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.
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.
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.
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.
