Truncated order-4 apeirogonal tiling

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Truncated order-4 apeirogonal tiling
Truncated order-4 apeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.∞.∞
Schläfli symbol t{∞,4}
tr{∞,∞} or
Wythoff symbol 2 4 | ∞
2 ∞ ∞ |
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.png or CDel node 1.pngCDel split1-ii.pngCDel nodes 11.png
Symmetry group [∞,4], (*∞42)
[∞,∞], (*∞∞2)
Dual Infinite-order tetrakis square tiling
Properties Vertex-transitive

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

Uniform colorings[edit]

A half symmetry coloring is tr{∞,∞}, has two types of apeirogons, shown red and yellow here. If the apeirogonal curvature is too large, it doesn't converge to a single ideal point, like the right image, red apeirogons below. Coxeter diagram are shown with dotted lines for these divergent, ultraparallel mirrors.

H2 tiling 2ii-7.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.png
(Vertex centered)
H2 tiling 2iu-7.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel ultra.pngCDel node 1.png
(Square centered)

Symmetry[edit]

From [∞,∞] symmetry, there are 15 small index subgroup by mirror removal and alternation. 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 as ∞42 symmetry by adding a mirror bisecting the fundamental domain. The subgroup index-8 group, [1+,∞,1+,∞,1+] (∞∞∞∞) is the commutator subgroup of [∞,∞].

