Order-3 apeirogonal tiling

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Order-3 apeirogonal tiling
Order-3 apeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex figure ∞.∞.∞
Schläfli symbol {∞,3}
t{∞,∞}
t{(∞,∞,∞)}
Wythoff symbol 3 | ∞ 2
2 ∞ | ∞
∞ ∞ ∞ |
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel labelinfin.pngCDel branch 11.pngCDel split2-ii.pngCDel node 1.png
Symmetry group [∞,3], (*∞32)
[∞,∞], (*∞∞2)
[(∞,∞,∞)], (*∞∞∞)
Dual Infinite-order triangular tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular apeirogons around each vertex. Each apeirogon is inscribed in a horocycle.

The order-2 apeirogonal tiling represents an infinite dihedron in the Euclidean plane as {∞,2}.

Images[edit]

Each apeirogon face is circumscribed by a horocycle, which looks like a circle in a Poincaré disk model, internally tangent to the projective circle boundary.

Order-3 apeirogonal tiling one cell horocycle.png

Uniform colorings[edit]

Like the Euclidean hexagonal tiling, there are 3 uniform colorings of the order-3 apeirogonal tiling, each from different reflective triangle group domains:

Regular Truncation Omnitruncation
H2 tiling 23i-1.png
{∞,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 2ii-3.png
t0,1{∞,∞}
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.png
H2 tiling 2ii-6.png
t1,2{∞,∞}
CDel node.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.png
H2 tiling iii-7.png
t0,1,2{(∞,∞,∞)}
CDel node 1.pngCDel split1-ii.pngCDel branch 11.pngCDel labelinfin.png
Hyperbolic triangle groups
H2checkers 23i.png
[∞,3]
H2checkers 2ii.png
[∞,∞]
H2checkers iii.png
[(∞,∞,∞)]

Symmetry[edit]

The dual to this tiling represents the fundamental domains of [(∞,∞,∞)] (*∞∞∞) symmetry. There are 15 small index subgroups (7 unique) constructed from [(∞,∞,∞)] 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 symmetry can be doubled as ∞∞2 symmetry by adding a mirror bisecting the fundamental domain. Dividing a fundamental domain by 3 mirrors creates a ∞32 symmetry.

Small index subgroups of [(∞,∞,∞)] (*∞∞∞)
Iii symmetry mirrors.png
Fundamental
domains
H2checkers iii.png H2chess iiif.png
H2chess iiic.png
H2chess iiid.png
H2chess iiia.png
H2chess iiie.png
H2chess iiib.png
H2chess iiic.png
H2chess iiia.png
H2chess iiib.png
Subgroup index 1 2 4
Coxeter
(orbifold)
[(∞,∞,∞)]
(*∞3)
[(1+,∞,∞,∞)]
(*∞4)
[(∞,1+,∞,∞)]
(*∞4)
[(∞,∞,1+,∞)]
(*∞4)
[(1+,∞,1+,∞,∞)]
(∞*∞4)
[(∞,1+,∞,1+,∞)]
(∞*∞4)
[(∞,∞+,∞)] [(∞+,∞,∞)] [(∞,∞,∞+)] [(1+,∞,∞,1+,∞)]
(∞*∞4)
[(∞+,∞+,∞)]
(∞∞∞×)
Rotational subgroups
Subgroup index 2 4 8
Coxeter
(orbifold)
[(∞,∞,∞)]+
(∞3)
[(1+,∞,∞+,∞)]
(∞4)
[(∞+,∞,1+,∞)]
(∞4)
[(∞,1+,∞,∞+)]
(∞4)
[(1+,∞,1+,∞,1+,∞)]
(∞6)

Related polyhedra and tilings[edit]

This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {n,3}.

