In geometry , the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square , one octagon , and one hexakaidecagon on each vertex . It has Schläfli symbol of tr{8,4}.
Dual tiling Symmetry File:Truncated tetraoctagonal tiling with mirrors.png Truncated tetraoctagonal tiling with *842, File:CDel node c2.png File:CDel 8.png File:CDel node c3.png File:CDel 4.png File:CDel node c1.png There are 15 subgroups constructed from [8,4] 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+ ,8,1+ ,4,1+ ] (4242) is the commutator subgroup of [8,4].
A larger subgroup is constructed as [8,4*], index 8, as [8,4+ ], (4*4) with gyration points removed, becomes (*4444) or (*44 ), and another [8*,4], index 16 as [8+ ,4], (8*2) with gyration points removed as (*22222222) or (*28 ). And their direct subgroups [8,4*]+ , [8*,4]+ , subgroup indices 16 and 32 respectively, can be given in orbifold notation as (4444) and (22222222).
Small index subgroups of [8,4] (*842)
Index
1
2
4
Diagram
File:842 symmetry mirrors.png File:842 symmetry 00a.png File:842 symmetry a00.png File:842 symmetry 0a0.png File:842 symmetry a0b.png File:842 symmetry xxx.png
Coxeter
[8,4]File:CDel node c2.png File:CDel 8.png File:CDel node c3.png File:CDel 4.png File:CDel node c1.png File:CDel node c3.png File:CDel split1-84.png File:CDel branch c2-1.png File:CDel label2.png
[1+ ,8,4]File:CDel node h0.png File:CDel 8.png File:CDel node c3.png File:CDel 4.png File:CDel node c1.png File:CDel label4.png File:CDel branch c3.png File:CDel split2-44.png File:CDel node c1.png
[8,4,1+ ]File:CDel node c2.png File:CDel 8.png File:CDel node c3.png File:CDel 4.png File:CDel node h0.png File:CDel node c2.png File:CDel split1-88.png File:CDel nodeab c3.png File:CDel node c2.png File:CDel split1-88.png File:CDel branch c3.png File:CDel label2.png
[8,1+ ,4]File:CDel node c2.png File:CDel 8.png File:CDel node h0.png File:CDel 4.png File:CDel node c1.png File:CDel label4.png File:CDel branch c2.png File:CDel 2a2b-cross.png File:CDel nodeab c1.png
[1+ ,8,4,1+ ]File:CDel node h0.png File:CDel 8.png File:CDel node c3.png File:CDel 4.png File:CDel node h0.png File:CDel label4.png File:CDel branch c3.png File:CDel 2a2b-cross.png File:CDel branch c3.png File:CDel label4.png
[8+ ,4+ ]File:CDel node h2.png File:CDel 8.png File:CDel node h4.png File:CDel 4.png File:CDel node h2.png
Orbifold
*842
*444
*882
*4222
*4242
42×
Semidirect subgroups
Diagram
File:842 symmetry bb0.png File:842 symmetry 0bb.png File:842 symmetry b0b.png File:842 symmetry ab0.png File:842 symmetry 0ab.png
Coxeter
[8,4+ ]File:CDel node c2.png File:CDel 8.png File:CDel node h2.png File:CDel 4.png File:CDel node h2.png
[8+ ,4]File:CDel node h2.png File:CDel 8.png File:CDel node h2.png File:CDel 4.png File:CDel node c1.png
[(8,4,2+ )]File:CDel node c3.png File:CDel split1-48.png File:CDel branch h2h2.png
[8,1+ ,4,1+ ]File:CDel node c2.png File:CDel 8.png File:CDel node h0.png File:CDel 4.png File:CDel node h0.png File:CDel node c2.png File:CDel 8.png File:CDel node h2.png File:CDel 4.png File:CDel node h0.png File:CDel node c2.png File:CDel split1-88.png File:CDel branch h2h2.png File:CDel label2.png File:CDel node c2.png File:CDel 8.png File:CDel node h0.png File:CDel 4.png File:CDel node h2.png File:CDel label4.png File:CDel branch c2.png File:CDel 2a2b-cross.png File:CDel branch h2h2.png File:CDel label2.png
