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Octagonal tiling | |
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![]() 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 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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}.
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 | Truncations | ||
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![]() {8,3} ![]() ![]() ![]() ![]() ![]() | ![]() t{4,8} ![]() ![]() ![]() ![]() ![]() | ![]() t{4[3]} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
Dual tiling | |||
![]() {3,8} ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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} | |||||||||||
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Spherical | Euclidean | Compact hyperb. | Paraco. | Noncompact hyperbolic | |||||||
![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
{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}.
Space | Spherical | Compact hyperbolic | Paracompact | |||||
---|---|---|---|---|---|---|---|---|
Tiling | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | |
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 | |||||||||||||
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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} | |||
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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 | |||
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Uniform octagonal/square tilings | |||||||||||
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[8,4], (*842) (with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries) (And [(∞,4,∞,4)] (*4242) index 4 subsymmetry) | |||||||||||
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{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 | |||||||||||
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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) | |||||
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h{8,4} | s{8,4} | hr{8,4} | s{4,8} | h{4,8} | hrr{8,4} | sr{8,4} | |||||
Alternation duals | |||||||||||
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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 | |||||||||||
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Symmetry: [(4,4,4)], (*444) | [(4,4,4)]+ (444) | [(1+,4,4,4)] (*4242) | [(4+,4,4)] (4*22) | ||||||||
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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 | |||||||||||
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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 |
![]() | Wikimedia Commons has media related to Order-3 octagonal 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.
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.
In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}.
In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,6}.
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}.
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}.
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}.
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}.
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.
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.
In geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,4}.
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}.
In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,8}.
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.
In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,4}.
In geometry, the order-5 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,5}.
In geometry, the order-6 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,6}.
In geometry, the order-6 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,6}.
In geometry, the rhombihexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,6}.
In geometry, the order-5 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,5}.