Order-8 square tiling

In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,8}.

Order-8 square tiling
Order-8 square tiling

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 CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
Symmetry group [8,4], (*842)
Dual Order-4 octagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

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 [1+,8,8,1+], (*4444 orbifold) removing two of three mirrors (passing through the square center) in the [8,8] 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[3]}, CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node 1.png:

Uniform tiling 444-t1 H2chess 248b

Related polyhedra and tiling

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).

See also

References

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, 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

Octagonal tiling

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}.

Order-4 octagonal tiling

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}.

Order-5 octahedral honeycomb

In the geometry of hyperbolic 3-space, the order-5 octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,5}. It has five octahedra {3,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an order-5 square tiling vertex arrangement.

Rhombitetraoctagonal tiling

In geometry, the rhombitetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,4}. It can be seen as constructed as a rectified tetraoctagonal tiling, r{8,4}, as well as an expanded order-4 octagonal tiling or expanded order-8 square tiling.

*n42 symmetry mutation of regular tilings: {4,n}
Spherical Euclidean Compact hyperbolic Paracompact
Uniform tiling 432-t0
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 44-t0
{4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
H2 tiling 245-4
{4,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
H2 tiling 246-4
{4,6}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png
H2 tiling 247-4
{4,7}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node.png
H2 tiling 248-4
{4,8}...
CDel node 1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 24i-4
{4,∞}
CDel node 1.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node.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 H2 tiling 248-3 H2 tiling 248-2 H2 tiling 248-6 H2 tiling 248-4 H2 tiling 248-5 H2 tiling 248-7
{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 H2chess 248f H2chess 248a H2chess 248e H2chess 248c H2chess 248d H2checkers 248
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 Uniform tiling 84-h01 Uniform tiling 443-t1 Uniform tiling 444-snub Uniform tiling 88-t0 Uniform tiling 54-t2 Uniform tiling 84-snub
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 Uniform tiling 66-t1 Uniform dual tiling 433-t0 Uniform tiling 88-t2 Uniform tiling 54-t0
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 label4.pngCDel branch 01rd.pngCDel split2-44.pngCDel node.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
CDel label4.pngCDel branch 01rd.pngCDel split2-44.pngCDel node 1.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node 1.png
CDel node h0.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel label4.pngCDel branch 10ru.pngCDel split2-44.pngCDel node 1.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel label4.pngCDel branch 10ru.pngCDel split2-44.pngCDel node.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node.png
CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.png
CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel label4.pngCDel branch hh.pngCDel split2-44.pngCDel node h.png
CDel node h0.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node h.png
CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node h1.png
CDel node h0.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node h1.png
CDel label4.pngCDel branch hh.pngCDel split2-44.pngCDel node.png
CDel node h0.pngCDel 8.pngCDel node h1.pngCDel 4.pngCDel node.png
H2 tiling 444-1 H2 tiling 444-3 H2 tiling 444-2 H2 tiling 444-6 H2 tiling 444-4 H2 tiling 444-5 H2 tiling 444-7 Uniform tiling 444-snub H2 tiling 288-4 H2 tiling 344-2
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
H2chess 444b H2chess 444f H2chess 444a H2chess 444e H2chess 444c H2chess 444d H2checkers 444 Uniform dual tiling 433-t0 H2 tiling 288-1 H2 tiling 266-2
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

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