Truncated triapeirogonal tiling

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Truncated triapeirogonal tiling
Truncated triapeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.6.∞
Schläfli symbol tr{∞,3}
Wythoff symbol 2 ∞ 3 |
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Symmetry group [∞,3], (*∞32)
Dual Order 3-infinite kisrhombille
Properties Vertex-transitive

In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr{∞,3}.

Symmetry[edit]

Truncated triapeirogonal tiling with mirrors

The dual of this tiling represents the fundamental domains of [∞,3], *∞32 symmetry. There are 3 small index subgroup constructed from [∞,3] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.

A special index 4 reflective subgroup, is [(∞,∞,3)], (*∞∞3), and its direct subgroup [(∞,∞,3)]+, (∞∞3), and semidirect subgroup [(∞,∞,3+)], (3*∞).[1] Given [∞,3] with generating mirrors {0,1,2}, then its index 4 subgroup has generators {0,121,212}.

An index 6 subgroup constructed as [∞,3*], becomes [(∞,∞,∞)], (*∞∞∞).

Small index subgroups of [∞,3], (*∞32)
Index 1 2 3 4 6 8 12 24
Diagrams I32 symmetry mirrors.png I32 symmetry a00.png I32 symmetry 0bb.png I32 symmetry mirrors-index3.png I32 symmetry mirrors-index4a.png I32 symmetry 0zz.png I32 symmetry mirrors-index6-i2i2.png I32 symmetry mirrors-index8a.png I32 symmetry mirrors-index12a.png I32 symmetry mirrors-index24a.png
Coxeter
(orbifold)
[∞,3]
CDel node c1.pngCDel infin.pngCDel node c2.pngCDel 3.pngCDel node c2.png = CDel node c2.pngCDel split1-i3.pngCDel branch c1-2.pngCDel label2.png
(*∞32)
[1+,∞,3]
CDel node h0.pngCDel infin.pngCDel node c2.pngCDel 3.pngCDel node c2.png = CDel labelinfin.pngCDel branch c2.pngCDel split2.pngCDel node c2.png
(*∞33)
[∞,3+]
CDel node c1.pngCDel infin.pngCDel node h2.pngCDel 3.pngCDel node h2.png
(3*∞)
[∞,∞]

(*∞∞2)
[(∞,∞,3)]

(*∞∞3)
[∞,3*]
CDel node c1.pngCDel infin.pngCDel node g.pngCDel 3sg.pngCDel node g.png = CDel labelinfin.pngCDel branch c1.pngCDel split2-ii.pngCDel node c1.png
(*∞3)
[∞,1+,∞]

(*(∞2)2)
[(∞,1+,∞,3)]

(*(∞3)2)
[1+,∞,∞,1+]

(*∞4)
[(∞,∞,3*)]

(*∞6)
Direct subgroups
Index 2 4 6 8 12 16 24 48
Diagrams I32 symmetry aaa.png I32 symmetry abb.png Ii2 symmetry aaa.png I32 symmetry mirrors-index4.png I32 symmetry azz.png Ii2 symmetry bab.png H2chess 26ia.png Ii2 symmetry abc.png H2chess 26ib.png
Coxeter
(orbifold)
[∞,3]+
CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 3.pngCDel node h2.png = CDel node h2.pngCDel split1-i3.pngCDel branch h2h2.pngCDel label2.png
(∞32)
[∞,3+]+
CDel node h0.pngCDel infin.pngCDel node h2.pngCDel 3.pngCDel node h2.png = CDel labelinfin.pngCDel branch h2h2.pngCDel split2.pngCDel node h2.png
(∞33)
[∞,∞]+

(∞∞2)
[(∞,∞,3)]+

(∞∞3)
[∞,3*]+
CDel node h2.pngCDel infin.pngCDel node g.pngCDel 3sg.pngCDel node g.png = CDel labelinfin.pngCDel branch h2h2.pngCDel split2-ii.pngCDel node h2.png
(∞3)
[∞,1+,∞]+

(∞2)2
[(∞,1+,∞,3)]+

(∞3)2
[1+,∞,∞,1+]+

(∞4)
[(∞,∞,3*)]+

(∞6)

Related polyhedra and tiling[edit]

Paracompact hyperbolic uniform tilings in [∞,3] family
Symmetry: [∞,3], (*∞32) [∞,3]+
(∞32)
[1+,∞,3]
(*∞33)
[∞,3+]
(3*∞)
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel infin.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png CDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel infin.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node h0.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel labelinfin.pngCDel branch 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel labelinfin.pngCDel branch 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel labelinfin.pngCDel branch.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel infin.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png =
CDel labelinfin.pngCDel branch 10ru.pngCDel split2.pngCDel node.png or CDel labelinfin.pngCDel branch 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel labelinfin.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png or CDel labelinfin.pngCDel branch 01rd.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel infin.pngCDel node h.pngCDel 3.pngCDel node h.png
= CDel labelinfin.pngCDel branch hh.pngCDel split2.pngCDel node h.png
H2 tiling 23i-1.png H2 tiling 23i-3.png H2 tiling 23i-2.png H2 tiling 23i-6.png H2 tiling 23i-4.png H2 tiling 23i-5.png H2 tiling 23i-7.png Uniform tiling i32-snub.png H2 tiling 33i-1.png
{∞,3} t{∞,3} r{∞,3} t{3,∞} {3,∞} rr{∞,3} tr{∞,3} sr{∞,3} h{∞,3} h2{∞,3} s{3,∞}
Uniform duals
CDel node f1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node fh.pngCDel 3.pngCDel node fh.png
H2 tiling 23i-4.png Ord-infin triakis triang til.png Ord3infin qreg rhombic til.png H2checkers 33i.png H2 tiling 23i-1.png Deltoidal triapeirogonal til.png H2checkers 23i.png Order-3-infinite floret pentagonal tiling.png Alternate order-3 apeirogonal tiling.png
V∞3 V3.∞.∞ V(3.∞)2 V6.6.∞ V3 V4.3.4.∞ V4.6.∞ V3.3.3.3.∞ V(3.∞)3 V3.3.3.3.3.∞

This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram CDel node 1.pngCDel p.pngCDel node 1.pngCDel 3.pngCDel node 1.png. For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

*n32 symmetry mutations of omnitruncated tilings: 4.6.2n
Sym.
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]
*∞32
[∞,3]
 
[12i,3]
 
[9i,3]
 
[6i,3]
 
[3i,3]
Figures Spherical truncated trigonal prism.png Uniform tiling 332-t012.png Uniform tiling 432-t012.png Uniform tiling 532-t012.png Uniform polyhedron-63-t012.png H2 tiling 237-7.png H2 tiling 238-7.png H2 tiling 23i-7.png H2 tiling 23j12-7.png H2 tiling 23j9-7.png H2 tiling 23j6-7.png H2 tiling 23j3-7.png
Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞ 4.6.24i 4.6.18i 4.6.12i 4.6.6i
Duals Spherical hexagonal bipyramid.png Spherical tetrakis hexahedron.png Spherical disdyakis dodecahedron.png Spherical disdyakis triacontahedron.png Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg H2checkers 237.png H2checkers 238.png H2checkers 23i.png H2 checkers 23j12.png H2 checkers 23j9.png H2 checkers 23j6.png H2 checkers 23j3.png
Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i

See also[edit]

References[edit]

  1. ^ Norman W. Johnson and Asia Ivic Weiss, Quadratic Integers and Coxeter Groups, Canad. J. Math. Vol. 51 (6), 1999 pp. 1307–1336 [1]

External links[edit]