Truncated infinite-order square tiling

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Infinite-order truncated square tiling
Truncated infinite-order square tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration ∞.8.8
Schläfli symbol t{4,∞}
Wythoff symbol 2 ∞ | 4
Coxeter diagram CDel node.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Symmetry group [∞,4], (*∞42)
Dual apeirokis apeirogonal tiling
Properties Vertex-transitive

In geometry, the truncated infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,∞}.

Uniform color[edit]

In (*∞44) symmetry this tiling has 3 colors. Bisecting the isosceles triangle domains can double the symmetry to *∞42 symmetry.

H2checkers 44i.pngH2 tiling 44i-7.png

Symmetry[edit]

The dual of the tiling represents the fundamental domains of (*∞44) orbifold symmetry. From [(∞,4,4)] (*∞44) symmetry, there are 15 small index subgroup (11 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled to *∞42 by adding a bisecting mirror across the fundamental domains. The subgroup index-8 group, [(1+,∞,1+,4,1+,4)] (∞22∞22) is the commutator subgroup of [(∞,4,4)].

Small index subgroups of [(∞,4,4)] (*∞44)
Fundamental
domains
H2checkers 44i.png H2chess 44ie.png
H2chess 44ib.png
H2chess 44if.png
H2chess 44ic.png
H2chess 44id.png
H2chess 44ia.png
H2chess 44ib.png
H2chess 44ic.png
H2chess 44ia.png
Subgroup index 1 2 4
Coxeter
(orbifold)
[(4,4,∞)]
CDel node c1.pngCDel split1-44.pngCDel branch c3-2.pngCDel labelinfin.png
(*∞44)
[(1+,4,4,∞)]
CDel node c1.pngCDel split1-44.pngCDel branch h0c2.pngCDel labelinfin.png
(*∞424)
[(4,4,1+,∞)]
CDel node c1.pngCDel split1-44.pngCDel branch c3h0.pngCDel labelinfin.png
(*∞424)
[(4,1+,4,∞)]
CDel labelh.pngCDel node.pngCDel split1-44.pngCDel branch c3-2.pngCDel labelinfin.png
(*∞2∞2)
[(4,1+,4,1+,∞)]
CDel labelh.pngCDel node.pngCDel split1-44.pngCDel branch c3h0.pngCDel labelinfin.png
2*∞2∞2
[(1+,4,4,1+,∞)]
CDel node c1.pngCDel split1-44.pngCDel branch h0h0.pngCDel labelinfin.png
(∞*2222)
[(4,4+,∞)]
CDel node h2.pngCDel split1-44.pngCDel branch c3h2.pngCDel labelinfin.png
(4*∞2)
[(4+,4,∞)]
CDel node h2.pngCDel split1-44.pngCDel branch h2c2.pngCDel labelinfin.png
(4*∞2)
[(4,4,∞+)]
CDel node.pngCDel split1-44.pngCDel branch h2h2.pngCDel labelinfin.png
(∞*22)
[(1+,4,1+,4,∞)]
CDel labelh.pngCDel node.pngCDel split1-44.pngCDel branch h0c2.pngCDel labelinfin.png
2*∞2∞2
[(4+,4+,∞)]
CDel node h4.pngCDel split1-44.pngCDel branch h2h2.pngCDel labelinfin.png
(∞22×)
Rotational subgroups
Subgroup index 2 4 8
Coxeter
(orbifold)
[(4,4,∞)]+
CDel node h2.pngCDel split1-44.pngCDel branch h2h2.pngCDel labelinfin.png
(∞44)
[(1+,4,4+,∞)]
CDel node h2.pngCDel split1-44.pngCDel branch h0h2.pngCDel labelinfin.png
(∞323)
[(4+,4,1+,∞)]
CDel node h2.pngCDel split1-44.pngCDel branch h2h0.pngCDel labelinfin.png
(∞424)
[(4,1+,4,∞+)]
CDel labelh.pngCDel node.pngCDel split1-44.pngCDel branch h2h2.pngCDel labelinfin.png
(∞434)
[(1+,4,1+,4,1+,∞)] = [(4+,4+,∞+)]
CDel node h4.pngCDel split1-44.pngCDel branch h4h4.pngCDel labelinfin.png
(∞22∞22)

Related polyhedra and tiling[edit]

See also[edit]

References[edit]

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
  • "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678. 

External links[edit]