Order-3 apeirogonal tiling

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Order-3 apeirogonal tiling
H2-I-3-dual.svg
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 3
Schläfli symbol {,3}
t{,}
t(,,)
Wythoff symbol 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.

Contents

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

Images

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

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

Regular Truncations
H2-I-3-dual.svg
{∞,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
t{∞[3]}
CDel node 1.pngCDel split1-ii.pngCDel branch 11.pngCDel labelinfin.png
Hyperbolic triangle groups
H2checkers 23i.png
[∞,3]
H2checkers 2ii.png
[∞,∞]
Infinite-order triangular tiling.svg
[(∞,∞,∞)]

Symmetry

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.

A larger subgroup is constructed [(∞,∞,∞*)], index 8, as (∞*∞) with gyration points removed, becomes (*∞).

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

*n32 symmetry mutation of regular tilings: {n,3}
Spherical Euclidean Compact hyperb.Paraco.Noncompact hyperbolic
Spherical trigonal hosohedron.png Uniform tiling 332-t0.png Uniform tiling 432-t0.png Uniform tiling 532-t0.png Uniform polyhedron-63-t0.png Heptagonal tiling.svg H2-8-3-dual.svg H2-I-3-dual.svg H2 tiling 23j12-1.png H2 tiling 23j9-1.png H2 tiling 23j6-1.png H2 tiling 23j3-1.png
{2,3} {3,3} {4,3} {5,3} {6,3} {7,3} {8,3} {∞,3} {12i,3}{9i,3}{6i,3}{3i,3}
Paracompact 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.pngCDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node h.pngCDel infin.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 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.pngCDel 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-I-3-dual.svg 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 H2 snub 33ia.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.pngCDel node f1.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node fh.pngCDel infin.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel node fh.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 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-I-3-dual.svg Deltoidal triapeirogonal til.png H2checkers 23i.png Order-3-infinite floret pentagonal tiling.png Alternate order-3 apeirogonal tiling.png
V3 V3..V(3.)2V6.6. V3 V4.3.4.V4.6.V3.3.3.3.V(3.)3V3.3.3.3.3.
Paracompact uniform tilings in [,] family
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.pngCDel node f1.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node.pngCDel node.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node.pngCDel node.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node f1.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node f1.pngCDel node f1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node f1.pngCDel 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
VV..V(.)2V..VV4..4.V4.4.
Alternations
[1+,,]
(*2)
[+,]
(*)
[,1+,]
(*)
[,+]
(*)
[,,1+]
(*2)
[(,,2+)]
(2*)
[,]+
(2)
CDel node h.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.pngCDel node h.pngCDel infin.pngCDel node h.pngCDel infin.pngCDel node.pngCDel node.pngCDel infin.pngCDel node h.pngCDel infin.pngCDel node.pngCDel node.pngCDel infin.pngCDel node h.pngCDel infin.pngCDel node h.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node h.pngCDel node h.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node h.pngCDel 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.pngCDel node fh.pngCDel infin.pngCDel node fh.pngCDel infin.pngCDel node.pngCDel node.pngCDel infin.pngCDel node fh.pngCDel infin.pngCDel node.pngCDel node.pngCDel infin.pngCDel node fh.pngCDel infin.pngCDel node fh.pngCDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node fh.pngCDel node fh.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node fh.pngCDel 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.)3V(.4)4V(3.)3VV(4..4)2V3.3..3.
Paracompact uniform tilings in [(,,)] family
CDel labelinfin.pngCDel branch 01rd.pngCDel split2-ii.pngCDel node.pngCDel labelinfin.pngCDel branch 01rd.pngCDel split2-ii.pngCDel node 1.pngCDel labelinfin.pngCDel branch.pngCDel split2-ii.pngCDel node 1.pngCDel labelinfin.pngCDel branch 10ru.pngCDel split2-ii.pngCDel node 1.pngCDel labelinfin.pngCDel branch 10ru.pngCDel split2-ii.pngCDel node.pngCDel labelinfin.pngCDel branch 11.pngCDel split2-ii.pngCDel node.pngCDel labelinfin.pngCDel branch 11.pngCDel split2-ii.pngCDel node 1.png
CDel node h1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.pngCDel node h1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node 1.pngCDel node h0.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node 1.pngCDel node h1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node 1.pngCDel node h1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.pngCDel node h0.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel node h0.pngCDel infin.pngCDel node 1.pngCDel infin.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
(,,)
h{,}
r(,,)
h2{,}
(,,)
h{,}
r(,,)
h2{,}
(,,)
h{,}
r(,,)
r{,}
t(,,)
t{,}
Dual tilings
H2chess iiia.png H2chess iiif.png H2chess iiib.png H2chess iiid.png H2chess iiic.png H2chess iiie.png Infinite-order triangular tiling.svg
VV...VV...VV...V..
Alternations
[(1+,,,)]
(*)
[+,,)]
(*)
[,1+,,)]
(*)
[,+,)]
(*)
[(,,,1+)]
(*)
[(,,+)]
(*)
[,,)]+
()
CDel labelinfin.pngCDel branch 0hr.pngCDel split2-ii.pngCDel node.pngCDel labelinfin.pngCDel branch 0hr.pngCDel split2-ii.pngCDel node h.pngCDel labelinfin.pngCDel branch.pngCDel split2-ii.pngCDel node h1.pngCDel labelinfin.pngCDel branch h0r.pngCDel split2-ii.pngCDel node h.pngCDel labelinfin.pngCDel branch h0r.pngCDel split2-ii.pngCDel node.pngCDel labelinfin.pngCDel branch hh.pngCDel split2-ii.pngCDel node.pngCDel labelinfin.pngCDel branch hh.pngCDel split2-ii.pngCDel node h.png
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
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)4V(.)V(.4)4V(.)V(.4)4V3..3..3.

