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Order-4 square tiling honeycomb

May 05, 2024

See also: Order-3 square tiling honeycomb
Order-4 square tiling honeycomb
Type Hyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbols {4,4,4}
h{4,4,4} ↔ {4,41,1}
{4[4]}
Coxeter diagrams
↔ ↔
↔ ↔
↔
↔
↔
↔
↔
Cells {4,4}
Faces square {4}
Edge figure square {4}
Vertex figure square tiling, {4,4}
Dual Self-dual
Coxeter groups , [4,4,4]
, [41,1,1]
, [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 edit

The order-4 square tiling honeycomb has many reflective symmetry constructions: as a regular honeycomb, ↔ with alternating types (colors) of square tilings, and 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: ↔ , and ↔ .

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 ; and [4,4*,4], with 4 orthogonal sets of ultra-parallel mirrors in an octahedral fundamental domain: .

Images edit

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.

It contains and that tile 2-hypercycle surfaces, which are similar to these paracompact order-4 apeirogonal tilings :

Related polytopes and honeycombs edit

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:

{p,4,4} honeycombs
Space E3 H3
Form Affine Paracompact Noncompact
Name {2,4,4} {3,4,4} {4,4,4} {5,4,4} {6,4,4} ..{∞,4,4}
Coxeter










 





Image
Cells
{2,4}
{3,4}
{4,4}
{5,4}
{6,4}
{∞,4}

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

{4,4,p} honeycombs
Space E3 H3
Form Affine Paracompact Noncompact
Name {4,4,2} {4,4,3} {4,4,4} {4,4,5} {4,4,6} ...{4,4,∞}
Coxeter












Image
Vertex
figure
{4,2}
{4,3}
{4,4}
{4,5}
{4,6}
{4,∞}

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

Quasiregular polychora and honeycombs: h{4,p,q}
Space Finite Affine Compact Paracompact
Schläfli
symbol 		h{4,3,3} h{4,3,4} h{4,3,5} h{4,3,6} h{4,4,3} h{4,4,4}
Coxeter
diagram 		↔ ↔ ↔ ↔ ↔ ↔
↔ ↔
Image
Vertex
figure
r{p,3}





Rectified order-4 square tiling honeycomb edit

Rectified order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols r{4,4,4} or t1{4,4,4}
Coxeter diagrams
↔
Cells {4,4}
r{4,4}
Faces square {4}
Vertex figure
cube
Coxeter groups , [4,4,4]
, [41,1,1]
Properties Quasiregular or regular, depending on symmetry

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

Truncated order-4 square tiling honeycomb edit

Truncated order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t{4,4,4} or t0,1{4,4,4}
Coxeter diagrams
↔
↔
↔
Cells {4,4}
t{4,4}
Faces square {4}
octagon {8}
Vertex figure
square pyramid
Coxeter groups , [4,4,4]
, [41,1,1]
Properties Vertex-transitive

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

Bitruncated order-4 square tiling honeycomb edit

Bitruncated order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols 2t{4,4,4} or t1,2{4,4,4}
Coxeter diagrams
↔
↔ ↔
Cells t{4,4}
Faces square {4}
octagon {8}
Vertex figure
tetragonal disphenoid
Coxeter groups , [[4,4,4]]
, [41,1,1]
, [4[4]]
Properties Vertex-transitive, edge-transitive, cell-transitive

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

Cantellated order-4 square tiling honeycomb edit

Cantellated order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols rr{4,4,4} or t0,2{4,4,4}
Coxeter diagrams
Cells {}x{4}
r{4,4}
rr{4,4}
Faces square {4}
Vertex figure
triangular prism
Coxeter groups , [4,4,4]
, [3,4,4]
Properties Vertex-transitive, edge-transitive

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

Cantitruncated order-4 square tiling honeycomb edit

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

↔
Cells {}x{4}
tr{4,4}
t{4,4}
Faces square {4}
octagon {8}
Vertex figure
mirrored sphenoid
Coxeter groups , [4,4,4]
, [3,4,4]
, [41,1,1]
Properties Vertex-transitive

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

It is the same as the truncated square tiling honeycomb, .

