Octagonal tiling

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Octagonal tiling
H2-8-3-dual.svg
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 83
Schläfli symbol {8,3}
t{4,8}
Wythoff symbol 8 2
2 8 | 4
4 4 4 |
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel node 1.pngCDel split1-44.pngCDel branch 11.pngCDel label4.png
Symmetry group [8,3], (*832)
[8,4], (*842)
[(4,4,4)], (*444)
Dual Order-8 triangular tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {8,3}, having three regular octagons around each vertex. It also has a construction as a truncated order-8 square tiling, t{4,8}.

Contents

Uniform colorings

Like the hexagonal tiling of the Euclidean plane, there are 3 uniform colorings of this hyperbolic tiling. The dual tiling V8.8.8 represents the fundamental domains of [(4,4,4)] symmetry.

RegularTruncations
H2-8-3-dual.svg
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 84-t12.png
t{4,8}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.png
Uniform tiling 444-t012.png
t{4[3]}
CDel node 1.pngCDel 8.pngCDel node g.pngCDel 3sg.pngCDel node g.png = CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node h0.png = CDel node 1.pngCDel split1-44.pngCDel branch 11.pngCDel label4.png
Dual tiling
H2-8-3-primal.svg
{3,8}
CDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png = CDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
Uniform tiling 433-t2.png
CDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.png = CDel node 1.pngCDel split1.pngCDel branch.pngCDel label4.png
H2checkers 444.png
CDel node f1.pngCDel 8.pngCDel node g.pngCDel 3sg.pngCDel node g.png = CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 8.pngCDel node h0.png = CDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.png

This tiling is topologically part of sequence of regular polyhedra and tilings 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}

And also is topologically part of sequence of regular tilings with Schläfli symbol {8,n}.

n82 symmetry mutations of regular tilings: 8n
SpaceSphericalCompact hyperbolicParacompact
Tiling H2-8-3-dual.svg H2 tiling 248-1.png H2 tiling 258-1.png H2 tiling 268-1.png H2 tiling 278-1.png H2 tiling 288-4.png H2 tiling 28i-4.png
Config. 8.8 83 84 85 86 87 88 ...8

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.

Uniform octagonal/triangular tilings
Symmetry: [8,3], (*832) [8,3]+
(832)
[1+,8,3]
(*443)
[8,3+]
(3*4)
{8,3} t{8,3} r{8,3} t{3,8} {3,8} rr{8,3}
s2{3,8}
tr{8,3} sr{8,3} h{8,3} h2{8,3} s{3,8}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node h.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel node.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel label4.pngCDel branch.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node.png or CDel label4.pngCDel branch 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png or CDel label4.pngCDel branch 01rd.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel label4.pngCDel branch hh.pngCDel split2.pngCDel node h.png
H2-8-3-dual.svg H2-8-3-trunc-dual.svg H2-8-3-rectified.svg
Uniform tiling 433-t01.png
H2-8-3-trunc-primal.svg
Uniform tiling 433-t012.png
H2-8-3-primal.svg
Uniform tiling 433-t2.png
H2-8-3-cantellated.svg H2-8-3-omnitruncated.svg H2-8-3-snub.svg Uniform tiling 433-t0.png Uniform tiling 433-t1.png Uniform tiling 433-t02.png Uniform tiling 433-t12.png Uniform tiling 433-snub1.png
Uniform tiling 433-snub2.png
Uniform duals
V83 V3.16.16V3.8.3.8V6.6.8 V38 V3.4.8.4 V4.6.16 V34.8V(3.4)3V8.6.6V35.4
CDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node fh.pngCDel 8.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel node fh.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel node fh.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node.pngCDel 8.pngCDel node fh.pngCDel 3.pngCDel node fh.png
H2-8-3-primal.svg H2-8-3-kis-primal.svg H2-8-3-rhombic.svg H2-8-3-kis-dual.svg H2-8-3-dual.svg H2-8-3-deltoidal.svg H2-8-3-kisrhombille.svg H2-8-3-floret.svg Uniform dual tiling 433-t0.png Uniform dual tiling 433-t01.png Uniform dual tiling 433-snub.png
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 (4,4,4) tilings
Symmetry: [(4,4,4)], (*444) [(4,4,4)]+
(444)
[(1+,4,4,4)]
(*4242)
[(4+,4,4)]
(4*22)
CDel label4.pngCDel branch 01rd.pngCDel split2-44.pngCDel node.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
CDel label4.pngCDel branch 01rd.pngCDel split2-44.pngCDel node 1.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node 1.png
CDel node h0.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel label4.pngCDel branch 10ru.pngCDel split2-44.pngCDel node 1.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel label4.pngCDel branch 10ru.pngCDel split2-44.pngCDel node.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node.png
CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.png
CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel label4.pngCDel branch hh.pngCDel split2-44.pngCDel node h.png
CDel node h0.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node h.png
CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node h1.png
CDel node h0.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node h1.png
CDel label4.pngCDel branch hh.pngCDel split2-44.pngCDel node.png
CDel node h0.pngCDel 8.pngCDel node h1.pngCDel 4.pngCDel node.png
H2 tiling 444-1.png H2 tiling 444-3.png H2 tiling 444-2.png H2 tiling 444-6.png H2 tiling 444-4.png H2 tiling 444-5.png H2 tiling 444-7.png Uniform tiling 444-snub.png H2 tiling 288-4.png H2 tiling 344-2.png
t0(4,4,4)
h{8,4}
t0,1(4,4,4)
h2{8,4}
t1(4,4,4)
{4,8}1/2
t1,2(4,4,4)
h2{8,4}
t2(4,4,4)
h{8,4}
t0,2(4,4,4)
r{4,8}1/2
t0,1,2(4,4,4)
t{4,8}1/2
s(4,4,4)
s{4,8}1/2
h(4,4,4)
h{4,8}1/2
hr(4,4,4)
hr{4,8}1/2
Uniform duals
H2chess 444b.png H2chess 444f.png H2chess 444a.png H2chess 444e.png H2chess 444c.png H2chess 444d.png H2checkers 444.png Uniform dual tiling 433-t0.png H2 tiling 288-1.png H2 tiling 266-2.png
V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V8.8.8 V3.4.3.4.3.4 V88 V(4,4)3

See also

Related Research Articles

Heptagonal tiling

In geometry, the heptagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {7,3}, having three regular heptagons around each vertex.

Order-4 pentagonal tiling Regular tiling of the hyperbolic plane

In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,4}. It can also be called a pentapentagonal tiling in a bicolored quasiregular form.

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}.

Snub trioctagonal tiling

In geometry, the order-3 snub octagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles, one octagon on each vertex. It has Schläfli symbol of sr{8,3}.

Trioctagonal tiling

In geometry, the trioctagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 octagonal tiling. There are two triangles and two octagons alternating on each vertex. It has Schläfli symbol of r{8,3}.

Truncated octagonal tiling

In geometry, the Truncated octagonal tiling is a semiregular tiling of the hyperbolic plane. There is one triangle and two hexakaidecagons on each vertex. It has Schläfli symbol of t{8,3}.

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}.

Rhombitrioctagonal tiling

In geometry, the rhombitrioctagonal tiling is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one octagon, alternating between two squares. The tiling has Schläfli symbol rr{8,3}. It can be seen as constructed as a rectified trioctagonal tiling, r{8,3}, as well as an expanded octagonal tiling or expanded order-8 triangular tiling.

Order-6 hexagonal tiling

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

Order-4 heptagonal tiling

In geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,4}.

Order-4 octagonal tiling

In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be called a octaoctagonal tiling, and Schläfli symbol of r{8,8}.

Order-8 square tiling

In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,8}.

Order-8 octagonal tiling

In geometry, the order-8 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,8} and is self-dual.

Order-4 apeirogonal tiling

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

Order-5 hexagonal tiling

In geometry, the order-5 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,5}.

Order-6 pentagonal tiling Regular tiling of the hyperbolic plane

In geometry, the order-6 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,6}.

Order-6 octagonal tiling

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

Rhombihexaoctagonal tiling

In geometry, the rhombihexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,6}.

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}.

References