Order-4 octagonal tiling
Order-4 octagonal tiling | |
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![]() Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic regular tiling |
Vertex configuration | 84 |
Schläfli symbol | {8,4} r{8,8} |
Wythoff symbol | 4 | 8 2 |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symmetry group | [8,4], (*842) [8,8], (*882) |
Dual | Order-8 square tiling |
Properties | Vertex-transitive, edge-transitive, face-transitive |
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}.
Uniform constructions
There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,8] kaleidoscope. Removing the mirror between the order 2 and 4 points, [8,8,1+], gives [(8,8,4)], (*884) symmetry. Removing two mirrors as [8,4*], leaves remaining mirrors *4444 symmetry.
Uniform Coloring | ![]() | ![]() | ![]() | ![]() |
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Symmetry | [8,4] (*842) ![]() ![]() ![]() ![]() ![]() | [8,8] (*882) ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | [(8,4,8)] = [8,8,1+] (*884) ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
| [1+,8,8,1+] (*4444) ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symbol | {8,4} | r{8,8} | r(8,4,8) = r{8,8}1⁄2 | r{8,4}1⁄8 = r{8,8}1⁄4 |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
| ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symmetry
This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting as edges of a regular hexagon. This symmetry by orbifold notation is called (*22222222) or (*28) with 8 order-2 mirror intersections. In Coxeter notation can be represented as [8*,4], removing two of three mirrors (passing through the octagon center) in the [8,4] symmetry. Adding a bisecting mirror through 2 vertices of an octagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 4 bisecting mirrors through the vertices defines *444 symmetry. Adding 4 bisecting mirrors through the edge defines *4222 symmetry. Adding all 8 bisectors leads to full *842 symmetry.
![]() *444 | ![]() *4222 | ![]() *832 |
The kaleidoscopic domains can be seen as bicolored octagonal tiling, representing mirror images of the fundamental domain. This coloring represents the uniform tiling r{8,8}, a quasiregular tiling and it can be called a octaoctagonal tiling.
![]() | ![]() |
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram , progressing to infinity.
*n42 symmetry mutation of regular tilings: {n,4}
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Spherical | Euclidean | Hyperbolic tilings | |||||
![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
24 | 34 | 44 | 54 | 64 | 74 | 84 | ...∞4 |
Regular tilings: {n,8}
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Spherical | Hyperbolic tilings | ||||||||||
![]() {2,8} ![]() ![]() ![]() ![]() ![]() | ![]() {3,8} ![]() ![]() ![]() ![]() ![]() | ![]() {4,8} ![]() ![]() ![]() ![]() ![]() | ![]() {5,8} ![]() ![]() ![]() ![]() ![]() | ![]() {6,8} ![]() ![]() ![]() ![]() ![]() | ![]() {7,8} ![]() ![]() ![]() ![]() ![]() | ![]() {8,8} ![]() ![]() ![]() ![]() ![]() | ... | ![]() {∞,8} ![]() ![]() ![]() ![]() ![]() |
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
![]() {3,4} ![]() ![]() ![]() ![]() ![]() | ![]() {4,4} ![]() ![]() ![]() ![]() ![]() | ![]() {5,4} ![]() ![]() ![]() ![]() ![]() | ![]() {6,4} ![]() ![]() ![]() ![]() ![]() | ![]() {7,4} ![]() ![]() ![]() ![]() ![]() | ![]() {8,4} ![]() ![]() ![]() ![]() ![]() | ... | ![]() {∞,4} ![]() ![]() ![]() ![]() ![]() |
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 octaoctagonal tilings
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Symmetry: [8,8], (*882) | |||||||||||
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{8,8} | t{8,8} | r{8,8} | 2t{8,8}=t{8,8} | 2r{8,8}={8,8} | rr{8,8} | tr{8,8} | |||||
Uniform duals | |||||||||||
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V88 | V8.16.16 | V8.8.8.8 | V8.16.16 | V88 | V4.8.4.8 | V4.16.16 | |||||
Alternations | |||||||||||
[1+,8,8] (*884) | [8+,8] (8*4) | [8,1+,8] (*4242) | [8,8+] (8*4) | [8,8,1+] (*884) | [(8,8,2+)] (2*44) | [8,8]+ (882) | |||||
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() ![]() | |||||
![]() | ![]() | ![]() | ![]() | ![]() | |||||||
h{8,8} | s{8,8} | hr{8,8} | s{8,8} | h{8,8} | hrr{8,8} | sr{8,8} | |||||
Alternation duals | |||||||||||
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![]() | ![]() | ||||||||||
V(4.8)8 | V3.4.3.8.3.8 | V(4.4)4 | V3.4.3.8.3.8 | V(4.8)8 | V46 | V3.3.8.3.8 |
See also
![](http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
- "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
- Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
- Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch