,1,1]
Properties Vertex-transitive

The truncated order-4 square tiling honeycomb, t0,1{4,4,4}, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png has square tiling and truncated square tiling facets, with a square pyramid vertex figure.

H3 444-1100.png

Bitruncated order-4 square tiling honeycomb

Bitruncated order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols 2t{4,4,4} or t1,2{4,4,4}
Coxeter diagrams CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel split1-44.pngCDel nodes 11.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node h0.png
CDel node 1.pngCDel split1-44.pngCDel nodes 11.pngCDel split2-44.pngCDel node 1.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel split1-44.pngCDel nodes 11.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node h0.png
Cells t{4,4} Uniform tiling 44-t01.png
Faces square {4}
octagon {8}
Vertex figure Bitruncated order-4 square tiling honeycomb verf.png
tetragonal disphenoid
Coxeter groups [math]\displaystyle{ 2\times\overline{N}_3 }[/math], 4,4,4
[math]\displaystyle{ \overline{M}_3 }[/math], [41,1,1]
[math]\displaystyle{ \widehat{RR}_3 }[/math], [4[4]]
Properties Vertex-transitive, edge-transitive, cell-transitive

The bitruncated order-4 square tiling honeycomb, t1,2{4,4,4}, CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png has truncated square tiling facets, with a tetragonal disphenoid vertex figure.

H3 444-0110.png

Cantellated order-4 square tiling honeycomb

Cantellated order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols rr{4,4,4} or t0,2{4,4,4}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Cells {}x{4} Tetragonal prism.png
r{4,4} 40px
rr{4,4} Uniform tiling 44-t02.svg
Faces square {4}
Vertex figure Cantellated order-4 square tiling honeycomb verf.png
triangular prism
Coxeter groups [math]\displaystyle{ \overline{N}_3 }[/math], [4,4,4]
[math]\displaystyle{ \overline{R}_3 }[/math], [3,4,4]
Properties Vertex-transitive, edge-transitive

The cantellated order-4 square tiling honeycomb, CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png is the same thing as the rectified square tiling honeycomb, CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png. It has cube and square tiling facets, with a triangular prism vertex figure.

H3 444-1010.png

Cantitruncated order-4 square tiling honeycomb

Cantitruncated order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols tr{4,4,4} or t0,1,2{4,4,4}
Coxeter diagrams 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 4.pngCDel node.pngCDel 3.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
Cells {}x{4} Tetragonal prism.png
tr{4,4} 40px
t{4,4} Uniform tiling 44-t01.svg
Faces square {4}
octagon {8}
Vertex figure Cantitruncated order-4 square tiling honeycomb verf.png
mirrored sphenoid
Coxeter groups [math]\displaystyle{ \overline{N}_3 }[/math], [4,4,4]
[math]\displaystyle{ \overline{R}_3 }[/math], [3,4,4]
[math]\displaystyle{ \overline{M}_3 }[/math], [41,1,1]
Properties Vertex-transitive

The cantitruncated order-4 square tiling honeycomb, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png is the same as the truncated square tiling honeycomb, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png. It contains cube and truncated square tiling facets, with a mirrored sphenoid vertex figure.

It is the same as the truncated square tiling honeycomb, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png.

H3 444-1110.png

Runcinated order-4 square tiling honeycomb

Runcinated order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t0,3{4,4,4}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel node 1.pngCDel split1-44.pngCDel nodes.pngCDel 2a2b-cross.pngCDel nodes 11.png
CDel node 1.pngCDel 4.pngCDel node g.pngCDel 4sg.pngCDel node g.pngCDel 4.pngCDel node 1.pngCDel branchu 11.pngCDel 2.pngCDel branchu 11.pngCDel 2.pngCDel branchu 11.pngCDel 2.pngCDel branchu 11.png
Cells {4,4} Uniform tiling 44-t0.svg
{}x{4} Tetragonal prism.png
Faces square {4}
Vertex figure Runcinated order-4 square tiling honeycomb verf.png
square antiprism
Coxeter groups [math]\displaystyle{ 2\times\overline{N}_3 }[/math], 4,4,4
Properties Vertex-transitive, edge-transitive

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

H3 444-1001.png

Runcitruncated order-4 square tiling honeycomb

Runcitruncated order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t0,1,3{4,4,4}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel node 1.pngCDel split1-44.pngCDel nodes 11.pngCDel 2a2b-cross.pngCDel nodes 11.png
Cells t{4,4} Uniform tiling 44-t01.png

rr{4,4} Uniform tiling 44-t02.png
{}x{4} Tetragonal prism.png
{8}x{} Octagonal prism.png

Faces square {4}
octagon {8}
Vertex figure Runcitruncated order-4 square tiling honeycomb verf.png
square pyramid
Coxeter groups [math]\displaystyle{ \overline{N}_3 }[/math], [4,4,4]
Properties Vertex-transitive

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

The runcicantellated order-4 square tiling honeycomb is equivalent to the runcitruncated order-4 square tiling honeycomb.

H3 444-1101.png

Omnitruncated order-4 square tiling honeycomb

Omnitruncated order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t0,1,2,3{4,4,4}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Cells tr{4,4} Uniform tiling 44-t012.png
{8}x{} Octagonal prism.png
Faces square {4}
octagon {8}
Vertex figure Omnitruncated order-4 square tiling honeycomb verf.png
digonal disphenoid
Coxeter groups [math]\displaystyle{ 2\times\overline{N}_3 }[/math], 4,4,4
Properties Vertex-transitive

The omnitruncated order-4 square tiling honeycomb, t0,1,2,3{4,4,4}, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png has truncated square tiling and octagonal prism facets, with a digonal disphenoid vertex figure.

H3 444-1111.png

Alternated order-4 square tiling honeycomb

The alternated order-4 square tiling honeycomb is a lower-symmetry construction of the order-4 square tiling honeycomb itself.

Cantic order-4 square tiling honeycomb

The cantic order-4 square tiling honeycomb is a lower-symmetry construction of the truncated order-4 square tiling honeycomb.

Runcic order-4 square tiling honeycomb

The runcic order-4 square tiling honeycomb is a lower-symmetry construction of the order-3 square tiling honeycomb.

Runcicantic order-4 square tiling honeycomb

The runcicantic order-4 square tiling honeycomb is a lower-symmetry construction of the bitruncated order-4 square tiling honeycomb.

Quarter order-4 square tiling honeycomb

Quarter order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols q{4,4,4}
Coxeter diagrams CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h1.png
CDel label4.pngCDel branch 10r.pngCDel 4a4b.pngCDel branch 10l.pngCDel label4.png
Cells t{4,4} Uniform tiling 44-t01.png
{4,4} Uniform tiling 44-t0.png
Faces square {4}
octagon {8}
Vertex figure Paracompact honeycomb 4444 1100 verf.png
square antiprism
Coxeter groups [math]\displaystyle{ \widehat{RR}_3 }[/math], [4[4]]
Properties Vertex-transitive, edge-transitive

The quarter order-4 square tiling honeycomb, q{4,4,4}, CDel label4.pngCDel branch 10r.pngCDel 4a4b.pngCDel branch 10l.pngCDel label4.png, or CDel node h1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h1.png, has truncated square tiling and square tiling facets, with a square antiprism vertex figure.

See also

  • Convex uniform honeycombs in hyperbolic space
  • Regular tessellations of hyperbolic 3-space
  • Paracompact uniform honeycombs

References

  1. Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III
  • Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
  • The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
  • Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I, II)
  • Norman Johnson Uniform Polytopes, Manuscript
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
    • N.W. Johnson: Geometries and Transformations, (2018) Chapter 13: Hyperbolic Coxeter groups
    • Norman W. Johnson and Asia Ivic Weiss Quadratic Integers and Coxeter Groups PDF Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336