Truncated order-4 octagonal tiling

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Truncated order-4 octagonal tiling
H2 tiling 248-3.png
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.16.16
Schläfli symbol t{8,4}
tr{8,8} or
Wythoff symbol 8
2 8 8 |
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node 1.png or CDel node 1.pngCDel split1-88.pngCDel nodes 11.png
Symmetry group [8,4], (*842)
[8,8], (*882)
Dual Order-8 tetrakis square tiling
Properties Vertex-transitive

In geometry, the truncated order-4 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,4}. A secondary construction t0,1,2{8,8} is called a truncated octaoctagonal tiling with two colors of hexakaidecagons.

Contents

Constructions

There are two uniform constructions of this tiling, first by the [8,4] kaleidoscope, and second by removing the last mirror, [8,4,1+], gives [8,8], (*882).

Two uniform constructions of 4.8.4.8
NameTetraoctagonalTruncated octaoctagonal
Image Uniform tiling 84-t01.png Uniform tiling 88-t012.png
Symmetry [8,4]
(*842)
CDel node c1.pngCDel 8.pngCDel node c2.pngCDel 4.pngCDel node c3.png
[8,8] = [8,4,1+]
(*882)
CDel node c1.pngCDel split1-88.pngCDel nodeab c2.png = CDel node c1.pngCDel 8.pngCDel node c2.pngCDel 4.pngCDel node h0.png
Symbolt{8,4}tr{8,8}
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node 1.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node 1.png

Dual tiling

Order-8 tetrakis square tiling.png Hyperbolic domains 882.png
The dual tiling, Order-8 tetrakis square tiling has face configuration V4.16.16, and represents the fundamental domains of the [8,8] symmetry group.

Symmetry

Truncated order-4 octagonal tiling with *882 mirror lines Truncated order-4 octagonal tiling with 882 mirrors.png
Truncated order-4 octagonal tiling with *882 mirror lines

The dual of the tiling represents the fundamental domains of (*882) orbifold symmetry. From [8,8] symmetry, there are 15 small index subgroup 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 unique mirrors are colored red, green, and blue, and alternatively colored triangles show the location of gyration points. The [8+,8+], (44×) subgroup has narrow lines representing glide reflections. The subgroup index-8 group, [1+,8,1+,8,1+] (4444) is the commutator subgroup of [8,8].

One larger subgroup is constructed as [8,8*], removing the gyration points of (8*4), index 16 becomes (*44444444), and its direct subgroup [8,8*]+, index 32, (44444444).

The [8,8] symmetry can be doubled by a mirror bisecting the fundamental domain, and creating *884 symmetry.

