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:
File:H2 tiling 488-7.png Related polyhedra and tiling
Uniform octaoctagonal tilings
Symmetry: [8,8], (*882)
File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel 8.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2-88.png File:CDel node.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 8.png File:CDel node.png File:CDel node 1.png File:CDel 8.png File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2-88.png File:CDel node 1.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 8.png File:CDel node 1.png File:CDel node.png File:CDel 8.png File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel nodes.png File:CDel split2-88.png File:CDel node 1.png File:CDel node h0.png File:CDel 4.png File:CDel node.png File:CDel 8.png File:CDel node 1.png File:CDel node.png File:CDel 8.png File:CDel node 1.png File:CDel 8.png File:CDel node 1.png File:CDel nodes 01rd.png File:CDel split2-88.png File:CDel node 1.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 8.png File:CDel node 1.png File:CDel node.png File:CDel 8.png File:CDel node.png File:CDel 8.png File:CDel node 1.png File:CDel nodes 01rd.png File:CDel split2-88.png File:CDel node.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 8.png File:CDel node.png File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel 8.png File:CDel node 1.png File:CDel nodes 11.png File:CDel split2-88.png File:CDel node.png File:CDel node h0.png File:CDel 4.png File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel node 1.png File:CDel 8.png File:CDel node 1.png File:CDel 8.png File:CDel node 1.png File:CDel nodes 11.png File:CDel split2-88.png File:CDel node 1.png File:CDel node h0.png File:CDel 4.png File:CDel node 1.png File:CDel 8.png File:CDel node 1.png
File:H2 tiling 288-1.png File:H2 tiling 288-3.png File:H2 tiling 288-2.png File:H2 tiling 288-6.png File:H2 tiling 288-4.png File:H2 tiling 288-5.png File:H2 tiling 288-7.png
{8,8}
t{8,8} r{8,8}
2t{8,8}=t{8,8}
2r{8,8}={8,8}
rr{8,8}
tr{8,8}
Uniform duals
File:CDel node f1.png File:CDel 8.png File:CDel node.png File:CDel 8.png File:CDel node.png File:CDel node f1.png File:CDel 8.png File:CDel node f1.png File:CDel 8.png File:CDel node.png File:CDel node.png File:CDel 8.png File:CDel node f1.png File:CDel 8.png File:CDel node.png File:CDel node.png File:CDel 8.png File:CDel node f1.png File:CDel 8.png File:CDel node f1.png File:CDel node.png File:CDel 8.png File:CDel node.png File:CDel 8.png File:CDel node f1.png File:CDel node f1.png File:CDel 8.png File:CDel node.png File:CDel 8.png File:CDel node f1.png File:CDel node f1.png File:CDel 8.png File:CDel node f1.png File:CDel 8.png File:CDel node f1.png
File:H2chess 288b.png File:H2chess 288f.png File:H2chess 288a.png File:H2chess 288e.png File:H2chess 288c.png File:H2chess 288d.png File:H2checkers 288.png
V88
V8.16.16
V8.8.8.8
V8.16.16
V88
V4.8.4.8
V4.16.16
Alternations
[1+ ,8,8]
[8+ ,8]
[8,1+ ,8]
[8,8+ ]
[8,8,1+ ]
[(8,8,2+ )]
[8,8]+
File:CDel node h1.png File:CDel 8.png File:CDel node.png File:CDel 8.png File:CDel node.png File:CDel label4.png File:CDel branch 10ru.png File:CDel split2-88.png File:CDel node.png File:CDel node h.png File:CDel 8.png File:CDel node h.png File:CDel 8.png File:CDel node.png File:CDel node.png File:CDel 8.png File:CDel node h1.png File:CDel 8.png File:CDel node.png File:CDel nodes 11.png File:CDel 4a4b-cross.png File:CDel nodes.png File:CDel node.png File:CDel 8.png File:CDel node h.png File:CDel 8.png File:CDel node h.png File:CDel node.png File:CDel 8.png File:CDel node.png File:CDel 8.png File:CDel node h1.png File:CDel node.png File:CDel split1-88.png File:CDel branch 01ld.png File:CDel node h.png File:CDel 8.png File:CDel node.png File:CDel 8.png File:CDel node h.png File:CDel nodes hh.png File:CDel split2-88.png File:CDel node.png File:CDel node h0.png File:CDel 4.png File:CDel node h.png File:CDel 8.png File:CDel node.png File:CDel node h.png File:CDel 8.png File:CDel node h.png File:CDel 8.png File:CDel node h.png File:CDel nodes hh.png File:CDel split2-88.png File:CDel node h.png File:CDel node h0.png File:CDel 4.png File:CDel node h.png File:CDel 8.png File:CDel node h.png
