Order-8 square tiling | |
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Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic regular tiling |
Vertex configuration | 4 |
Schläfli symbol | {4,8} |
Wythoff symbol | 8 | 4 2 |
Coxeter diagram | |
Symmetry group | , (*842) |
Dual | Order-4 octagonal tiling |
Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,8}.
Symmetry
This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with eight squares around every vertex. This symmetry by orbifold notation is called (*4444) with 4 order-4 mirror intersections. In Coxeter notation can be represented as , (*4444 orbifold) removing two of three mirrors (passing through the square center) in the symmetry. The *4444 symmetry can be doubled by bisecting the fundamental domain (square) by a mirror, creating *884 symmetry.
This bicolored square tiling shows the even/odd reflective fundamental square domains of this symmetry. This bicolored tiling has a wythoff construction (4,4,4), or {4}, :
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4).
*n42 symmetry mutation of regular tilings: {4,n} | |||||||||||
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Spherical | Euclidean | Compact hyperbolic | Paracompact | ||||||||
{4,3} |
{4,4} |
{4,5} |
{4,6} |
{4,7} |
{4,8}... |
{4,∞} |
Uniform octagonal/square tilings | |||||||||||
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, (*842) (with (*882), (*444) , (*4222) index 2 subsymmetries) (And (*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 | |||||||||||
V8 | V4.16.16 | V(4.8) | V8.8.8 | V4 | V4.4.4.8 | V4.8.16 | |||||
Alternations | |||||||||||
(*444) |
(8*2) |
(*4222) |
(4*4) |
(*882) |
(2*42) |
(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 | |||||||||||
V(4.4) | V3.(3.8) | V(4.4.4) | V(3.4) | V8 | V4.4 | V3.3.4.3.8 |
Uniform (4,4,4) tilings | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Symmetry: , (*444) | (444) |
(*4242) |
(4*22) | ||||||||
t0(4,4,4) h{8,4} |
t0,1(4,4,4) h2{8,4} |
t1(4,4,4) {4,8}/2 |
t1,2(4,4,4) h2{8,4} |
t2(4,4,4) h{8,4} |
t0,2(4,4,4) r{4,8}/2 |
t0,1,2(4,4,4) t{4,8}/2 |
s(4,4,4) s{4,8}/2 |
h(4,4,4) h{4,8}/2 |
hr(4,4,4) hr{4,8}/2 | ||
Uniform duals | |||||||||||
V(4.4) | V4.8.4.8 | V(4.4) | V4.8.4.8 | V(4.4) | V4.8.4.8 | V8.8.8 | V3.4.3.4.3.4 | V8 | V(4,4) |
See also
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
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