Small index subgroups of [8,8] (*882)
Index 124
Diagram 882 symmetry 000.png 882 symmetry a00.png 882 symmetry 00a.png 882 symmetry 0a0.png 882 symmetry a0b.png 882 symmetry xxx.png
Coxeter [8,8]
CDel node c1.pngCDel 8.pngCDel node c3.pngCDel 8.pngCDel node c2.png
[1+,8,8]
CDel node h0.pngCDel 8.pngCDel node c3.pngCDel 8.pngCDel node c2.png = CDel label4.pngCDel branch c3.pngCDel split2-88.pngCDel node c2.png
[8,8,1+]
CDel node c1.pngCDel 8.pngCDel node c3.pngCDel 8.pngCDel node h0.png = CDel node c1.pngCDel split1-88.pngCDel branch c3.pngCDel label4.png
[8,1+,8]
CDel node c1.pngCDel 8.pngCDel node h0.pngCDel 8.pngCDel node c2.png = CDel label4.pngCDel branch c1.pngCDel 2a2b-cross.pngCDel branch c2.pngCDel label4.png
[1+,8,8,1+]
CDel node h0.pngCDel 8.pngCDel node c3.pngCDel 8.pngCDel node h0.png = CDel label4.pngCDel branch c3.pngCDel 4a4b-cross.pngCDel branch c3.pngCDel label4.png
[8+,8+]
CDel node h2.pngCDel 8.pngCDel node h4.pngCDel 8.pngCDel node h2.png
Orbifold *882 *884 *4242 *4444 44×
Semidirect subgroups
Diagram 882 symmetry 0aa.png 882 symmetry aa0.png 882 symmetry a0a.png 882 symmetry 0ab.png 882 symmetry ab0.png
Coxeter[8,8+]
CDel node c1.pngCDel 8.pngCDel node h2.pngCDel 8.pngCDel node h2.png
[8+,8]
CDel node h2.pngCDel 8.pngCDel node h2.pngCDel 8.pngCDel node c2.png
[(8,8,2+)]
CDel node c3.pngCDel split1-88.pngCDel branch h2h2.pngCDel label2.png
[8,1+,8,1+]
CDel node c1.pngCDel 8.pngCDel node h0.pngCDel 8.pngCDel node h0.png = CDel node c1.pngCDel 8.pngCDel node h2.pngCDel 8.pngCDel node h0.png = CDel node c1.pngCDel split1-88.pngCDel branch h2h2.pngCDel label4.png
= CDel node c1.pngCDel 8.pngCDel node h0.pngCDel 8.pngCDel node h2.png = CDel label4.pngCDel branch c1.pngCDel 2a2b-cross.pngCDel branch h2h2.pngCDel label4.png
[1+,8,1+,8]
CDel node h0.pngCDel 8.pngCDel node h0.pngCDel 8.pngCDel node c2.png = CDel node h0.pngCDel 8.pngCDel node h2.pngCDel 8.pngCDel node c2.png = CDel label4.pngCDel branch h2h2.pngCDel split2-88.pngCDel node c2.png
= CDel node h2.pngCDel 8.pngCDel node h0.pngCDel 8.pngCDel node c2.png = CDel label4.pngCDel branch h2h2.pngCDel 2a2b-cross.pngCDel branch c2.pngCDel label4.png
Orbifold8*42*444*44
Direct subgroups
Index248
Diagram 882 symmetry aaa.png 882 symmetry abb.png 882 symmetry bba.png 882 symmetry bab.png 882 symmetry abc.png
Coxeter[8,8]+
CDel node h2.pngCDel 8.pngCDel node h2.pngCDel 8.pngCDel node h2.png
[8,8+]+
CDel node h0.pngCDel 8.pngCDel node h2.pngCDel 8.pngCDel node h2.png = CDel label4.pngCDel branch h2h2.pngCDel split2-88.pngCDel node h2.png
[8+,8]+
CDel node h2.pngCDel 8.pngCDel node h2.pngCDel 8.pngCDel node h0.png = CDel node h2.pngCDel split1-88.pngCDel branch h2h2.pngCDel label4.png
[8,1+,8]+
CDel labelh.pngCDel node.pngCDel split1-88.pngCDel branch h2h2.pngCDel label2.png = CDel label4.pngCDel branch h2h2.pngCDel 2xa2xb-cross.pngCDel branch h2h2.pngCDel label4.png
[8+,8+]+ = [1+,8,1+,8,1+]
CDel node h4.pngCDel split1-88.pngCDel branch h4h4.pngCDel label2.png = CDel node h0.pngCDel 8.pngCDel node h0.pngCDel 8.pngCDel node h0.png = CDel node h0.pngCDel 8.pngCDel node h2.pngCDel 8.pngCDel node h0.png = CDel label4.pngCDel branch h2h2.pngCDel 4a4b-cross.pngCDel branch h2h2.pngCDel label4.png
Orbifold88288442424444
Radical subgroups
Index1632
Diagram 882-m0.png 882 symmetry zz0.png 882 symmetry zza.png 882 symmetry azz.png
Coxeter[8,8*]
CDel node c1.pngCDel 8.pngCDel node g.pngCDel 8.pngCDel 3sg.pngCDel node g.png
[8*,8]
CDel node g.pngCDel 8.pngCDel 3sg.pngCDel node g.pngCDel 8.pngCDel node c2.png
[8,8*]+
CDel node h0.pngCDel 8.pngCDel node g.pngCDel 8.pngCDel 3sg.pngCDel node g.png
[8*,8]+
CDel node g.pngCDel 8.pngCDel 3sg.pngCDel node g.pngCDel 8.pngCDel node h0.png
Orbifold*4444444444444444
*n42 symmetry mutation of truncated tilings: 4.2n.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolicParacomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*42
[,4]
Truncated
figures
Spherical square prism.png Uniform tiling 432-t12.png Uniform tiling 44-t01.png H2-5-4-trunc-dual.svg H2 tiling 246-3.png H2 tiling 247-3.png H2 tiling 248-3.png H2 tiling 24i-3.png
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4..
n-kis
figures
Spherical square bipyramid.png Spherical tetrakis hexahedron.png 1-uniform 2 dual.svg H2-5-4-kis-primal.svg Order-6 tetrakis square tiling.png Hyperbolic domains 772.png Order-8 tetrakis square tiling.png H2checkers 2ii.png
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10V4.12.12V4.14.14V4.16.16V4..
Uniform octagonal/square tilings
[8,4], (*842)
(with [8,8] (*882), [(4,4,4)] (*444) , [,4,] (*4222) index 2 subsymmetries)
(And [(,4,,4)] (*4242) index 4 subsymmetry)
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-88.pngCDel nodes.png
CDel 2.png
= CDel label4.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel nodes.png
= CDel label4.pngCDel branch 11.pngCDel 4a4b-cross.pngCDel branch 11.pngCDel label4.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-88.pngCDel nodes 11.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node.pngCDel split1-88.pngCDel nodes 11.png
= CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node.png
CDel 2.png
= CDel label4.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel branch 11.pngCDel label4.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node 1.png
= CDel label4.pngCDel branch.pngCDel 2a2b-cross.pngCDel nodes 11.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
CDel 2.png
= CDel label4.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
H2 tiling 248-1.png H2 tiling 248-3.png H2 tiling 248-2.png H2 tiling 248-6.png H2 tiling 248-4.png H2 tiling 248-5.png H2 tiling 248-7.png
{8,4} t{8,4}
r{8,4} 2t{8,4}=t{4,8} 2r{8,4}={4,8} rr{8,4} tr{8,4}
Uniform duals
CDel node f1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.pngCDel node f1.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel node.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel node.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel node f1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel node f1.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node f1.png
H2chess 248b.png H2chess 248f.png H2chess 248a.png H2chess 248e.png H2chess 248c.png H2chess 248d.png H2checkers 248.png
V84 V4.16.16V(4.8)2 V8.8.8 V48 V4.4.4.8 V4.8.16
Alternations
[1+,8,4]
(*444)
[8+,4]
(8*2)
[8,1+,4]
(*4222)
[8,4+]
(4*4)
[8,4,1+]
(*882)
[(8,4,2+)]
(2*42)
[8,4]+
(842)
CDel node h1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
= CDel label4.pngCDel branch 10ru.pngCDel split2-44.pngCDel node.png
CDel node h.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node.png
= CDel node h.pngCDel split1-88.pngCDel nodes hh.png
CDel node.pngCDel 8.pngCDel node h1.pngCDel 4.pngCDel node.png
= CDel label4.pngCDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 10.png
CDel node.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node h.png
= CDel label4.pngCDel branch hh.pngCDel split2-44.pngCDel node h.png
CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node h1.png
= CDel node.pngCDel split1-88.pngCDel nodes 10lu.png
CDel node h.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node h.png
= CDel label4.pngCDel branch hh.pngCDel 2a2b-cross.pngCDel nodes hh.png
CDel node h.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node h.png
Uniform tiling 444-t0.png Uniform tiling 84-h01.png Uniform tiling 443-t1.png Uniform tiling 444-snub.png Uniform tiling 88-t0.png H2-5-4-primal.svg Uniform tiling 84-snub.png
h{8,4} s{8,4} hr{8,4} s{4,8} h{4,8} hrr{8,4} sr{8,4}
Alternation duals
CDel node fh.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.pngCDel node fh.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel node.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel node.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node fh.pngCDel node fh.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node fh.pngCDel node fh.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node fh.png