Runcinated order-4 square tiling honeycomb edit

Runcinated order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t0,3{4,4,4}
Coxeter diagrams
↔
↔
Cells {4,4}
{}x{4}
Faces square {4}
Vertex figure
square antiprism
Coxeter groups , [[4,4,4]]
Properties Vertex-transitive, edge-transitive

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

Runcitruncated order-4 square tiling honeycomb edit

Runcitruncated order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t0,1,3{4,4,4}
Coxeter diagrams
↔
Cells t{4,4}

rr{4,4}
{}x{4}
{8}x{}

Faces square {4}
octagon {8}
Vertex figure
square pyramid
Coxeter groups , [4,4,4]
Properties Vertex-transitive

The runcitruncated order-4 square tiling honeycomb, t0,1,3{4,4,4}, 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.

Omnitruncated order-4 square tiling honeycomb edit

Omnitruncated order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t0,1,2,3{4,4,4}
Coxeter diagrams
Cells tr{4,4}
{8}x{}
Faces square {4}
octagon {8}
Vertex figure
digonal disphenoid
Coxeter groups , [[4,4,4]]
Properties Vertex-transitive

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

Alternated order-4 square tiling honeycomb edit

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 edit

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 edit

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 edit

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 edit

Quarter order-4 square tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols q{4,4,4}
Coxeter diagrams
Cells t{4,4}
{4,4}
Faces square {4}
octagon {8}
Vertex figure
square antiprism
Coxeter groups , [4[4]]
Properties Vertex-transitive, edge-transitive

The quarter order-4 square tiling honeycomb, q{4,4,4}, , or , has truncated square tiling and square tiling facets, with a square antiprism vertex figure.