See also

Related Research Articles

Order-4 hexagonal tiling

In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}.

Order-6 square tiling

In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,6}.

Truncated tetrahexagonal tiling

In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{6,4}.

Rhombitetrahexagonal tiling

In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{6,4}. It can be seen as constructed as a rectified tetrahexagonal tiling, r{6,4}, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling.

Truncated order-4 hexagonal tiling

In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,4}. A secondary construction tr{6,6} is called a truncated hexahexagonal tiling with two colors of dodecagons.

Truncated order-6 square tiling

In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}.

Order-8 triangular tiling

In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {3,8}, having eight regular triangles around each vertex.

Truncated order-8 triangular tiling

In geometry, the truncated order-8 triangular tiling is a semiregular tiling of the hyperbolic plane. There are two hexagons and one octagon on each vertex. It has Schläfli symbol of t{3,8}.

Truncated order-4 octagonal tiling

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.

Truncated tetraoctagonal tiling Semiregular tiling in geometry

In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,4}.

Truncated order-8 octagonal tiling

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}.

Tetraapeirogonal tiling

In geometry, the tetraapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of r{∞,4}.

Truncated order-4 apeirogonal tiling

In geometry, the truncated order-4 apeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{∞,4}.

Truncated infinite-order square tiling

In geometry, the truncated infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,∞}.

Truncated tetraapeirogonal tiling

In geometry, the truncated tetraapeirogonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one apeirogon on each vertex. It has Schläfli symbol of tr{∞,4}.

Infinite-order apeirogonal tiling

In geometry, the infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,∞}, which means it has countably infinitely many apeirogons around all its ideal vertices.

Truncated hexaoctagonal tiling

In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,6}.

Truncated order-8 hexagonal tiling

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

Order-5 apeirogonal tiling

In geometry, the order-5 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,5}.

Order-6 apeirogonal tiling

In geometry, the order-6 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,6}.

References