File:Uniform tiling 88-h0.png
File:Uniform tiling 444-t0.png
File:Uniform tiling 88-h0.png File:Uniform tiling 443-t1.png File:Uniform tiling 88-snub.png
h{8,8}
s{8,8}
hr{8,8}
s{8,8}
h{8,8}
hrr{8,8}
sr{8,8}
Alternation duals
File:CDel node fh.png File:CDel 8.png File:CDel node.png File:CDel 8.png File:CDel node.png File:CDel node fh.png File:CDel 8.png File:CDel node fh.png File:CDel 8.png File:CDel node.png File:CDel node.png File:CDel 8.png File:CDel node fh.png File:CDel 8.png File:CDel node.png File:CDel node.png File:CDel 8.png File:CDel node fh.png File:CDel 8.png File:CDel node fh.png File:CDel node.png File:CDel 8.png File:CDel node.png File:CDel 8.png File:CDel node fh.png File:CDel node fh.png File:CDel 8.png File:CDel node.png File:CDel 8.png File:CDel node fh.png File:CDel node fh.png File:CDel 8.png File:CDel node fh.png File:CDel 8.png File:CDel node fh.png
File:Uniform tiling 88-t1.png
File:Uniform tiling 66-t1.png
V(4.8)8
V3.4.3.8.3.8
V(4.4)4
V3.4.3.8.3.8
V(4.8)8
V46
V3.3.8.3.8
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
File:H2checkers 488.png File:H2chess 488e.png File:H2chess 488b.png File:H2chess 488f.png File:H2chess 488c.png File:H2chess 488d.png File:H2chess 488a.png File:H2chess 488b.png File:H2chess 488c.png File:H2chess 488a.png
Subgroup index
1
2
4
Coxeter
[(8,8,4)]File:CDel node.png File:CDel split1-88.png File:CDel branch.png File:CDel label4.png
[(1+ ,8,8,4)]File:CDel node c1.png File:CDel split1-88.png File:CDel branch h0c2.png File:CDel label4.png
[(8,8,1+ ,4)]File:CDel node c1.png File:CDel split1-88.png File:CDel branch c3h0.png File:CDel label4.png
[(8,1+ ,8,4)]File:CDel labelh.png File:CDel node.png File:CDel split1-88.png File:CDel branch c3-2.png File:CDel label4.png
[(1+ ,8,8,1+ ,4)]File:CDel labelh.png File:CDel node.png File:CDel split1-88.png File:CDel branch c3h0.png File:CDel label4.png
[(8+ ,8+ ,4)]File:CDel node c1.png File:CDel split1-88.png File:CDel branch h0h0.png File:CDel label4.png
orbifold
*884
*8482
*4444
2*4444
442×
Coxeter
[(8,8+ ,4)]File:CDel node h2.png File:CDel split1-88.png File:CDel branch c3h2.png File:CDel label4.png
[(8+ ,8,4)]File:CDel node h2.png File:CDel split1-88.png File:CDel branch h2c2.png File:CDel label4.png
[(8,8,4+ )]File:CDel node c1.png File:CDel split1-88.png File:CDel branch h2h2.png File:CDel label4.png
[(8,1+ ,8,1+ ,4)]File:CDel labelh.png File:CDel node.png File:CDel split1-88.png File:CDel branch h0c2.png File:CDel label4.png
[(1+ ,8,1+ ,8,4)]File:CDel node h4.png File:CDel split1-88.png File:CDel branch h2h2.png File:CDel label4.png
Orbifold
8*42
4*44
4*4242
Direct subgroups
Subgroup index
2
4
8
Coxeter
[(8,8,4)]+ File:CDel node h2.png File:CDel split1-88.png File:CDel branch h2h2.png File:CDel label4.png
[(1+ ,8,8+ ,4)]File:CDel node h2.png File:CDel split1-88.png File:CDel branch h0h2.png File:CDel label4.png
[(8+ ,8,1+ ,4)]File:CDel node h2.png File:CDel split1-88.png File:CDel branch h2h0.png File:CDel label4.png
[(8,1+ ,8,4+ )]File:CDel labelh.png File:CDel node.png File:CDel split1-88.png File:CDel branch h2h2.png File:CDel label4.png
[(1+ ,8,1+ ,8,1+ ,4)] = [(8+ ,8+ ,4+ )]File:CDel node h4.png File:CDel split1-88.png File:CDel branch h4h4.png File:CDel label4.png
Orbifold
844
8482
4444
442442
References See also External links
