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Truncated order-4 apeirogonal tiling#Symmetry

{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|Ui4_01}}

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

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.

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|200px
{{CDD|node_1|infin|node_1|infin|node_1}}
(Vertex centered)

|200px
{{CDD|node_1|infin|node_1|ultra|node_1}}
(Square centered)

Symmetry

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 [∞,∞].

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|+ Small index subgroups of [∞,∞] (*∞∞2)

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!Index

!1

!colspan=3|2

!colspan=2|4

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!Diagram

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|120px

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!Coxeter

|[∞,∞]
{{CDD|node_c1|infin|node_c3|infin|node_c2}} = {{CDD|node_c3|split1-ii|branch_c1-2|label2}}

|[1+,∞,∞]
{{CDD|node_h0|infin|node_c3|infin|node_c2}} = {{CDD|labelinfin|branch_c3|split2-ii|node_c2}}

|[∞,∞,1+]
{{CDD|node_c1|infin|node_c3|infin|node_h0}} = {{CDD|node_c1|split1-ii|branch_c3|labelinfin}}

|[∞,1+,∞]
{{CDD|node_c1|infin|node_h0|infin|node_c2}} = {{CDD|labelinfin|branch_c1|2a2b-cross|branch_c2|labelinfin}}

|[1+,∞,∞,1+]
{{CDD|node_h0|infin|node_c3|infin|node_h0}} = {{CDD|labelinfin|branch_c3|iaib-cross|branch_c3|labelinfin}}

|[∞+,∞+]
{{CDD|node_h2|infin|node_h4|infin|node_h2}}

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!Orbifold

|*∞∞2

|colspan=2|*∞∞∞

|*∞2∞2

|*∞∞∞∞

|∞∞×

colspan=7|Semidirect subgroups
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!Diagram

|

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|120px

|120px

|120px

|120px

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!Coxeter

|

|[∞,∞+]
{{CDD|node_c1|infin|node_h2|infin|node_h2}}

|[∞+,∞]
{{CDD|node_h2|infin|node_h2|infin|node_c2}}

|[(∞,∞,2+)]
{{CDD|node_c3|split1-ii|branch_h2h2|label2}}

|[∞,1+,∞,1+]
{{CDD|node_c1|infin|node_h0|infin|node_h0}} = {{CDD|node_c1|infin|node_h2|infin|node_h0}} = {{CDD|node_c1|split1-ii|branch_h2h2|labelinfin}}
= {{CDD|node_c1|infin|node_h0|infin|node_h2}} = {{CDD|labelinfin|branch_c1|iaib-cross|branch_h2h2|labelinfin}}

|[1+,∞,1+,∞]
{{CDD|node_h0|infin|node_h0|infin|node_c2}} = {{CDD|node_h0|infin|node_h2|infin|node_c2}} = {{CDD|labelinfin|branch_h2h2|split2-ii|node_c2}}
= {{CDD|node_h2|infin|node_h0|infin|node_c2}} = {{CDD|labelinfin|branch_h2h2|iaib-cross|branch_c2|labelinfin}}

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!Orbifold

|

|colspan=2|∞*∞

|2*∞∞

|colspan=2|∞*∞∞

colspan=7|Direct subgroups
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!Index

!2

!colspan=3|4

!colspan=2|8

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!Diagram

|120px

|120px

|120px

|120px

|colspan=2|120px

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!Coxeter

|[∞,∞]+
{{CDD|node_h2|infin|node_h2|infin|node_h2}} = {{CDD|node_h2|split1-ii|branch_h2h2|label2}}

|[∞,∞+]+
{{CDD|node_h0|infin|node_h2|infin|node_h2}} = {{CDD|labelinfin|branch_h2h2|split2-ii|node_h2}}

|[∞+,∞]+
{{CDD|node_h2|infin|node_h2|infin|node_h0}} = {{CDD|node_h2|split1-ii|branch_h2h2|labelinfin}}

|[∞,1+,∞]+
{{CDD|labelh|node|split1-ii|branch_h2h2|label2}} = {{CDD|labelinfin|branch_h2h2|2xa2xb-cross|branch_h2h2|labelinfin}}

|colspan=2|[∞+,∞+]+ = [1+,∞,1+,∞,1+]
{{CDD|node_h4|split1-ii|branch_h4h4|label2}} = {{CDD|node_h0|infin|node_h0|infin|node_h0}} = {{CDD|node_h0|infin|node_h2|infin|node_h0}} = {{CDD|labelinfin|branch_h2h2|iaib-cross|branch_h2h2|labelinfin}}

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!Orbifold

|∞∞2

|colspan=2|∞∞∞

|∞2∞2

|colspan=2|∞∞∞∞

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!colspan=7|Radical subgroups

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!Index

!

!colspan=2|∞

!colspan=2|∞

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!Diagram

|

|120px

|120px

|120px

|120px

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!Coxeter

|

|[∞,∞*]
{{CDD|node_c1|infin|node_g|ig|3sg|node_g}}

|[∞*,∞]
{{CDD|node_g|ig|3sg|node_g|infin|node_c2}}

|[∞,∞*]+
{{CDD|node_h0|infin|node_g|ig|3sg|node_g}}

|[∞*,∞]+
{{CDD|node_g|ig|3sg|node_g|infin|node_h0}}

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!Orbifold

|

|colspan=2|*∞∞

|colspan=2|∞∞

Related polyhedra and tiling

{{Truncated_figure3_table}}

{{Order i-4 tiling table}}

{{Order i-i tiling table}}

See also

{{Commons category|Uniform tiling 4-i-i}}

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]

{{Tessellation}}

Category:Apeirogonal tilings

Category:Hyperbolic tilings

Category:Isogonal tilings

Category:Order-4 tilings

Category:Truncated tilings

Category:Uniform tilings