Spherical
Polyhedra
Polyhedra Euclidean Hyperbolic tilings
Spherical trigonal hosohedron.png
{2,3}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-33-t0.png
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t0.png
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-53-t0.png
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-63-t0.png
{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 237-1.png
{7,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 238-1.png
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
... H2 tiling 23i-1.png
(∞,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
Noncompact 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.∞
Noncompact hyperbolic uniform tilings in [∞,∞] family
Symmetry: [∞,∞], (*∞∞2)
CDel node 1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node.png
= CDel node 1.pngCDel split1-ii.pngCDel branch.pngCDel labelinfin.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node 1.png
= CDel node 1.pngCDel split1-ii.pngCDel branch 11.pngCDel labelinfin.png
CDel node.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.png
= CDel node h0.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node 1.png
= CDel labelinfin.pngCDel branch 11.pngCDel split2-ii.pngCDel node.png
CDel node.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node 1.png
= CDel labelinfin.pngCDel branch 11.pngCDel split2-ii.pngCDel node 1.png
CDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node.png
= CDel labelinfin.pngCDel branch.pngCDel split2-ii.pngCDel node 1.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel infin.pngCDel node 1.png
H2 tiling 2ii-1.png H2 tiling 2ii-3.png H2 tiling 2ii-2.png H2 tiling 2ii-6.png H2 tiling 2ii-4.png H2 tiling 2ii-5.png H2 tiling 2ii-7.png
{∞,∞} t{∞,∞} r{∞,∞} 2t{∞,∞}=t{∞,∞} 2r{∞,∞}={∞,∞} rr{∞,∞} tr{∞,∞}
Dual tilings
CDel node f1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node f1.png CDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node f1.png
H2chess 2iib.png H2chess 2iif.png H2chess 2iia.png H2chess 2iie.png H2chess 2iic.png H2chess 2iid.png H2checkers 2ii.png
V∞ V∞.∞.∞ V(∞.∞)2 V∞.∞.∞ V∞ V4.∞.4.∞ V4.4.∞
Alternations
[1+,∞,∞]
(*∞∞2)
[∞+,∞]
(∞*∞)
[∞,1+,∞]
(*∞∞∞∞)
[∞,∞+]
(∞*∞)
[∞,∞,1+]
(*∞∞2)
[(∞,∞,2+)]
(2*∞∞)
[∞,∞]+
(2∞∞)
CDel node h.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.png CDel node h.pngCDel infin.pngCDel node h.pngCDel infin.pngCDel node.png CDel node.pngCDel infin.pngCDel node h.pngCDel infin.pngCDel node.png CDel node.pngCDel infin.pngCDel node h.pngCDel infin.pngCDel node h.png CDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node h.png CDel node h.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node h.png CDel node h.pngCDel infin.pngCDel node h.pngCDel infin.pngCDel node h.png
H2 tiling 2ii-1.png H2 tiling 33i-1.png H2 tiling 44i-1.png H2 tiling 33i-2.png H2 tiling 2ii-4.png Uniform tiling ii2-snub.png
h{∞,∞} s{∞,∞} hr{∞,∞} s{∞,∞} h2{∞,∞} hrr{∞,∞} sr{∞,∞}
Alternation duals
CDel node fh.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel infin.pngCDel node.png CDel node.pngCDel infin.pngCDel node fh.pngCDel infin.pngCDel node.png CDel node.pngCDel infin.pngCDel node fh.pngCDel infin.pngCDel node fh.png CDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel infin.pngCDel node fh.png
H2 tiling 2ii-4.png H2chess 44ib.png H2 tiling 2ii-1.png Infinitely-infinite-order floret pentagonal tiling.png
V(∞.∞) V(3.∞)3 V(∞.4)4 V(3.∞)3 V∞ V(4.∞.4)2 V3.3.∞.3.∞
Noncompact hyperbolic uniform tilings in [(∞,∞,∞)] family
Symmetry: [(∞,∞,∞)], (*∞∞∞)
CDel 3.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.png CDel 3.pngCDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel infin.png CDel 3.pngCDel node.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel infin.png CDel 3.pngCDel node.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.png CDel 3.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node 1.pngCDel infin.png CDel 3.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node 1.pngCDel infin.png CDel 3.pngCDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.png
