Editing Hyperbolic geometry (section)

== In art ==

[[M. C. Escher]]'s famous prints ''[[Circle Limit III]]'' and ''Circle Limit IV''
illustrate the conformal disc model ([[Poincaré disk model]]) quite well. The white lines in ''III'' are not quite geodesics (they are [[hypercycle (hyperbolic geometry)|hypercycles]]), but are close to them. It is also possible to see the negative [[curvature]] of the hyperbolic plane, through its effect on the sum of angles in triangles and squares.

For example, in ''Circle Limit III'' every vertex belongs to three triangles and three squares. In the Euclidean plane, their angles would sum to 450°; i.e., a circle and a quarter. From this, we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is [[exponential growth]]. In ''Circle Limit III'', for example, one can see that the number of fishes within a distance of ''n'' from the center rises exponentially. The fishes have an equal hyperbolic area, so the area of a ball of radius ''n'' must rise exponentially in ''n''.

The art of [[crochet]] has [[Mathematics and fiber arts#Knitting and crochet|been used]] to demonstrate hyperbolic planes (pictured above) with the first being made by [[Daina Taimiņa]],<ref name="hyperbolicspace" /> whose book ''[[Crocheting Adventures with Hyperbolic Planes]]'' won the 2009 [[Bookseller/Diagram Prize for Oddest Title of the Year|''Bookseller''/Diagram Prize for Oddest Title of the Year]].<ref>{{Cite journal
 | last = Bloxham | first = Andy
 | date = March 26, 2010
 | journal = [[The Daily Telegraph|The Telegraph]]
 | title = Crocheting Adventures with Hyperbolic Planes wins oddest book title award
 | url = https://www.telegraph.co.uk/culture/books/bookprizes/7520047/Crocheting-Adventures-with-Hyperbolic-Planes-wins-oddest-book-title-award.html
 }}</ref>

''[[HyperRogue]]'' is a [[roguelike]] game set on various tilings of the hyperbolic plane.