Octagonal tiling

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For other uses, see truncated square tiling.
Octagonal tiling
Octagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex figure 8.8.8
Schläfli symbol {8,3}
t{4,8}
Wythoff symbol 3 | 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.

Uniform colorings[edit]

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.

Regular Truncation Omnitruncation
Uniform tiling 83-t0.png
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 84-t12.png
t1,2{8,4}
CDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Uniform tiling 444-t012.png
t0,1,2(4,4,4)
CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png = CDel label4.pngCDel node 1.pngCDel split1-44.pngCDel branch 11.png
Dual tiling
Uniform tiling 83-t2.png
{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
f0,1,2(4,4,4)
CDel node h0.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.png = CDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.png

Symmetry[edit]

From [(4,4,4)] symmetry, there are 15 small index subgroups (7 unique) by mirror removal and alternation operators. 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 to 842 symmetry by adding a bisecting mirror across the fundamental domains. Adding 3 bisecting mirrors across each fundamental domains creates 832 symmetry. The subgroup index-8 group, [(1+,4,1+,4,1+,4)] (222222) is the commutator subgroup of [(4,4,4)].

A larger subgroup is constructed [(4,4,4*)], index 8, as (2*2222) with gyration points removed, becomes (*22222222).

Small index subgroup symmetries of [(4,4,4)] (*444)
444 symmetry mirrors.png H2checkers 444.png H2chess 444e.png
H2chess 444b.png
H2chess 444f.png
H2chess 444c.png
H2chess 444d.png
H2chess 444a.png
H2chess 444b.png
H2chess 444c.png
H2chess 444a.png
Subgroup index 1 2 4
Coxeter
(orbifold)
[(4,4,4)]
(*444)
[(1+,4,4,4)]
(*4242)
[(4,4,1+,4)]
(*4242)
[(4,1+,4,4)]
(*4242)
[(4,1+,4,1+,4)]
2*2222
[(1+,4,4,1+,4)]
(2*2222)
[(4,4+,4)]
(4*22)
[(4+,4,4)]
(4*22)
[(4,4,4+)]
(4*22)
[(1+,4,1+,4,4)]
2*2222
[(4+,4+,4)]
(222×)
Rotational subgroups
Subgroup index 2 4 8
Coxeter
(orbifold)
[(4,4,4)]+
(444)
[(1+,4,4+,4)]
(4242)
[(4+,4,1+,4)]
(4242)
[(4,1+,4,4+)]
(4242)
[(1+,4,1+,4,1+,4)]
(222222)

Related polyhedra and tilings[edit]

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

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

{8,2}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node.png
Uniform tiling 83-t0.png
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 84-t0.png
{8,4}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 85-t0.png
{8,5}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 86-t0.png
{8,6}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 87-t0.png
{8,7}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 7.pngCDel node.png
Uniform tiling 88-t2.png
{8,8}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png
... H2 tiling 28i-4.png
{8,∞}
CDel node 1.pngCDel 8.pngCDel node.pngCDel infin.pngCDel node.png

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.png CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png CDel 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 1.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png CDel 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
Uniform tiling 83-t0.png Uniform tiling 83-t01.png Uniform tiling 83-t1.png
Uniform tiling 433-t02.png
Uniform tiling 83-t12.png
Uniform tiling 433-t012.png
Uniform tiling 83-t2.png Uniform tiling 83-t02.png Uniform tiling 83-t012.png Uniform tiling 83-snub.png Uniform tiling 433-t0.pngUniform tiling 433-t1.png Uniform tiling 433-t02.pngUniform tiling 433-t12.png Uniform tiling 433-snub1.png
Uniform tiling 433-snub2.png
Uniform duals
V83 V3.16.16 V3.8.3.8 V6.6.8 V38 V3.4.8.4 V4.6.16 V34.8 V(3.4)3 V8.6.6 V35.4
CDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Uniform tiling 83-t2.png Ord8 triakis triang til.png Uniform dual tiling 433-t01-yellow.png Uniform dual tiling 433-t012.png Uniform tiling 83-t0.png Deltoidal trioctagonal til.png Order-3 octakis octagonal tiling.png 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.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node f1.png CDel 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 V4.16.16 V(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 Uniform tiling 54-t2.png 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.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node fh.png CDel 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 Uniform tiling 54-t0.png
V(4.4)4 V3.(3.8)2 V(4.4.4)2 V(3.4)3 V88 V4.44 V3.3.4.3.8
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 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.png CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.png CDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 4.png CDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.png CDel 3.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 4.png CDel 3.pngCDel node.pngCDel 4.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h.pngCDel 4.png
CDel label4.pngCDel branch 01rd.pngCDel split2-44.pngCDel node.png CDel label4.pngCDel branch 01rd.pngCDel split2-44.pngCDel node 1.png CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node 1.png CDel label4.pngCDel branch 10ru.pngCDel split2-44.pngCDel node 1.png CDel label4.pngCDel branch 10ru.pngCDel split2-44.pngCDel node.png CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node.png CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.png CDel label4.pngCDel branch hh.pngCDel split2-44.pngCDel node h.png CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node h1.png CDel label4.pngCDel branch hh.pngCDel split2-44.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)} t0,1{(4,4,4)} t1{(4,4,4)} t1,2{(4,4,4)} t2{(4,4,4)} t0,2{(4,4,4)} t0,1,2{(4,4,4)} s{(4,4,4)} h{(4,4,4)} hr{(4,4,4)}
Uniform duals
CDel 3.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.png CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 4.png CDel 3.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 4.png CDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.png CDel 3.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 4.png CDel 3.pngCDel node.pngCDel 4.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 4.png CDel 3.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node fh.pngCDel 4.png
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[edit]

References[edit]

External links[edit]