geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the tritetragonal tiling or alternated octagonal tiling is a

uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...

tiling Tiling may refer to: *The physical act of laying tiles *Tessellations Computing *The compiler optimization of loop tiling * Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People * Heinrich Sylvester ...

of the

hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

. It has

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mor ...

s of or h.

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.

Dual tiling

In art

Circle Limit III ''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".Escher, as quoted by . It is one of a series of fo ...

is a

woodcut Woodcut is a relief printing technique in printmaking Printmaking is the process of creating artworks by printing, normally on paper, but also on fabric, wood, metal, and other surfaces. "Traditional printmaking" normally covers only t ...

made in 1959 by Dutch artist

M. C. Escher Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, and mezzotints. Despite wide popular interest, Escher was for most of his life neglected in th ...

, 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.

Related polyhedra and tiling

References

John H. Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branch ...

, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations) *

External links

* Douglas Dunham Department of Computer Science University of Minnesota, Duluth *
Examples Based on Circle Limits III and IV
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More “Circle Limit III” Patterns
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A “Circle Limit III” Calculation
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A “Circle Limit III” Backbone Arc Formula
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Hyperbolic and Spherical Tiling Gallery

* ttp://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch Hyperbolic tilings Isogonal tilings Uniform tilings Octagonal tilings {{geometry-stub