View source for Alternated octagonal tiling

{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U433_0}}
In [[geometry]], the '''tritetragonal tiling''' or '''alternated octagonal tiling''' is a [[Uniform tilings in hyperbolic plane|uniform]] [[tessellation|tiling]] of the [[Hyperbolic geometry|hyperbolic plane]]. It has [[Schläfli symbol]]s 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. 
{| class=wikitable width=480
|- align=center
|[[File:Uniform tiling 433-t0 3-fold.png|160px]]<BR>Triangle-centered<BR>hyperbolic straight edges
|[[File:Uniform tiling 433-t0 edgecenter.png|160px]]<BR>Edge-centered<BR>projective straight edges
|[[File:Uniform tiling 433-t0 point.png|160px]]<BR>Point-centered<BR>projective straight edges
|}

== Dual tiling==
[[File:Uniform dual tiling 433-t0.png|240px]]

== 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 [[Hypercycle (geometry)|hypercycle]]s.

== Related polyhedra and tiling ==
{{Order 4-3-3 tiling table}}
{{Order 4-4-4 tiling table}}

==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, {{ISBN|978-1-56881-220-5}} (Chapter 19, The Hyperbolic Archimedean Tessellations)
* {{Cite book|title=The Beauty of Geometry: Twelve Essays|year=1999|publisher=Dover Publications|lccn=99035678|isbn=0-486-40919-8|chapter=Chapter 10: Regular honeycombs in hyperbolic space}}

== External links ==
{{Commonscat|Uniform tiling 3-4-3-4-3-4}}
* Douglas Dunham Department of Computer Science University of Minnesota, Duluth
** [http://www.d.umn.edu/~ddunham/isis4/section6.html Examples Based on Circle Limits III and IV], 2006:[http://www.d.umn.edu/~ddunham/dunbrid06.pdf More “Circle Limit III” Patterns], 2007:[http://www.d.umn.edu/~ddunham/dunbrid07.pdf A “Circle Limit III” Calculation], 2008:[http://www.d.umn.edu/~ddunham/bridges08.pdf A “Circle Limit III” Backbone Arc Formula]
*{{MathWorld | urlname= HyperbolicTiling | title = Hyperbolic tiling}}
*{{MathWorld | urlname=PoincareHyperbolicDisk | title = Poincaré hyperbolic disk }}
* [http://bork.hampshire.edu/~bernie/hyper/ Hyperbolic and Spherical Tiling Gallery] {{Webarchive|url=https://web.archive.org/web/20130324095520/http://bork.hampshire.edu/~bernie/hyper/ |date=2013-03-24 }}
* [http://geometrygames.org/KaleidoTile/index.html KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings]
* [http://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]

{{Tessellation}}

[[Category:Hyperbolic tilings]]
[[Category:Isogonal tilings]]
[[Category:Uniform tilings]]
[[Category:Octagonal tilings]]