Square tiling honeycomb

Square tiling honeycombTypeHyperbolic regular honeycombParacompact uniform honeycombSchläfli symbols{4,4,3}r{4,4,4}{41,1,1}Coxeter diagrams ↔ ↔ ↔ Cells{4,4} Facessquare {4}Edge figuretriangle {3}Vertex figurecube, {4,3}DualOrder-4 octahedral honeycombCoxeter groups, [4,4,3], , [41,1,1]PropertiesRegular 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}, it has three square tilings, {4,4}, around each edge, and six square tilings around each vertex, in a cubic {4,3} vertex figure.Coxeter, The Beauty of Geometry, 1999, Chapter 10, Table III

Rectified order-4 square tiling 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} =

Symmetry The square tiling honeycomb has three reflective symmetry constructions: as a regular honeycomb, a half symmetry construction ↔ , and lastly a construction with three types (colors) of checkered square tilings ↔ .

It also contains an index 6 subgroup [4,4,3*] ↔ [41,1,1], and a radial subgroup [4,(4,3)*] of index 48, with a right dihedral-angled octahedral fundamental domain, and four pairs of ultraparallel mirrors: .

This honeycomb contains that tile 2-hypercycle surfaces, which are similar to the paracompact order-3 apeirogonal tiling :

Related polytopes and honeycombs The square tiling honeycomb is a regular hyperbolic honeycomb in 3-space. It is one of eleven regular paracompact honeycombs.

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}.

The square tiling honeycomb is part of the order-4 square tiling honeycomb family, as it can be seen as a rectified order-4 square tiling honeycomb.

It is related to the 24-cell, {3,4,3}, which also has a cubic vertex figure. It is also part of a sequence of honeycombs with square tiling cells:

Rectified square tiling honeycomb Rectified square tiling honeycomb TypeParacompact uniform honeycombSemiregular honeycombSchläfli symbolsr{4,4,3} or t1{4,4,3}2r{3,41,1}r{41,1,1}Coxeter diagrams ↔ ↔ ↔ Cells{4,3} r{4,4}Facessquare {4}Vertex figuretriangular prismCoxeter groups, [4,4,3], [3,41,1], [41,1,1]PropertiesVertex-transitive, edge-transitive The rectified square tiling honeycomb, t1{4,4,3}, has cube and square tiling facets, with a triangular prism vertex figure.

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

Truncated square tiling honeycomb Truncated square tiling honeycomb TypeParacompact uniform honeycombSchläfli symbolst{4,4,3} or t0,1{4,4,3}Coxeter diagrams ↔ ↔ Cells{4,3} t{4,4}Facessquare {4}octagon {8}Vertex figuretriangular pyramidCoxeter groups, [4,4,3], , [41,1,1]PropertiesVertex-transitive The truncated square tiling honeycomb, t{4,4,3}, 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}, .

Bitruncated square tiling honeycomb Bitruncated square tiling honeycomb TypeParacompact uniform honeycombSchläfli symbols2t{4,4,3} or t1,2{4,4,3}Coxeter diagramCellst{4,3} t{4,4}Facestriangle {3}square {4}octagon {8}Vertex figuredigonal disphenoidCoxeter groups, [4,4,3]PropertiesVertex-transitive The bitruncated square tiling honeycomb, 2t{4,4,3}, has truncated cube and truncated square tiling facets, with a digonal disphenoid vertex figure.

Cantellated square tiling honeycomb Cantellated square tiling honeycomb TypeParacompact uniform honeycombSchläfli symbolsrr{4,4,3} or t0,2{4,4,3}Coxeter diagrams ↔ Cellsr{4,3} rr{4,4}{}x{3}Facestriangle {3}square {4}Vertex figureisosceles triangular prismCoxeter groups, [4,4,3]PropertiesVertex-transitive The cantellated square tiling honeycomb, rr{4,4,3}, has cuboctahedron, square tiling, and triangular prism facets, with an isosceles triangular prism vertex figure.

Cantitruncated square tiling honeycomb Cantitruncated square tiling honeycomb TypeParacompact uniform honeycombSchläfli symbolstr{4,4,3} or t0,1,2{4,4,3}Coxeter diagramCellst{4,3} tr{4,4}{}x{3} Facestriangle {3}square {4}octagon {8}Vertex figureisosceles triangular pyramidCoxeter groups, [4,4,3]PropertiesVertex-transitive The cantitruncated square tiling honeycomb, tr{4,4,3}, has truncated cube, truncated square tiling, and triangular prism facets, with an isosceles triangular pyramid vertex figure.

Runcinated square tiling honeycomb Runcinated square tiling honeycomb TypeParacompact uniform honeycombSchläfli symbolt0,3{4,4,3}Coxeter diagrams ↔ Cells{3,4} {4,4}{}x{4} {}x{3} Facestriangle {3}square {4}Vertex figureirregular triangular antiprismCoxeter groups, [4,4,3]PropertiesVertex-transitive The runcinated square tiling honeycomb, t0,3{4,4,3}, has octahedron, triangular prism, cube, and square tiling facets, with an irregular triangular antiprism vertex figure.

Runcitruncated square tiling honeycomb Runcitruncated square tiling honeycomb TypeParacompact uniform honeycombSchläfli symbolst0,1,3{4,4,3}s2,3{3,4,4}Coxeter diagramsCellsrr{4,3} t{4,4}{}x{3} {}x{8} Facestriangle {3}square {4}octagon {8}Vertex figureisosceles-trapezoidal pyramidCoxeter groups, [4,4,3]PropertiesVertex-transitive The runcitruncated square tiling honeycomb, t0,1,3{4,4,3}, has rhombicuboctahedron, octagonal prism, triangular prism and truncated square tiling facets, with an isosceles-trapezoidal pyramid vertex figure.

Runcicantellated square tiling honeycomb The runcicantellated square tiling honeycomb is the same as the runcitruncated order-4 octahedral honeycomb.

Omnitruncated square tiling honeycomb Omnitruncated square tiling honeycomb TypeParacompact uniform honeycombSchläfli symbolt0,1,2,3{4,4,3}Coxeter diagramCellstr{4,4} {}x{6} {}x{8} tr{4,3} Facessquare {4}hexagon {6}octagon {8}Vertex figureirregular tetrahedronCoxeter groups, [4,4,3]PropertiesVertex-transitive The omnitruncated square tiling honeycomb, t0,1,2,3{4,4,3}, has truncated square tiling, truncated cuboctahedron, hexagonal prism, and octagonal prism facets, with an irregular tetrahedron vertex figure.

Omnisnub square tiling honeycomb Omnisnub square tiling honeycomb TypeParacompact uniform honeycombSchläfli symbolh(t0,1,2,3{4,4,3})Coxeter diagramCellssr{4,4} sr{2,3} sr{2,4} sr{4,3} Facestriangle {3}square {4}Vertex figureirregular tetrahedronCoxeter group[4,4,3]+PropertiesNon-uniform, vertex-transitive The alternated omnitruncated square tiling honeycomb (or omnisnub square tiling honeycomb), h(t0,1,2,3{4,4,3}), has snub square tiling, snub cube, triangular antiprism, square antiprism, and tetrahedron cells, with an irregular tetrahedron vertex figure.

Alternated square tiling honeycomb Alternated square tiling honeycombTypeParacompact uniform honeycombSemiregular honeycombSchläfli symbolh{4,4,3}hr{4,4,4}{(4,3,3,4)}h{41,1,1}Coxeter diagrams ↔ ↔ ↔ ↔ ↔ ↔ Cells{4,4} {4,3} Facessquare {4}Vertex figurecuboctahedronCoxeter groups, [3,41,1][4,1+,4,4] ↔ [∞,4,4,∞], [(4,4,3,3)][1+,41,1,1] ↔ [∞]PropertiesVertex-transitive, edge-transitive, quasiregular The alternated square tiling honeycomb, h{4,4,3}, is a quasiregular paracompact uniform honeycomb in hyperbolic 3-space. It has cube and square tiling facets in a cuboctahedron vertex figure.

Cantic square tiling honeycomb Cantic square tiling honeycombTypeParacompact uniform honeycombSchläfli symbolh2{4,4,3}Coxeter diagrams ↔ Cellst{4,4} r{4,3} t{4,3} Facestriangle {3}square {4}octagon {8}Vertex figurerectangular pyramidCoxeter groups, [3,41,1]PropertiesVertex-transitive The cantic square tiling honeycomb, h2{4,4,3}, is a paracompact uniform honeycomb in hyperbolic 3-space. It has truncated square tiling, truncated cube, and cuboctahedron facets, with a rectangular pyramid vertex figure.

Runcic square tiling honeycomb Runcic square tiling honeycombTypeParacompact uniform honeycombSchläfli symbolh3{4,4,3}Coxeter diagrams ↔ Cells{4,4} r{4,3} {3,4} Facestriangle {3}square {4}Vertex figuresquare frustumCoxeter groups, [3,41,1]PropertiesVertex-transitive The runcic square tiling honeycomb, h3{4,4,3}, is a paracompact uniform honeycomb in hyperbolic 3-space. It has square tiling, rhombicuboctahedron, and octahedron facets in a square frustum vertex figure.

Runcicantic square tiling honeycomb Runcicantic square tiling honeycombTypeParacompact uniform honeycombSchläfli symbolh2,3{4,4,3}Coxeter diagrams ↔ Cellst{4,4} tr{4,3} t{3,4} Facessquare {4}hexagon {6}octagon {8}Vertex figuremirrored sphenoidCoxeter groups, [3,41,1]PropertiesVertex-transitive The runcicantic square tiling honeycomb, h2,3{4,4,3}, ↔ , is a paracompact uniform honeycomb in hyperbolic 3-space. It has truncated square tiling, truncated cuboctahedron, and truncated octahedron facets in a mirrored sphenoid vertex figure.

Alternated rectified square tiling honeycomb Alternated rectified square tiling honeycombTypeParacompact uniform honeycombSchläfli symbolhr{4,4,3}Coxeter diagrams ↔ CellsFacesVertex figuretriangular prismCoxeter groups[4,1+,4,3] = [∞,3,3,∞]PropertiesNonsimplectic, vertex-transitive The alternated rectified square tiling honeycomb is a paracompact uniform honeycomb in hyperbolic 3-space.

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

References

Coxeter, Regular Polytopes, 3rd ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296) Coxeter, The Beauty of Geometry: Twelve Essays (1999), Dover Publications, , , (Chapter 10, Regular Honeycombs in Hyperbolic Space), Table III Jeffrey R. Weeks, The Shape of Space, 2nd edition, , (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 Ivić Weiss, Quadratic Integers and Coxeter Groups PDF, Can. J. Math. Vol. 51 (6), 1999, pp. 1307–1336