Truncated infinite-order square tiling

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Truncated infinite-order square tiling

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

More information Infinite-order truncated square tiling ...
Infinite-order truncated square tiling
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Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration.8.8
Schläfli symbolt{4,}
Wythoff symbol2 | 4
Coxeter diagram
Symmetry group[,4], (*42)
Dualapeirokis apeirogonal tiling
PropertiesVertex-transitive
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Uniform color

Summarize
Perspective

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

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Symmetry

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)].

More information Fundamental domains, Subgroup index ...
Small index subgroups of [(∞,4,4)] (*∞44)
Fundamental
domains
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Subgroup index 1 2 4
Coxeter
(orbifold)
[(4,4,∞)]

(*∞44)
[(1+,4,4,∞)]

(*∞424)
[(4,4,1+,∞)]

(*∞424)
[(4,1+,4,∞)]

(*∞2∞2)
[(4,1+,4,1+,∞)]

2*∞2∞2
[(1+,4,4,1+,∞)]

(∞*2222)
[(4,4+,∞)]

(4*∞2)
[(4+,4,∞)]

(4*∞2)
[(4,4,∞+)]

(∞*22)
[(1+,4,1+,4,∞)]

2*∞2∞2
[(4+,4+,∞)]

(∞22×)
Rotational subgroups
Subgroup index 2 4 8
Coxeter
(orbifold)
[(4,4,∞)]+

(∞44)
[(1+,4,4+,∞)]

(∞323)
[(4+,4,1+,∞)]

(∞424)
[(4,1+,4,∞+)]

(∞434)
[(1+,4,1+,4,1+,∞)] = [(4+,4+,∞+)]

(∞22∞22)
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More information Symmetry*n42 [n,4], Spherical ...
*n42 symmetry mutation of truncated tilings: n.8.8
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*42
[,4]
Truncated
figures
Config. 2.8.8 3.8.8 4.8.8 5.8.8 6.8.8 7.8.8 8.8.8 .8.8
n-kis
figures
Config. V2.8.8 V3.8.8 V4.8.8 V5.8.8 V6.8.8 V7.8.8 V8.8.8 V.8.8
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More information Dual figures, Alternations ...
Paracompact uniform tilings in [,4] family
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{,4} t{,4} r{,4} 2t{,4}=t{4,} 2r{,4}={4,} rr{,4} tr{,4}
Dual figures
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V4 V4.. V(4.)2 V8.8. V4 V43. V4.8.
Alternations
[1+,,4]
(*44)
[+,4]
(*2)
[,1+,4]
(*22)
[,4+]
(4*)
[,4,1+]
(*2)
[(,4,2+)]
(2*2)
[,4]+
(42)

=

=
h{,4} s{,4} hr{,4} s{4,} h{4,} hrr{,4} s{,4}
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Alternation duals
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V(.4)4 V3.(3.)2 V(4..4)2 V3..(3.4)2 V V.44 V3.3.4.3.
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See also

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)
  • "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
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