Uniform tiling 88-t1.png Uniform tiling 66-t1.png Uniform dual tiling 433-t0.png Uniform tiling 88-t2.png H2-5-4-dual.svg
V(4.4)4V3.(3.8)2V(4.4.4)2V(3.4)3V88V4.44V3.3.4.3.8
Uniform octaoctagonal tilings
Symmetry: [8,8], (*882)
CDel node 1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png = CDel nodes 10ru.pngCDel split2-88.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node.png = CDel nodes 10ru.pngCDel split2-88.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node.png = CDel nodes.pngCDel split2-88.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node 1.png = CDel nodes 01rd.pngCDel split2-88.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node 1.png = CDel nodes 01rd.pngCDel split2-88.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node 1.png = CDel nodes 11.pngCDel split2-88.pngCDel node.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node 1.png = CDel nodes 11.pngCDel split2-88.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node 1.png
H2 tiling 288-1.png H2 tiling 288-3.png H2 tiling 288-2.png H2 tiling 288-6.png H2 tiling 288-4.png H2 tiling 288-5.png 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
CDel node f1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.pngCDel node f1.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node.pngCDel node.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node.pngCDel node.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node f1.pngCDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node f1.pngCDel node f1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node f1.pngCDel node f1.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node f1.png
H2chess 288b.png H2chess 288f.png H2chess 288a.png H2chess 288e.png H2chess 288c.png H2chess 288d.png H2checkers 288.png
V88 V8.16.16 V8.8.8.8 V8.16.16 V88 V4.8.4.8V4.16.16
Alternations
[1+,8,8]
(*884)
[8+,8]
(8*4)
[8,1+,8]
(*4242)
[8,8+]
(8*4)
[8,8,1+]
(*884)
[(8,8,2+)]
(2*44)
[8,8]+
(882)
CDel node h1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png = CDel label4.pngCDel branch 10ru.pngCDel split2-88.pngCDel node.pngCDel node h.pngCDel 8.pngCDel node h.pngCDel 8.pngCDel node.pngCDel node.pngCDel 8.pngCDel node h1.pngCDel 8.pngCDel node.png = CDel nodes 11.pngCDel 4a4b-cross.pngCDel nodes.pngCDel node.pngCDel 8.pngCDel node h.pngCDel 8.pngCDel node h.pngCDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node h1.png = CDel node.pngCDel split1-88.pngCDel branch 01ld.pngCDel node h.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node h.png = CDel nodes hh.pngCDel split2-88.pngCDel node.png
= CDel node h0.pngCDel 4.pngCDel node h.pngCDel 8.pngCDel node.png
CDel node h.pngCDel 8.pngCDel node h.pngCDel 8.pngCDel node h.png = CDel nodes hh.pngCDel split2-88.pngCDel node h.png
= CDel node h0.pngCDel 4.pngCDel node h.pngCDel 8.pngCDel node h.png
Uniform tiling 88-h0.png Uniform tiling 444-t0.png Uniform tiling 88-h0.png Uniform tiling 443-t1.png 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
CDel node fh.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.pngCDel node fh.pngCDel 8.pngCDel node fh.pngCDel 8.pngCDel node.pngCDel node.pngCDel 8.pngCDel node fh.pngCDel 8.pngCDel node.pngCDel node.pngCDel 8.pngCDel node fh.pngCDel 8.pngCDel node fh.pngCDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node fh.pngCDel node fh.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node fh.pngCDel node fh.pngCDel 8.pngCDel node fh.pngCDel 8.pngCDel node fh.png
Uniform tiling 88-t1.png Uniform tiling 66-t1.png
V(4.8)8V3.4.3.8.3.8 V(4.4)4 V3.4.3.8.3.8V(4.8)8 V46 V3.3.8.3.8

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

Truncated order-8 hexagonal tiling Semiregular tiling of the hyperbolic plane

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

References

See also