Hyperbolic Geometry
Although Euclidean geometry, in which every line has exactly one parallel through any point, is most familiar to us, many other geometries are possible. Particularly important is hyperbolic geometry, in which infinitely many parallels to a line can go through the same point.
Are most manifolds hyperbolic? From Dave Rusin's known math pages.
Area of hyperbolic triangles. From the Geometry Center's Java gallery of interactive geometry.
Embedding the hyperbolic plane in higher dimensional Euclidean spaces. D. Rusin summarizes what's known; the existence of an isometric immersion into R4 is apparently open.
Géometriés non euclidiennes. Description of several models of the hyperbolic plane and some interesting hyperbolic constructions. From the Cabri geometry site. (In French.)
The golden ratio in an equilateral triangle. If one inscribes a circle in an ideal hyperbolic triangle, its points of tangency form an equilateral triangle with side length 4 ln phi! One can then place horocycles centered on the ideal triangle's vertices and tangent to each side of the inner equilateral triangle. From the Cabri geometry site. (In French.)
Hyperbolic crochet coral reef, the Institute for Figuring. Daina Taimina's technique for crocheting yarn into hyperbolic surfaces forms the basis for an exhibit of woolen undersea fauna and flora.
Hyperbolic games. Freeware multiplatform software for games such as Sudoku on hyperbolic surfaces, intended as a way for students to gain familiarity with hyperbolic geometry. By Jeff Weeks.
Hyperbolic geometry. Visualizations and animations including several pictures of hyperbolic tessellations.
Hyperbolic Knot. From Eric Weisstein's treasure trove of mathematics.
Hyperbolic packing of convex bodies. William Thurston answers a question of Greg Kuperberg, on whether there is a constant C such that every convex body in the hyperbolic plane can be packed with density C. The answer is no -- long skinny bodies can not be packed efficiently.
The hyperbolic surface activity page. Tom Holroyd describes hyperbolic surfaces occurring in nature, and explains how to make a paper model of a hyperbolic surface based on a tiling by heptagons.
Ideal hyperbolic polyhedra ray-traced by Matthias Weber.
The Institute for Figuring's online exhibit on hyperbolic space.
Kleinian Groups. Rather incomprehensible exposition of hyperbolic symmetry, but plenty of pretty pictures.
3-Manifolds from regular solids. Brent Everitt lists the finite volume orientable hyperbolic and spherical 3-manifolds obtained by identifying the faces of regular solids.
Mutations and knots. Connections between knot theory and dissection of hyperbolic polyhedra.
Non-Euclidean geometry with LOGO. A project at Cardiff, Wales, for using the LOGO programming language to help mathematics students visualise non-Euclidean geometry.
Packing circles in the hyperbolic plane, Java animation by Kevin Pilgrim illustrating the effects of changing radii in the hyperbolic plane.
Parabolic isometry of an ideal hyperbolic triangulation. Animation by John Griffin.
Wilson Stothers' Cabri pages. Geometric animations teaching projective conics, hyperbolic geometry, and the Klein view of geometry as symmetry.
The tractrix and the pseudosphere, hyperbolic surfaces modeled in Cabri.
Triangles and squares. Slides from a talk I gave relating a simple 2d puzzle, Escher's drawings of 3d polyhedra, and the combinatorics of 4d polytopes, via angles in hyperbolic space. Warning: very large file (~8Mb). For more technical details see my paper with Kuperberg and Ziegler.
Visualization of a hyperbolic universe, Martin Bucher.
The world of hyperbolic geometry, Colleen Robles.