Small index subgroups of [∞,∞] (*∞∞2)
Index 1 2 4
Diagram Ii2 symmetry 000.png Ii2 symmetry a00.png Ii2 symmetry 00a.png Ii2 symmetry 0a0.png Ii2 symmetry z0z.png Ii2 symmetry xxx.png
Coxeter [∞,∞]
CDel node c1.pngCDel infin.pngCDel node c3.pngCDel infin.pngCDel node c2.png = CDel node c3.pngCDel split1-ii.pngCDel branch c1-2.pngCDel label2.png
[1+,∞,∞]
CDel node h0.pngCDel infin.pngCDel node c3.pngCDel infin.pngCDel node c2.png = CDel labelinfin.pngCDel branch c3.pngCDel split2-ii.pngCDel node c2.png
[∞,∞,1+]
CDel node c1.pngCDel infin.pngCDel node c3.pngCDel infin.pngCDel node h0.png = CDel node c1.pngCDel split1-ii.pngCDel branch c3.pngCDel labelinfin.png
[∞,1+,∞]
CDel node c1.pngCDel infin.pngCDel node h0.pngCDel infin.pngCDel node c2.png = CDel labelinfin.pngCDel branch c1.pngCDel 2a2b-cross.pngCDel branch c2.pngCDel labelinfin.png
[1+,∞,∞,1+]
CDel node h0.pngCDel infin.pngCDel node c3.pngCDel infin.pngCDel node h0.png = CDel labelinfin.pngCDel branch c3.pngCDel iaib-cross.pngCDel branch c3.pngCDel labelinfin.png
[∞+,∞+]
CDel node h2.pngCDel infin.pngCDel node h4.pngCDel infin.pngCDel node h2.png
Orbifold *∞∞2 *∞∞∞ *∞2∞2 *∞∞∞∞ ∞∞×
Semidirect subgroups
Diagram Ii2 symmetry 0bb.png Ii2 symmetry aa0.png Ii2 symmetry a0a.png Ii2 symmetry 0ab.png Ii2 symmetry ab0.png
Coxeter [∞,∞+]
CDel node c1.pngCDel infin.pngCDel node h2.pngCDel infin.pngCDel node h2.png
[∞+,∞]
CDel node h2.pngCDel infin.pngCDel node h2.pngCDel infin.pngCDel node c2.png
[(∞,∞,2+)]
CDel node c3.pngCDel split1-ii.pngCDel branch h2h2.pngCDel label2.png
[∞,1+,∞,1+]
CDel node c1.pngCDel infin.pngCDel node h0.pngCDel infin.pngCDel node h0.png = CDel node c1.pngCDel infin.pngCDel node h2.pngCDel infin.pngCDel node h0.png = CDel node c1.pngCDel split1-ii.pngCDel branch h2h2.pngCDel labelinfin.png
= CDel node c1.pngCDel infin.pngCDel node h0.pngCDel infin.pngCDel node h2.png = CDel labelinfin.pngCDel branch c1.pngCDel iaib-cross.pngCDel branch h2h2.pngCDel labelinfin.png
[1+,∞,1+,∞]
CDel node h0.pngCDel infin.pngCDel node h0.pngCDel infin.pngCDel node c2.png = CDel node h0.pngCDel infin.pngCDel node h2.pngCDel infin.pngCDel node c2.png = CDel labelinfin.pngCDel branch h2h2.pngCDel split2-ii.pngCDel node c2.png
= CDel node h2.pngCDel infin.pngCDel node h0.pngCDel infin.pngCDel node c2.png = CDel labelinfin.pngCDel branch h2h2.pngCDel iaib-cross.pngCDel branch c2.pngCDel labelinfin.png
Orbifold ∞*∞ 2*∞∞ ∞*∞∞
Direct subgroups
Index 2 4 8
Diagram Ii2 symmetry aaa.png Ii2 symmetry abb.png Ii2 symmetry bba.png Ii2 symmetry bab.png Ii2 symmetry abc.png
Coxeter [∞,∞]+
CDel node h2.pngCDel infin.pngCDel node h2.pngCDel infin.pngCDel node h2.png = CDel node h2.pngCDel split1-ii.pngCDel branch h2h2.pngCDel label2.png
[∞,∞+]+
CDel node h0.pngCDel infin.pngCDel node h2.pngCDel infin.pngCDel node h2.png = CDel labelinfin.pngCDel branch h2h2.pngCDel split2-ii.pngCDel node h2.png
[∞+,∞]+
CDel node h2.pngCDel infin.pngCDel node h2.pngCDel infin.pngCDel node h0.png = CDel node h2.pngCDel split1-ii.pngCDel branch h2h2.pngCDel labelinfin.png
[∞,1+,∞]+
CDel labelh.pngCDel node.pngCDel split1-ii.pngCDel branch h2h2.pngCDel label2.png = CDel labelinfin.pngCDel branch h2h2.pngCDel 2xa2xb-cross.pngCDel branch h2h2.pngCDel labelinfin.png
[∞+,∞+]+ = [1+,∞,1+,∞,1+]
CDel node h4.pngCDel split1-ii.pngCDel branch h4h4.pngCDel label2.png = CDel node h0.pngCDel infin.pngCDel node h0.pngCDel infin.pngCDel node h0.png = CDel node h0.pngCDel infin.pngCDel node h2.pngCDel infin.pngCDel node h0.png = CDel labelinfin.pngCDel branch h2h2.pngCDel iaib-cross.pngCDel branch h2h2.pngCDel labelinfin.png
Orbifold ∞∞2 ∞∞∞ ∞2∞2 ∞∞∞∞
Radical subgroups
Index
Diagram Ii2 symmetry 0zz.png Ii2 symmetry zz0.png Ii2 symmetry azz.png Ii2 symmetry zza.png
Coxeter [∞,∞*]
CDel node c1.pngCDel infin.pngCDel node g.pngCDel ig.pngCDel 3sg.pngCDel node g.png
[∞*,∞]
CDel node g.pngCDel ig.pngCDel 3sg.pngCDel node g.pngCDel infin.pngCDel node c2.png
[∞,∞*]+
CDel node h0.pngCDel infin.pngCDel node g.pngCDel ig.pngCDel 3sg.pngCDel node g.png
[∞*,∞]+
CDel node g.pngCDel ig.pngCDel 3sg.pngCDel node g.pngCDel infin.pngCDel node h0.png
Orbifold *∞

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]