Truncated order-8 octagonal tiling

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Truncated order-8 octagonal tiling
Truncated order-8 octagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 8.16.16
Schläfli symbol t{8,8}
t{(8,8,4)}
Wythoff symbol 2 8 | 4
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node.png
CDel 3.pngCDel node 1.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel 3.png
Symmetry group [8,8], (*882)
[(8,8,4)], (*884)
Dual Order-8 octakis octagonal tiling
Properties Vertex-transitive

In geometry, the truncated order-8 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,8}.

Uniform colorings

This tiling can also be constructed in *884 symmetry with 3 colors of faces:

240px

Related polyhedra and tiling

Symmetry

The dual of the tiling represents the fundamental domains of (*884) orbifold symmetry. From [(8,8,4)] (*884) 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 882 symmetry by adding a bisecting mirror across the fundamental domains. The subgroup index-8 group, [(1+,8,1+,8,1+,4)] (442442) is the commutator subgroup of [(8,8,4)].

Small index subgroups of [(8,8,4)] (*884)
Fundamental
domains		 80px 80px
80px		 80px
80px		 80px
80px		 80px
80px		 80px
Subgroup index 1 2 4
Coxeter [(8,8,4)]
CDel node.pngCDel split1-88.pngCDel branch.pngCDel label4.png		 [(1+,8,8,4)]
CDel node c1.pngCDel split1-88.pngCDel branch h0c2.pngCDel label4.png		 [(8,8,1+,4)]
CDel node c1.pngCDel split1-88.pngCDel branch c3h0.pngCDel label4.png		 [(8,1+,8,4)]
CDel labelh.pngCDel node.pngCDel split1-88.pngFile:CDel branch c3-2.pngCDel label4.png		 [(1+,8,8,1+,4)]
CDel labelh.pngCDel node.pngCDel split1-88.pngCDel branch c3h0.pngCDel label4.png		 [(8+,8+,4)]
CDel node c1.pngCDel split1-88.pngCDel branch h0h0.pngCDel label4.png
orbifold *884 *8482 *4444 2*4444 442×
Coxeter [(8,8+,4)]
CDel node h2.pngCDel split1-88.pngCDel branch c3h2.pngCDel label4.png		 [(8+,8,4)]
CDel node h2.pngCDel split1-88.pngCDel branch h2c2.pngCDel label4.png		 [(8,8,4+)]
CDel node c1.pngCDel split1-88.pngCDel branch h2h2.pngCDel label4.png		 [(8,1+,8,1+,4)]
CDel labelh.pngCDel node.pngCDel split1-88.pngCDel branch h0c2.pngCDel label4.png		 [(1+,8,1+,8,4)]
CDel node h4.pngCDel split1-88.pngCDel branch h2h2.pngCDel label4.png
Orbifold 8*42 4*44 4*4242
Direct subgroups
Subgroup index 2 4 8
Coxeter [(8,8,4)]+
CDel node h2.pngCDel split1-88.pngCDel branch h2h2.pngCDel label4.png		 [(1+,8,8+,4)]
CDel node h2.pngCDel split1-88.pngCDel branch h0h2.pngCDel label4.png		 [(8+,8,1+,4)]
CDel node h2.pngCDel split1-88.pngCDel branch h2h0.pngCDel label4.png		 [(8,1+,8,4+)]
CDel labelh.pngCDel node.pngCDel split1-88.pngCDel branch h2h2.pngCDel label4.png		 [(1+,8,1+,8,1+,4)] = [(8+,8+,4+)]
CDel node h4.pngCDel split1-88.pngCDel branch h4h4.pngCDel label4.png
Orbifold 844 8482 4444 442442

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)
  • Lua error in package.lua at line 80: module 'strict' not found.

See also

External links

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