Truncated order-8 hexagonal tiling

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Truncated order-8 hexagonal tiling
Truncated order-8 hexagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 8.12.12
Schläfli symbol t{6,8}
Wythoff symbol 2 8 | 6
Coxeter diagram CDel node.pngCDel 8.pngCDel node 1.pngCDel 6.pngCDel node 1.png
Symmetry group [8,6], (*862)
Dual Order-6 octakis octagonal tiling
Properties Vertex-transitive

In geometry, the truncated order-8 hexagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of t{6,8}.

Uniform colorings[edit]

This tiling can also be constructed from *664 symmetry, as t{(6,6,4)}.

H2 tiling 466-7.png

Related polyhedra and tilings[edit]

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 octagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [8,6] symmetry, and 7 with subsymmetry.

Symmetry[edit]

The dual of the tiling represents the fundamental domains of (*664) orbifold symmetry. From [(6,6,4)] (*664) 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 862 symmetry by adding a bisecting mirror across the fundamental domains. The subgroup index-8 group, [(1+,6,1+,6,1+,4)] (332332) is the commutator subgroup of [(6,6,4)].

A large subgroup is constructed [(6,6,4*)], index 8, as (4*33) with gyration points removed, becomes (*38), and another large subgroup is constructed [(6,6*,4)], index 12, as (6*32) with gyration points removed, becomes (*(32)6).

Small index subgroups of [(6,6,4)] (*664)
Fundamental
domains
H2checkers 466.png H2chess 466e.png
H2chess 466b.png
H2chess 466f.png
H2chess 466c.png
H2chess 466d.png
H2chess 466a.png
H2chess 466b.png
H2chess 466c.png
H2chess 466a.png
Subgroup index 1 2 4
Coxeter [(6,6,4)]
CDel node c1.pngCDel split1-66.pngCDel branch c3-2.pngCDel label4.png
[(1+,6,6,4)]
CDel node c1.pngCDel split1-66.pngCDel branch h0c2.pngCDel label4.png
[(6,6,1+,4)]
CDel node c1.pngCDel split1-66.pngCDel branch c3h0.pngCDel label4.png
[(6,1+,6,4)]
CDel labelh.pngCDel node.pngCDel split1-66.pngCDel branch c3-2.pngCDel label4.png
[(1+,6,6,1+,4)]
CDel node c1.pngCDel split1-66.pngCDel branch h0h0.pngCDel label4.png
[(6+,6+,4)]
CDel node h4.pngCDel split1-66.pngCDel branch h2h2.pngCDel label4.png
Orbifold *664 *6362 *4343 2*3333 332×
Coxeter [(6,6+,4)]
CDel node h2.pngCDel split1-66.pngCDel branch c3h2.pngCDel label4.png
[(6+,6,4)]
CDel node h2.pngCDel split1-66.pngCDel branch h2c2.pngCDel label4.png
[(6,6,4+)]
CDel node c1.pngCDel split1-66.pngCDel branch h2h2.pngCDel label4.png
[(6,1+,6,1+,4)]
CDel labelh.pngCDel node.pngCDel split1-66.pngCDel branch c3h0.pngCDel label4.png
[(1+,6,1+,6,4)]
CDel labelh.pngCDel node.pngCDel split1-66.pngCDel branch h0c2.pngCDel label4.png
Orbifold 6*32 4*33 3*3232
Direct subgroups
Subgroup index 2 4 8
Coxeter [(6,6,4)]+
CDel node h2.pngCDel split1-66.pngCDel branch h2h2.pngCDel label4.png
[(1+,6,6+,4)]
CDel node h2.pngCDel split1-66.pngCDel branch h0h2.pngCDel label4.png
[(6+,6,1+,4)]
CDel node h2.pngCDel split1-66.pngCDel branch h2h0.pngCDel label4.png
[(6,1+,6,4+)]
CDel labelh.pngCDel node.pngCDel split1-66.pngCDel branch h2h2.pngCDel label4.png
[(6+,6+,4+)] = [(1+,6,1+,6,1+,4)]
CDel node h4.pngCDel split1-66.pngCDel branch h4h4.pngCDel label4.png = CDel labelh.pngCDel node.pngCDel split1-66.pngCDel branch h0h0.pngCDel label4.png
Orbifold 664 6362 4343 332332

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]