“Truncated tetraapeirogonal tiling”的意思、由来-开放百科全书

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Truncated tetraapeirogonal tiling

释义

Related polyhedra and tilings
Symmetry
See also
References
External links

{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|Ui4_012}}

In geometry, the truncated tetraapeirogonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one apeirogon on each vertex. It has Schläfli symbol of tr{∞,4}.

Related polyhedra and tilings

Symmetry

The dual of this tiling represents the fundamental domains of [∞,4], (*∞42) symmetry. There are 15 small index subgroups constructed from [∞,4] 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 subgroup index-8 group, [1⁺,∞,1⁺,4,1⁺] (∞2∞2) is the commutator subgroup of [∞,4].

A larger subgroup is constructed as [∞,4*], index 8, as [∞,4⁺], (4*∞) with gyration points removed, becomes (*∞∞∞∞) or (*∞⁴), and another [∞*,4], index ∞ as [∞⁺,4], (∞*2) with gyration points removed as (*2^∞). And their direct subgroups [∞,4*]⁺, [∞*,4]⁺, subgroup indices 16 and ∞ respectively, can be given in orbifold notation as (∞∞∞∞) and (2^∞).

Small index subgroups of [∞,4], (*∞42)
Index	1	2			4
Diagram
Coxeter	{{CDD>node_c1\|infin\|node_c3\|4\|node_c2}}	+,∞,4] {{CDD>node_h0\|infin\|node_c3\|4\|node_c2}} = {{CDD\|labelinfin\|branch_c2\|split2-44\|node_c3}}	+] {{CDD>node_c1\|infin\|node_c3\|4\|node_h0}} = {{CDD\|node_c1\|split1-ii\|branch_c3\|label2}}	+,4] {{CDD>node_c1\|infin\|node_h0\|4\|node_c2}} = {{CDD\|labelinfin\|branch_c1\|2xa2xb-cross\|branch_c2\|label2}}	+,∞,4,1⁺] {{CDD>node_h0\|infin\|node_c3\|4\|node_h0}} = {{CDD\|labelinfin\|branch_c3\|2xa2xb-cross\|branch_c3\|labelinfin}}	+,4⁺] {{CDD>node_h2\|infin\|node_h4\|4\|node_h2}}
Orbifold	*∞42	*∞44	*∞∞2	*∞222	*∞2∞2	∞2×
Semidirect subgroups
Diagram
Coxeter	+] {{CDD>node_c1\|infin\|node_h2\|4\|node_h2}}	+,4] {{CDD>node_h2\|infin\|node_h2\|4\|node_c2}}	+)] {{CDD>node_c3\|split1-i4\|branch_h2h2\|label2}}	+,∞,1⁺,4] {{CDD>node_h0\|infin\|node_h0\|4\|node_c2}} = {{CDD\|node_h0\|infin\|node_h2\|4\|node_c2}} = {{CDD\|labelinfin\|branch_h2h2\|split2-44\|node_c2}} = {{CDD\|node_h2\|infin\|node_h0\|4\|node_c2}} = {{CDD\|labelinfin\|branch_h2h2\|2xa2xb-cross\|branch_c2\|label2}}	+,4,1⁺] {{CDD>node_c1\|infin\|node_h0\|4\|node_h0}} = {{CDD\|node_c1\|infin\|node_h2\|4\|node_h0}} = {{CDD\|node_c1\|split1-ii\|branch_h2h2\|label2}} = {{CDD\|node_c1\|infin\|node_h0\|4\|node_h2}} = {{CDD\|labelinfin\|branch_c1\|2xa2xb-cross\|branch_h2h2\|label2}}
Orbifold	4*∞	∞*2	2*∞2	∞*22	2*∞∞
Direct subgroups
Index	2	4			8
Diagram
Coxeter	+ {{CDD>node_h2\|infin\|node_h2\|4\|node_h2}} = {{CDD\|node_h2\|split1-i4\|branch_h2h2\|label2}}	+]⁺ {{CDD>node_h0\|infin\|node_h2\|4\|node_h2}} = {{CDD\|labelinfin\|branch_h2h2\|split2-44\|node_h2}}	+,4]⁺ {{CDD>node_h2\|infin\|node_h2\|4\|node_h0}} = {{CDD\|node_h2\|split1-ii\|branch_h2h2\|label2}}	+,4]⁺ {{CDD>labelh\|node\|split1-i4\|branch_h2h2\|label2}} = {{CDD\|labelinfin\|branch_h2h2\|2xa2xb-cross\|branch_h2h2\|label2}}	[∞⁺,4⁺]⁺ = [1⁺,∞,1⁺,4,1⁺] {{CDD\|node_h4\|split1-i4\|branch_h4h4\|label2}} = {{CDD\|node_h0\|infin\|node_h0\|4\|node_h0}} = {{CDD\|node_h0\|infin\|node_h2\|4\|node_h0}} = {{CDD\|labelinfin\|branch_h2h2\|2xa2xb-cross\|branch_h2h2\|labelinfin}}
Orbifold	∞42	∞44	∞∞2	∞222	∞2∞2
Radical subgroups
Index	8	∞	16	∞
Diagram
Coxeter	{{CDD>node_c1\|infin\|node_g\|4sg\|node_g}} = {{CDD\|labelinfin\|branch_c1\|iaib-cross\|branch_c1\|labelinfin}}	{{CDD>node_g\|ig\|3sg\|node_g\|4\|node_c2}}	+ {{CDD>node_h0\|infin\|node_g\|4sg\|node_g}} = {{CDD\|labelinfin\|branch_h2h2\|iaib-cross\|branch_h2h2\|labelinfin}}	+ {{CDD>node_g\|ig\|3sg\|node_g\|4\|node_h0}}
Orbifold	*∞∞∞∞	*2^∞	∞∞∞∞	2^∞

References

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)
{{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 }}
Hyperbolic and Spherical Tiling Gallery

5 : Apeirogonal tilings|Hyperbolic tilings|Isogonal tilings|Semiregular tilings|Truncated tilings

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Related polyhedra and tilings

Symmetry

See also

References

External links