Alternated octagonal tiling
Poincaré disk model of the hyperbolic plane
Type
Hyperbolic uniform tiling
Vertex configuration (3.4)3
Schläfli symbol (4,3,3)
Wythoff symbol 3 | 3 4
Coxeter diagram File:CDel branch hh.png File:CDel split2-44.png File:CDel node h.png
Symmetry group [(4,3,3)], (*433)+ , (444)
Dual Alternated octagonal tiling#Dual tiling
Properties
Vertex-transitive
In geometry , the tritetragonal tiling or alternated octagonal tiling is a uniform tiling of the hyperbolic plane . It has Schläfli symbols of {(4,3,3)} or h{8,3}.
Although a sequence of edges seem to represent straight lines (projected into curves), careful attention will show they are not straight, as can be seen by looking at it from different projective centers.
File:Uniform dual tiling 433-t0.png
Error creating thumbnail: File missing The alternated octagonal tiling, a hyperbolic tiling of squares and equilateral triangles, overlaid on Escher's image Circle Limit III is a woodcut made in 1959 by Dutch artist M. C. Escher , in which "strings of fish shoot up like rockets from infinitely far away" and then "fall back again whence they came". White curves within the figure, through the middle of each line of fish, divide the plane into squares and triangles in the pattern of the tritetragonal tiling. However, in the tritetragonal tiling, the corresponding curves are chains of hyperbolic line segments, with a slight angle at each vertex, while in Escher's woodcut they appear to be smooth hypercycles .
Uniform (4,3,3) tilings
Symmetry: [(4,3,3)], (*433)
[(4,3,3)]+ , (433)
File:CDel branch 01rd.png File:CDel branch 11.png File:CDel node 1.png File:CDel branch.png File:CDel node 1.png File:CDel branch 01rd.png File:CDel node 1.png File:CDel branch 11.png File:CDel node 1.png File:CDel branch hh.png File:CDel node h.png
File:CDel node h1.png File:CDel 8.png File:CDel 3.png File:CDel node h0.png File:CDel 8.png File:CDel node 1.png File:CDel 3.png File:CDel node h1.png File:CDel 8.png File:CDel 3.png File:CDel node h1.png File:CDel 8.png File:CDel 3.png File:CDel node 1.png File:CDel node h0.png File:CDel 8.png File:CDel 3.png File:CDel node 1.png File:CDel node h1.png File:CDel 8.png File:CDel 3.png File:CDel node 1.png File:CDel node h0.png File:CDel 8.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel node h0.png File:CDel 8.png File:CDel node h.png File:CDel 3.png File:CDel node h.png
File:H2 tiling 334-1.png File:H2 tiling 334-3.png File:H2 tiling 334-2.png File:H2 tiling 334-6.png File:H2 tiling 334-4.png File:H2 tiling 334-5.png File:H2 tiling 334-7.png File:Uniform tiling 433-snub2.png
h{8,3} 0 (4,3,3)
r{3,8}1 /2 0,1 (4,3,3)
h{8,3} 1 (4,3,3)
h2 {8,3} 1,2 (4,3,3)
{3,8}1 /2 2 (4,3,3)
h2 {8,3} 0,2 (4,3,3)
t{3,8}1 /2 0,1,2 (4,3,3)
s{3,8}1 /2
Uniform duals
File:Uniform dual tiling 433-t0.png File:Uniform dual tiling 433-t01.png File:Uniform dual tiling 433-t0.png File:Uniform dual tiling 433-t12.png File:H2-8-3-dual.svg File:Uniform dual tiling 433-t12.png File:H2-8-3-kis-dual.svg File:Uniform dual tiling 433-snub.png
V(3.4)3
V3.8.3.8
V(3.4)3
V3.6.4.6
V(3.3)4
V3.6.4.6
V6.6.8
V3.3.3.3.3.4
Uniform (4,4,4) tilings
Symmetry: [(4,4,4)], (*444)
[(4,4,4)]+
[(1+ ,4,4,4)]
[(4+ ,4,4)]
File:CDel branch 01rd.png File:CDel split2-44.png File:CDel node h1.png File:CDel 8.png File:CDel 4.png File:CDel branch 01rd.png File:CDel split2-44.png File:CDel node 1.png File:CDel node h1.png File:CDel 8.png File:CDel 4.png File:CDel node 1.png File:CDel branch.png File:CDel split2-44.png File:CDel node 1.png File:CDel node h0.png File:CDel 8.png File:CDel 4.png File:CDel node 1.png File:CDel split2-44.png File:CDel node 1.png File:CDel node h1.png File:CDel 8.png File:CDel 4.png File:CDel node 1.png File:CDel split2-44.png File:CDel node h1.png File:CDel 8.png File:CDel 4.png File:CDel branch 11.png File:CDel split2-44.png File:CDel node h0.png File:CDel 8.png File:CDel node 1.png File:CDel 4.png File:CDel branch 11.png File:CDel split2-44.png File:CDel node 1.png File:CDel node h0.png File:CDel 8.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel branch hh.png File:CDel split2-44.png File:CDel node h.png File:CDel node h0.png File:CDel 8.png File:CDel node h.png File:CDel 4.png File:CDel node h.png File:CDel branch.png File:CDel split2-44.png File:CDel node h1.png File:CDel node h0.png File:CDel 8.png File:CDel 4.png File:CDel node h1.png File:CDel branch hh.png File:CDel split2-44.png File:CDel node h0.png File:CDel 8.png File:CDel node h1.png File:CDel 4.png
File:H2 tiling 444-1.png File:H2 tiling 444-3.png File:H2 tiling 444-2.png File:H2 tiling 444-6.png File:H2 tiling 444-4.png File:H2 tiling 444-5.png File:H2 tiling 444-7.png File:Uniform tiling 444-snub.png File:H2 tiling 288-4.png File:H2 tiling 344-2.png
t0 (4,4,4) t0,1 (4,4,4) 2 {8,4}
t1 (4,4,4) 1 /2
t1,2 (4,4,4) 2 {8,4}
t2 (4,4,4) t0,2 (4,4,4) 1 /2
t0,1,2 (4,4,4) 1 /2
s(4,4,4) 1 /2
h(4,4,4) 1 /2
hr(4,4,4) 1 /2
Uniform duals
File:H2chess 444b.png File:H2chess 444f.png File:H2chess 444a.png File:H2chess 444e.png File:H2chess 444c.png File:H2chess 444d.png File:H2checkers 444.png File:Uniform dual tiling 433-t0.png File:H2 tiling 288-1.png File:H2 tiling 266-2.png
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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