Truncated hexaoctagonal tiling

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Truncated hexaoctagonal tiling
Truncated hexaoctagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.12.16
Schläfli symbol tr{8,6} or
Wythoff symbol 2 8 6 |
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 6.pngCDel node 1.png or CDel node 1.pngCDel split1-86.pngCDel nodes 11.png
Symmetry group [8,6], (*862)
Dual Order-6-8 kisrhombille tiling
Properties Vertex-transitive

In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon on each vertex, it has Schläfli symbol of tr{8,6}.

Dual tiling[edit]

Hyperbolic domains 862.png H2checkers 268.png
The dual tiling is called an order-6-8 kisrhombille tiling, made as a complete bisection of the order-6 octagonal tiling, here with triangles are shown with alternating colors. This tiling represents the fundamental triangular domains of [8,6] (*862) symmetry.

Symmetry[edit]

Truncated hexaoctagonal tiling with mirror lines

There are six reflective subgroup kaleidoscopic constructed from [8,6] by removing one or two of three mirrors. 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+,8,1+,6,1+] (4343) is the commutator subgroup of [8,6].

A radical subgroup is constructed as [8,6*], index 12, as [8,6+], (6*4) with gyration points removed, becomes (*444444), and another [8*,6], index 16 as [8+,6], (8*3) with gyration points removed as (*33333333).


Small index subgroups of [8,6] (*862)
Index 1 2 4
Diagram 862 symmetry mirrors.png 862 symmetry 00a.png 862 symmetry a00.png 862 symmetry 0a0.png 862 symmetry z0z.png 862 symmetry xxx.png
Coxeter [8,6]
CDel node c2.pngCDel 8.pngCDel node c3.pngCDel 6.pngCDel node c1.png = CDel node c3.pngCDel split1-86.pngCDel branch c2-1.png
[1+,8,6]
CDel node h0.pngCDel 8.pngCDel node c3.pngCDel 6.pngCDel node c1.png = CDel label4.pngCDel branch c3.pngCDel split2-66.pngCDel node c1.png
[8,6,1+]
CDel node c2.pngCDel 8.pngCDel node c3.pngCDel 6.pngCDel node h0.png = CDel node c2.pngCDel split1-88.pngCDel branch c3.png = CDel node c2.pngCDel split1-88.pngCDel branch c3.png
[8,1+,6]
CDel node c2.pngCDel 8.pngCDel node h0.pngCDel 6.pngCDel node c1.png = CDel label4.pngCDel branch c2.pngCDel 2a2b-cross.pngCDel branch c1.png
[1+,8,6,1+]
CDel node h0.pngCDel 8.pngCDel node c3.pngCDel 6.pngCDel node h0.png = CDel label4.pngCDel branch c3.pngCDel 3a3b-cross.pngCDel branch c3.pngCDel label4.png
[8+,6+]
CDel node h2.pngCDel 8.pngCDel node h4.pngCDel 6.pngCDel node h2.png
Orbifold *862 *664 *883 *4232 *4343 43×
Semidirect subgroups
Diagram 862 symmetry bb0.png 862 symmetry 0bb.png 862 symmetry b0b.png 862 symmetry ab0.png 862 symmetry 0ab.png
Coxeter [8,6+]
CDel node c2.pngCDel 8.pngCDel node h2.pngCDel 6.pngCDel node h2.png
[8+,6]
CDel node h2.pngCDel 8.pngCDel node h2.pngCDel 6.pngCDel node c1.png
[(8,6,2+)]
CDel node c3.pngCDel split1-86.pngCDel branch h2h2.png
[8,1+,6,1+]
CDel node c2.pngCDel 8.pngCDel node h0.pngCDel 6.pngCDel node h0.png = CDel node c2.pngCDel 8.pngCDel node h2.pngCDel 6.pngCDel node h0.png = CDel node c2.pngCDel split1-88.pngCDel branch h2h2.png
= CDel node c2.pngCDel 8.pngCDel node h0.pngCDel 6.pngCDel node h2.png = CDel label4.pngCDel branch c2.pngCDel 2a2b-cross.pngCDel branch h2h2.png
[1+,8,1+,6]
CDel node h0.pngCDel 8.pngCDel node h0.pngCDel 6.pngCDel node c1.png = CDel node h0.pngCDel 8.pngCDel node h2.pngCDel 6.pngCDel node c1.png = CDel label4.pngCDel branch h2h2.pngCDel split2-66.pngCDel node c1.png
= CDel node h2.pngCDel 8.pngCDel node h0.pngCDel 6.pngCDel node c1.png = CDel label4.pngCDel branch h2h2.pngCDel 2a2b-cross.pngCDel branch c1.png
Orbifold 6*4 8*3 2*43 3*44 4*33
Direct subgroups
Index 2 4 8
Diagram 862 symmetry aaa.png 862 symmetry bba.png 862 symmetry abb.png 862 symmetry bab.png 862 symmetry abc.png
Coxeter [8,6]+
CDel node h2.pngCDel 8.pngCDel node h2.pngCDel 6.pngCDel node h2.png = CDel node h2.pngCDel split1-86.pngCDel branch h2h2.pngCDel label2.png
[8,6+]+
CDel node h0.pngCDel 8.pngCDel node h2.pngCDel 6.pngCDel node h2.png = CDel label4.pngCDel branch h2h2.pngCDel split2-66.pngCDel node h2.png
[8+,6]+
CDel node h2.pngCDel 8.pngCDel node h2.pngCDel 6.pngCDel node h0.png = CDel node h2.pngCDel split1-88.pngCDel branch h2h2.png
[8,1+,6]+
CDel labelh.pngCDel node.pngCDel split1-86.pngCDel branch h2h2.png = CDel label4.pngCDel branch h2h2.pngCDel 2xa2xb-cross.pngCDel branch h2h2.png
[8+,6+]+ = [1+,8,1+,6,1+]
CDel node h4.pngCDel split1-86.pngCDel branch h4h4.pngCDel label2.png = CDel node h0.pngCDel 8.pngCDel node h0.pngCDel 6.pngCDel node h0.png = CDel node h0.pngCDel 8.pngCDel node h2.pngCDel 6.pngCDel node h0.png = CDel label4.pngCDel branch h2h2.pngCDel 3a3b-cross.pngCDel branch h2h2.pngCDel label4.png
Orbifold 862 664 883 4232 4343
Radical subgroups
Index 12 24 16 32
Diagram 862 symmetry zz0.png 862 symmetry 0zz.png 862 symmetry zza.png 862 symmetry azz.png
Coxeter [8,6*]
CDel node c2.pngCDel 8.pngCDel node g.pngCDel 6g.pngCDel 3sg.pngCDel node g.png
[8*,6]
CDel node g.pngCDel 8g.pngCDel 3sg.pngCDel node g.pngCDel 6.pngCDel node c1.png
[8,6*]+
CDel node h0.pngCDel 8.pngCDel node g.pngCDel 6g.pngCDel 3sg.pngCDel node g.png
[8*,6]+
CDel node g.pngCDel 8g.pngCDel 3sg.pngCDel node g.pngCDel 6.pngCDel node h0.png
Orbifold *444444 *33333333 444444 33333333

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.

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]