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	Template:CDD Template:CDD
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.

File:Uniform tiling 433-t0 3-fold.png Triangle-centered hyperbolic straight edges

File:Uniform tiling 433-t0 edgecenter.png Edge-centered projective straight edges

File:Uniform tiling 433-t0 point.png Point-centered projective straight edges

Dual tiling

File:Uniform dual tiling 433-t0.png

In art

File:Circle limits III with overlay.png

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.

See also

• Circle Limit III
• Square tiling
• Uniform tilings in hyperbolic plane
• List of regular polytopes

References

• John Horton Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Template:ISBN (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

• Douglas Dunham Department of Computer Science University of Minnesota, Duluth
  • Examples Based on Circle Limits III and IV, 2006:More “Circle Limit III” Patterns, 2007:A “Circle Limit III” Calculation, 2008:A “Circle Limit III” Backbone Arc Formula
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch