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order-4 apeirogonal tiling

{{Short description|Regular tiling in geometry}}

{{technical|date=July 2013}}

{{Uniform hyperbolic tiles db|Reg hyperbolic tiling stat table|Ui4_0}}

In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,4}.

Symmetry

This tiling represents the mirror lines of *2∞ symmetry. Its dual tiling represents the fundamental domains of orbifold notation *∞∞∞∞ symmetry, a square domain with four ideal vertices.

: 120px

Uniform colorings

Like the Euclidean square tiling there are 9 uniform colorings for this tiling, with 3 uniform colorings generated by triangle reflective domains. A fourth can be constructed from an infinite square symmetry (*∞∞∞∞) with 4 colors around a vertex. The checker board, r{∞,∞}, coloring defines the fundamental domains of [(∞,4,4)], (*∞44) symmetry, usually shown as black and white domains of reflective orientations.

class=wikitable

!1 color

!2 color

!colspan=2|3 and 2 colors

!colspan=3|4, 3 and 2 colors

align=center

|[∞,4], (*∞42)

|[∞,∞], (*∞∞2)

|colspan=2|[(∞,∞,∞)], (*∞∞∞)

|colspan=3|(*∞∞∞∞)

align=center

|{∞,4}

|r{∞,∞}
= {∞,4}{{frac|1|2}}

|colspan=2|t0,2(∞,∞,∞)
= r{∞,∞}{{frac|1|2}}

|colspan=3|t0,1,2,3(∞,∞,∞,∞)
= r{∞,∞}{{frac|1|4}} = {∞,4}{{frac|1|8}}

align=center

|80px
(1111)

|80px
(1212)

|80px
(1213)

|80px
(1112)

|80px
(1234)

|80px
(1123)

|80px
(1122)

align=center

|{{CDD|node_1|infin|node|4|node}}

|{{CDD|node_1|split1-ii|nodes}} = {{CDD|node_1|infin|node|4|node_h0}}

|colspan=2|{{CDD|labelinfin|branch_11|split2-ii|node}} = {{CDD|node_h0|infin|node_1|infin|node}}
{{CDD|node_1|infin|node_h0|4|node}} = {{CDD|labelinfin|branch_11|2a2b-cross|nodes}}

|colspan=3|{{CDD|labelinfin|branch_11|iaib-cross|branch_11|labelinfin}} = {{CDD|labelinfin|branch_11|split2-ii|node|labelh}} = {{CDD|node_h0|infin|node_1|infin|node_h0}}

Related polyhedra and tiling

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram {{CDD|node_1|n|node|4|node}}, with n progressing to infinity.

{{Order-4 regular tilings}}

{{Order i-4 tiling table}}

{{Order i-i tiling table}}

{{Order i-i-i tiling table}}

See also

{{Commons category|Order-4 apeirogonal tiling}}

References

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, {{isbn|978-1-56881-220-5}} (Chapter 19, The Hyperbolic Archimedean Tessellations)
  • {{Cite book|title=The Beauty of Geometry: Twelve Essays|year=1999|publisher=Dover Publications|lccn=99035678|isbn=0-486-40919-8|chapter=Chapter 10: Regular honeycombs in hyperbolic space}}

External links

  • {{MathWorld | urlname= HyperbolicTiling | title = Hyperbolic tiling}}
  • {{MathWorld | urlname=PoincareHyperbolicDisk | title = Poincaré hyperbolic disk }}
  • [http://bork.hampshire.edu/~bernie/hyper/ Hyperbolic and Spherical Tiling Gallery]
  • [http://geometrygames.org/KaleidoTile/index.html KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings]
  • [http://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]

{{Tessellation}}

Category:Apeirogonal tilings

Category:Hyperbolic tilings

Category:Isogonal tilings

Category:Isohedral tilings

Category:Order-4 tilings

Category:Regular tilings