[1+ ,8,1+ ,4]File:CDel node h0.png File:CDel 8.png File:CDel node h0.png File:CDel 4.png File:CDel node c1.png File:CDel node h0.png File:CDel 8.png File:CDel node h2.png File:CDel 4.png File:CDel node c1.png File:CDel label4.png File:CDel branch h2h2.png File:CDel split2-44.png File:CDel node c1.png File:CDel node h2.png File:CDel 8.png File:CDel node h0.png File:CDel 4.png File:CDel node c1.png File:CDel label4.png File:CDel branch h2h2.png File:CDel 2a2b-cross.png File:CDel nodeab c1.png
Orbifold
4*4
8*2
2*42
2*44
4*22
Direct subgroups
Index
2
4
8
Diagram
File:842 symmetry aaa.png File:842 symmetry bba.png File:842 symmetry abb.png File:842 symmetry bab.png File:842 symmetry abc.png
Coxeter
[8,4]+ File:CDel node h2.png File:CDel 8.png File:CDel node h2.png File:CDel 4.png File:CDel node h2.png File:CDel node h2.png File:CDel split1-84.png File:CDel branch h2h2.png File:CDel label2.png
[8,4+ ]+ File:CDel node h0.png File:CDel 8.png File:CDel node h2.png File:CDel 4.png File:CDel node h2.png File:CDel label4.png File:CDel branch h2h2.png File:CDel split2-44.png File:CDel node h2.png
[8+ ,4]+ File:CDel node h2.png File:CDel 8.png File:CDel node h2.png File:CDel 4.png File:CDel node h0.png File:CDel node h2.png File:CDel split1-88.png File:CDel branch h2h2.png File:CDel label2.png
[8,1+ ,4]+ File:CDel labelh.png File:CDel split1-48.png File:CDel branch h2h2.png File:CDel label4.png File:CDel branch h2h2.png File:CDel 2xa2xb-cross.png File:CDel branch h2h2.png File:CDel label2.png
[8+ ,4+ ]+ = [1+ ,8,1+ ,4,1+ ]File:CDel node h4.png File:CDel split1-48.png File:CDel branch h4h4.png File:CDel label2.png File:CDel node h0.png File:CDel 8.png File:CDel node h0.png File:CDel 4.png File:CDel node h0.png File:CDel node h0.png File:CDel 8.png File:CDel node h2.png File:CDel 4.png File:CDel node h0.png File:CDel label4.png File:CDel branch h2h2.png File:CDel 2xa2xb-cross.png File:CDel branch h2h2.png File:CDel label4.png
Orbifold
842
444
882
4222
4242
Radical subgroups
Index
8
16
32
Diagram
File:842 symmetry zz0.png File:842 symmetry 0zz.png File:842 symmetry zza.png File:842 symmetry azz.png
Coxeter
[8,4*]File:CDel node c2.png File:CDel 8.png File:CDel node g.png File:CDel 4sg.png File:CDel node g.png File:CDel label4.png File:CDel branch c2.png File:CDel 4a4b-cross.png File:CDel branch c2.png File:CDel label4.png
[8*,4]File:CDel node g.png File:CDel 8g.png File:CDel 3sg.png File:CDel node g.png File:CDel 4.png File:CDel node c1.png
[8,4*]+ File:CDel node h0.png File:CDel 8.png File:CDel node g.png File:CDel 4sg.png File:CDel node g.png File:CDel label4.png File:CDel branch h2h2.png File:CDel 4a4b-cross.png File:CDel branch h2h2.png File:CDel label4.png
[8*,4]+ File:CDel node g.png File:CDel 8g.png File:CDel 3sg.png File:CDel node g.png File:CDel 4.png File:CDel node h0.png
Orbifold
*4444
*22222222
4444
22222222
Related polyhedra and tilings From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-4 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 [8,4] symmetry, and 7 with subsymmetry.
Template:Order 8-4 tiling table
Template:Omnitruncated4 table
Template:Omnitruncated symmetric table
See also References External links
This article "Truncated tetraoctagonal tiling" is from Wikipedia . The list of its authors can be seen in its historical and/or the page Edithistory:Truncated tetraoctagonal tiling . Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.
File:CDel 8.png
. (Chapter 19, The Hyperbolic Archimedean Tessellations)
