Alternated octagonal tiling

Alternated octagonal tiling
Alternated octagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (3.4)3
Schläfli symbol (4,3,3)
s(4,4,4)
Wythoff symbol 3 | 3 4
Coxeter diagram
Symmetry group [(4,3,3)], (*433)
[(4,4,4)]+, (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}.

Geometry

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.


Triangle-centered
hyperbolic straight edges
Edge-centered
projective straight edges
Point-centered
projective straight edges

Dual tiling

In art

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)
h{8,3}
t0(4,3,3) 		r{3,8}1/2
t0,1(4,3,3) 		h{8,3}
t1(4,3,3) 		h2{8,3}
t1,2(4,3,3) 		{3,8}1/2
t2(4,3,3) 		h2{8,3}
t0,2(4,3,3) 		t{3,8}1/2
t0,1,2(4,3,3) 		s{3,8}1/2
s(4,3,3)
Uniform duals
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)]+
(444) 		[(1+,4,4,4)]
(*4242) 		[(4+,4,4)]
(4*22)










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
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

References

  • John Horton 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.
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