Order-8 square tiling
Appearance
Order-8 square tiling | |
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![]() Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic regular tiling |
Vertex configuration | 48 |
Schläfli symbol | {4,8} |
Wythoff symbol | 8 | 4 2 |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() |
Symmetry group | [8,4], (*842) |
Dual | Order-4 octagonal tiling |
Properties | face-transitive
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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
*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[3]}, :
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Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).
*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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[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}
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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)
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[(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 |
See also
![](http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
Wikimedia Commons has media related to Order-8 square tiling.
References
- 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. LCCN 99035678.