See also edit

References edit

  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

May 05, 2024
order, square, tiling, honeycomb, also, order, square, tiling, honeycomb, type, hyperbolic, regular, honeycombparacompact, uniform, honeycomb, schläfli, symbols, coxeter, diagrams, cells, faces, square, edge, figure, square, vertex, figure, square, tiling, dua. See also Order 3 square tiling honeycomb Order 4 square tiling honeycomb Type Hyperbolic regular honeycombParacompact uniform honeycomb Schlafli symbols 4 4 4 h 4 4 4 4 41 1 4 4 Coxeter diagrams Cells 4 4 Faces square 4 Edge figure square 4 Vertex figure square tiling 4 4 Dual Self dual Coxeter groups N x00AF 3 displaystyle overline N 3 4 4 4 M x00AF 3 displaystyle overline M 3 41 1 1 R R x005E 3 displaystyle widehat RR 3 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 Schlafli 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 91 1 93 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 Contents 1 Symmetry 2 Images 3 Related polytopes and honeycombs 3 1 Rectified order 4 square tiling honeycomb 3 2 Truncated order 4 square tiling honeycomb 3 3 Bitruncated order 4 square tiling honeycomb 3 4 Cantellated order 4 square tiling honeycomb 3 5 Cantitruncated order 4 square tiling honeycomb 3 6 Runcinated order 4 square tiling honeycomb 3 7 Runcitruncated order 4 square tiling honeycomb 3 8 Omnitruncated order 4 square tiling honeycomb 3 9 Alternated order 4 square tiling honeycomb 3 10 Cantic order 4 square tiling honeycomb 3 11 Runcic order 4 square tiling honeycomb 3 12 Runcicantic order 4 square tiling honeycomb 3 13 Quarter order 4 square tiling honeycomb 4 See also 5 References Symmetry edit The order 4 square tiling honeycomb has many reflective symmetry constructions as a regular honeycomb with alternating types colors of square tilings and 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 and 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 and 4 4 4 with 4 orthogonal sets of ultra parallel mirrors in an octahedral fundamental domain Images edit 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 It contains and that tile 2 hypercycle surfaces which are similar to these paracompact order 4 apeirogonal tilings Related polytopes and honeycombs edit The order 4 square tiling honeycomb is a regular hyperbolic honeycomb in 3 space It is one of eleven regular paracompact honeycombs 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 nine uniform honeycombs in the 4 4 4 Coxeter group family including this regular form 4 4 4 family honeycombs 4 4 4 r 4 4 4 t 4 4 4 rr 4 4 4 t0 3 4 4 4 2t 4 4 4 tr 4 4 4 t0 1 3 4 4 4 t0 1 2 3 4 4 4 It is part of a sequence of honeycombs with a square tiling vertex figure p 4 4 honeycombs vte Space E3 H3 Form Affine Paracompact Noncompact Name 2 4 4 3 4 4 4 4 4 5 4 4 6 4 4 8734 4 4 Coxeter 160 Image Cells 2 4 3 4 4 4 5 4 6 4 8734 4 It is part of a sequence of honeycombs with square tiling cells 4 4 p honeycombs vte Space E3 H3 Form Affine Paracompact Noncompact Name 4 4 2 4 4 3 4 4 4 4 4 5 4 4 6 4 4 8734 Coxeter Image Vertexfigure 4 2 4 3 4 4 4 5 4 6 4 8734 It is part of a sequence of quasiregular polychora and honeycombs Quasiregular polychora and honeycombs h 4 p q Space Finite Affine Compact Paracompact Schlaflisymbol h 4 3 3 h 4 3 4 h 4 3 5 h 4 3 6 h 4 4 3 h 4 4 4 3 3 3 displaystyle left 3 3 atop 3 right 3 4 3 displaystyle left 3 4 atop 3 right 3 5 3 displaystyle left 3 5 atop 3 right 3 6 3 displaystyle left 3 6 atop 3 right 4 4 3 displaystyle left 4 4 atop 3 right 4 4 4 displaystyle left 4 4 atop 4 right Coxeterdiagram Image Vertexfigurer p 3 Rectified order 4 square tiling honeycomb edit Rectified order 4 square tiling honeycomb Type Paracompact uniform honeycomb Schlafli symbols r 4 4 4 or t1 4 4 4 Coxeter diagrams Cells 4 4 r 4 4 Faces square 4 Vertex figure cube Coxeter groups N x00AF 3 displaystyle overline N 3 4 4 4 M x00AF 3 displaystyle overline M 3 41 1 1 Properties Quasiregular or regular depending on symmetry The rectified order 4 hexagonal tiling honeycomb t1 4 4 4 has square tiling facets with a cubic vertex figure It is the same as the regular square tiling honeycomb 4 4 3 Truncated order 4 square tiling honeycomb edit Truncated order 4 square tiling honeycomb Type Paracompact uniform honeycomb Schlafli symbols t 4 4 4 or t0 1 4 4 4 Coxeter diagrams Cells 4 4 t 4 4 Faces square 4 octagon 8 Vertex figure square pyramid Coxeter groups N x00AF 3 displaystyle overline N 3 4 4 4 M x00AF 3 displaystyle overline M 3 41 1 1 Properties Vertex transitive The truncated order 4 square tiling honeycomb t0 1 4 4 4 has square tiling and truncated square tiling facets with a square pyramid vertex figure Bitruncated order 4 square tiling honeycomb edit Bitruncated order 4 square tiling honeycomb Type Paracompact uniform honeycomb Schlafli symbols 2t 4 4 4 or t1 2 4 4 4 Coxeter diagrams Cells t 4 4 Faces square 4 octagon 8 Vertex figure tetragonal disphenoid Coxeter groups 2 x00D7 N x00AF 3 displaystyle 2 times overline N 3 4 4 4 M x00AF 3 displaystyle overline M 3 41 1 1 R R x005E 3 displaystyle widehat RR 3 4 4 Properties Vertex transitive edge transitive cell transitive The bitruncated order 4 square tiling honeycomb t1 2 4 4 4 has truncated square tiling facets with a tetragonal disphenoid vertex figure Cantellated order 4 square tiling honeycomb edit Cantellated order 4 square tiling honeycomb Type Paracompact uniform honeycomb Schlafli symbols rr 4 4 4 or t0 2 4 4 4 Coxeter diagrams Cells x 4 r 4 4 rr 4 4 Faces square 4 Vertex figure triangular prism Coxeter groups N x00AF 3 displaystyle overline N 3 4 4 4 R x00AF 3 displaystyle overline R 3 3 4 4 Properties Vertex transitive edge transitive The cantellated order 4 square tiling honeycomb is the same thing as the rectified square tiling honeycomb It has cube and square tiling facets with a triangular prism vertex figure Cantitruncated order 4 square tiling honeycomb edit Cantitruncated order 4 square tiling honeycomb Type Paracompact uniform honeycomb Schlafli symbols tr 4 4 4 or t0 1 2 4 4 4 Coxeter diagrams Cells x 4 tr 4 4 t 4 4 Faces square 4 octagon 8 Vertex figure mirrored sphenoid Coxeter groups N x00AF 3 displaystyle overline N 3 4 4 4 R x00AF 3 displaystyle overline R 3 3 4 4 M x00AF 3 displaystyle overline M 3 41 1 1 Properties Vertex transitive The cantitruncated order 4 square tiling honeycomb is the same as the truncated square tiling honeycomb It contains cube and truncated square tiling facets with a mirrored sphenoid vertex figure It is the same as the truncated square tiling honeycomb Runcinated order 4 square tiling honeycomb edit Runcinated order 4 square tiling honeycomb Type Paracompact uniform honeycomb Schlafli symbols t0 3 4 4 4 Coxeter diagrams Cells 4 4 x 4 Faces square 4 Vertex figure square antiprism Coxeter groups 2 x00D7 N x00AF 3 displaystyle 2 times overline N 3 4 4 4 Properties Vertex transitive edge transitive The runcinated order 4 square tiling honeycomb t0 3 4 4 4 has square tiling and cube facets with a square antiprism vertex figure Runcitruncated order 4 square tiling honeycomb edit Runcitruncated order 4 square tiling honeycomb Type Paracompact uniform honeycomb Schlafli symbols t0 1 3 4 4 4 Coxeter diagrams Cells t 4 4 rr 4 4 x 4 8 x Faces square 4 octagon 8 Vertex figure square pyramid Coxeter groups N x00AF 3 displaystyle overline N 3 4 4 4 Properties Vertex transitive The runcitruncated order 4 square tiling honeycomb t0 1 3 4 4 4 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 Omnitruncated order 4 square tiling honeycomb edit Omnitruncated order 4 square tiling honeycomb Type Paracompact uniform honeycomb Schlafli symbols t0 1 2 3 4 4 4 Coxeter diagrams Cells tr 4 4 8 x Faces square 4 octagon 8 Vertex figure digonal disphenoid Coxeter groups 2 x00D7 N x00AF 3 displaystyle 2 times overline N 3 4 4 4 Properties Vertex transitive The omnitruncated order 4 square tiling honeycomb t0 1 2 3 4 4 4 has truncated square tiling and octagonal prism facets with a digonal disphenoid vertex figure Alternated order 4 square tiling honeycomb edit 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 edit 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 edit 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 edit 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 edit Quarter order 4 square tiling honeycomb Type Paracompact uniform honeycomb Schlafli symbols q 4 4 4 Coxeter diagrams Cells t 4 4 4 4 Faces square 4 octagon 8 Vertex figure square antiprism Coxeter groups R R x005E 3 displaystyle widehat RR 3 4 4 Properties Vertex transitive edge transitive The quarter order 4 square tiling honeycomb q 4 4 4 or has truncated square tiling and square tiling facets with a square antiprism vertex figure See also edit Convex uniform honeycombs in hyperbolic space Regular tessellations of hyperbolic 3 space Paracompact uniform honeycombs References edit Coxeter The Beauty of Geometry 1999 Chapter 10 Table III Coxeter Regular Polytopes 3rd ed Dover Publications 1973 ISBN 160 0 486 61480 8 Tables I and II Regular polytopes and honeycombs pp 160 294 296 The Beauty of Geometry Twelve Essays 1999 Dover Publications LCCN 160 99 35678 ISBN 160 0 486 40919 8 Chapter 10 Regular Honeycombs in Hyperbolic Space Table III Jeffrey R Weeks The Shape of Space 2nd edition ISBN 160 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 160 1307 1336 Retrieved from https en wikipedia org w index php title Order 4 square tiling honeycomb amp oldid 1199789303 Cantellated order 4 square tiling honeycomb, wikipedia, wiki, book, books, library,

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