Truncated infinite-order triangular tiling

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Infinite-order truncated triangular tiling
Truncated infinite-order triangular tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration ∞.6.6
Schläfli symbol t{3,∞}
Wythoff symbol 2 ∞ | 3
Coxeter diagram CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Symmetry group [∞,3], (*∞32)
Dual apeirokis apeirogonal tiling
Properties Vertex-transitive

In geometry, the truncated infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{3,∞}.

Symmetry[edit]

Truncated infinite-order triangular tiling with mirror lines

The dual of this tiling represents the fundamental domains of *∞33 symmetry. There are no mirror removal subgroups of [(∞,3,3)], but this symmetry group can be doubled to ∞32 symmetry by adding a mirror.

Small index subgroups of [(∞,3,3)], (*∞33)
Type Reflectional Rotational
Index 1 2
Diagram I33 symmetry 000.png I33 symmetry aaa.png
Coxeter
(orbifold)
[(∞,3,3)]
CDel node c1.pngCDel split1.pngCDel branch c1.pngCDel labelinfin.png
(*∞33)
[(∞,3,3)]+
CDel node h2.pngCDel split1.pngCDel branch h2h2.pngCDel labelinfin.png
(∞33)

Related polyhedra and tiling[edit]

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (6.n.n), and [n,3] Coxeter group symmetry.

*n32 symmetry mutation of truncated tilings: n.6.6
Sym.
*n42
[n,3]
Spherical Euclid. Compact hyperb. Parac. 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]
Truncated
figures
Hexagonal dihedron.png Uniform tiling 332-t12.png Uniform tiling 432-t12.png Uniform tiling 532-t12.png Uniform tiling 63-t12.png H2 tiling 237-6.png H2 tiling 238-6.png H2 tiling 23i-6.png H2 tiling 23j12-6.png H2 tiling 23j9-6.png H2 tiling 23j-6.png
Config. 2.6.6 3.6.6 4.6.6 5.6.6 6.6.6 7.6.6 8.6.6 ∞.6.6 12i.6.6 9i.6.6 6i.6.6
n-kis
figures
Hexagonal Hosohedron.svg Spherical triakis tetrahedron.png Spherical tetrakis hexahedron.png Spherical pentakis dodecahedron.png Uniform tiling 63-t2.png Order3 heptakis heptagonal til.png H2checkers 334.png H2checkers 33i.png
Config. V2.6.6 V3.6.6 V4.6.6 V5.6.6 V6.6.6 V7.6.6 V8.6.6 V∞.6.6 V12i.6.6 V9i.6.6 V6i.6.6
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.∞
Paracompact hyperbolic uniform tilings in [(∞,3,3)] family
Symmetry: [(∞,3,3)], (*∞33) [(∞,3,3)]+, (∞33)
CDel labelinfin.pngCDel branch 01rd.pngCDel split2.pngCDel node.png CDel labelinfin.pngCDel branch 11.pngCDel split2.pngCDel node.png CDel labelinfin.pngCDel branch 10ru.pngCDel split2.pngCDel node.png CDel labelinfin.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png CDel labelinfin.pngCDel branch.pngCDel split2.pngCDel node 1.png CDel labelinfin.pngCDel branch 01rd.pngCDel split2.pngCDel node 1.png CDel labelinfin.pngCDel branch 11.pngCDel split2.pngCDel node 1.png CDel labelinfin.pngCDel branch hh.pngCDel split2.pngCDel node h.png
CDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png CDel node h0.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.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 h0.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel infin.pngCDel node h.pngCDel 3.pngCDel node h.png
H2 tiling 33i-1.png H2 tiling 33i-3.png H2 tiling 33i-2.png H2 tiling 33i-6.png H2 tiling 33i-4.png H2 tiling 33i-5.png H2 tiling 33i-7.png
{(∞,∞,3)} t0,1{(∞,3,3)} t1(∞,3,3) t1,2(∞,3,3) t2{(∞,3,3)} t0,2(∞,3,3) t0,1,2{(∞,3,3)} s(∞,3,3)
Dual tilings
CDel 3.pngCDel node f1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node fh.pngCDel infin.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.png
Ord3infin qreg rhombic til.png H2 tiling 23i-1.png H2checkers 33i.png
V(3.∞)3 V3.∞.3.∞ V(3.∞)3 V3.6.∞.6 V(3.3) V3.6.∞.6 V6.6.∞ V3.3.3.3.3.∞

See also[edit]

References[edit]

External links[edit]