CDel labelinfin.pngCDel branch 01rd.pngCDel split2-ii.pngCDel node.png CDel labelinfin.pngCDel branch 01rd.pngCDel split2-ii.pngCDel node 1.png CDel labelinfin.pngCDel branch.pngCDel split2-ii.pngCDel node 1.png CDel labelinfin.pngCDel branch 10ru.pngCDel split2-ii.pngCDel node 1.png CDel labelinfin.pngCDel branch 10ru.pngCDel split2-ii.pngCDel node.png CDel labelinfin.pngCDel branch 11.pngCDel split2-ii.pngCDel node.png CDel labelinfin.pngCDel branch 11.pngCDel split2-ii.pngCDel node 1.png
H2 tiling iii-1.png H2 tiling iii-3.png H2 tiling iii-2.png H2 tiling iii-6.png H2 tiling iii-4.png H2 tiling iii-5.png H2 tiling iii-7.png
(∞,∞,∞) r(∞,∞,∞) (∞,∞,∞) r(∞,∞,∞) (∞,∞,∞) r(∞,∞,∞) t(∞,∞,∞)
Dual tilings
CDel 3.pngCDel node f1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.png CDel 3.pngCDel node f1.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node.pngCDel infin.png CDel 3.pngCDel node.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node.pngCDel infin.png CDel 3.pngCDel node.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node f1.pngCDel infin.png CDel 3.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node f1.pngCDel infin.png CDel 3.pngCDel node f1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node f1.pngCDel infin.png CDel 3.pngCDel node f1.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node f1.pngCDel infin.png
H2chess iiia.png H2chess iiif.png H2chess iiib.png H2chess iiid.png H2chess iiic.png H2chess iiie.png H2checkers iii.png
V∞ V∞.∞.∞.∞ V∞ V∞.∞.∞.∞ V∞ V∞.∞.∞.∞ V∞.∞.∞
Alternations
[(1+,∞,∞,∞)]
(*∞∞∞∞)
[∞+,∞,∞)]
(∞*∞)
[∞,1+,∞,∞)]
(*∞∞∞∞)
[∞,∞+,∞)]
(∞*∞)
[(∞,∞,∞,1+)]
(*∞∞∞∞)
[(∞,∞,∞+)]
(∞*∞)
[∞,∞,∞)]+
(∞∞∞)
CDel 3.pngCDel node h1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.png CDel 3.pngCDel node h.pngCDel infin.pngCDel node h.pngCDel infin.pngCDel node.pngCDel infin.png CDel 3.pngCDel node.pngCDel infin.pngCDel node h1.pngCDel infin.pngCDel node.pngCDel infin.png CDel 3.pngCDel node.pngCDel infin.pngCDel node h.pngCDel infin.pngCDel node h.pngCDel infin.png CDel 3.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node h1.pngCDel infin.png CDel 3.pngCDel node h.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node h.pngCDel infin.png CDel 3.pngCDel node h.pngCDel infin.pngCDel node h.pngCDel infin.pngCDel node h.pngCDel infin.png
CDel labelinfin.pngCDel branch 0hr.pngCDel split2-ii.pngCDel node.png CDel labelinfin.pngCDel branch 0hr.pngCDel split2-ii.pngCDel node h.png CDel labelinfin.pngCDel branch.pngCDel split2-ii.pngCDel node h1.png CDel labelinfin.pngCDel branch h0r.pngCDel split2-ii.pngCDel node h.png CDel labelinfin.pngCDel branch h0r.pngCDel split2-ii.pngCDel node.png CDel labelinfin.pngCDel branch hh.pngCDel split2-ii.pngCDel node.png CDel labelinfin.pngCDel branch hh.pngCDel split2-ii.pngCDel node h.png
h(∞,∞,∞) hr(∞,∞,∞) h(∞,∞,∞) hr(∞,∞,∞) h(∞,∞,∞) hr(∞,∞,∞) s(∞,∞,∞)
H2 tiling 2ii-1.png H2 tiling 44i-1.png H2 tiling 2ii-1.png H2 tiling 44i-1.png H2 tiling 2ii-1.png H2 tiling 44i-1.png Uniform tiling iii-snub.png
Alternation duals
CDel 3.pngCDel node fh.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.png CDel 3.pngCDel node fh.pngCDel infin.pngCDel node fh.pngCDel infin.pngCDel node.pngCDel infin.png CDel 3.pngCDel node.pngCDel infin.pngCDel node fh.pngCDel infin.pngCDel node.pngCDel infin.png CDel 3.pngCDel node.pngCDel infin.pngCDel node fh.pngCDel infin.pngCDel node fh.pngCDel infin.png CDel 3.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node fh.pngCDel infin.png CDel 3.pngCDel node fh.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node fh.pngCDel infin.png CDel 3.pngCDel node fh.pngCDel infin.pngCDel node fh.pngCDel infin.pngCDel node fh.pngCDel infin.png
H2 tiling 2ii-4.png H2chess 44ib.png H2 tiling 2ii-4.png H2chess 44ib.png H2 tiling 2ii-4.png H2chess 44ib.png
V(∞.∞) V(∞.4)4 V(∞.∞) V(∞.4)4 V(∞.∞) V(∞.4)4 V3.∞.3.∞.3.∞

See also[edit]

References[edit]

External links[edit]