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Truncated infinite-order square tiling

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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
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

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

Symmetry

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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




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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*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
Paracompact uniform tilings in [∞,4] family
{∞,4} t{∞,4} r{∞,4} 2t{∞,4}=t{4,∞} 2r{∞,4}={4,∞} rr{∞,4} tr{∞,4}
Dual figures
V∞4 V4.∞.∞ V(4.∞)2 V8.8.∞ V4 V43.∞ V4.8.∞
Alternations
[1+,∞,4]
(*44∞)
[∞+,4]
(∞*2)
[∞,1+,4]
(*2∞2∞)
[∞,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}
Alternation duals
V(∞.4)4 V3.(3.∞)2 V(4.∞.4)2 V3.∞.(3.4)2 V∞ V∞.44 V3.3.4.3.∞

See also

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References

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  • 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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