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• Tilings of Hyperbolic Spaces .
• Cellular automata on hyperbolic tilings? Message to CAS mailing list from B. ...
• Dehn invariants of hyperbolic tiles. ... But as Doug Zare describes, there are hyperbolic tiles with nonzero Dehn invariant. ...
• Do buckyballs fill hyperbolic space? .
• The Geometry Center's collection includes programs for generating Penrose tilings, making periodic drawings a la Escher in the Euclidean and hyperbolic planes, playing pinball in negatively curved spaces, viewing 3d objects, exploring the space of angle geometries, and visualizing Riemann surfaces. ...
• Stephen Fortescue discusses some connections between basic number-theoretic algorithms and the geometry of tilings of 2d and 3d hyperbolic spaces. ...
• Huson's favorite hyperbolic tiling. ...
• Hyperbolic geometry. Visualizations and animations including several pictures of hyperbolic tessellations. ...
• 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. ...
• Hyperbolic Tessellations, David Joyce, Clark U. ...
• Hyperbolic tiles. John Conway answers a question of Doug Zare on the polyhedra that can form periodic tilings of 3-dimensional hyperbolic space. ...
• Hyperbolic and spherical tiling gallery, Bernie Freidin. ...

• Hyperbolic Animations.
• Hyperbolic animation #1.
• Hyperbolic animation #2 .
• Hyperbolic animation #3 .
• Hyperbolic animation #4 .
• Hyperbolic animation #5 .
• Hyperbolic animation #6 .
• Hyperbolic Penrose Tiling .
• Hyperbolic Dance .

• Hyperbolic Geometry .
• 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. ...
• Dense sphere-packings in hyperbolic space. ...
• Embedding the hyperbolic plane in higher dimensional Euclidean spaces. ...
• Poritz and coworkers investigate the fundamental domains of cyclic group actions on hyperbolic 3-space, resulting in lots of pretty pictures of overlapping spheres. ...
• Description of several models of the hyperbolic plane and some interesting hyperbolic constructions. ...
• 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. ...
• Hyperbolic geometry. Visualizations and animations including several pictures of hyperbolic tessellations. ...
• Hyperbolic Knot. ...
• 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 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. ...

• Java Demos of the EBI Hyperbolic Viewer .
• An image displayed in the hyperbolic viewer.
• The Eukaryotae kingdom phylogenetic tree displayed in the Hyperbolic viewer.
• NEW! The EBI Hyperbolic viewer used with CORBA to display the NCBI taxonomic tree located in our Sybase database. ...
• The EBI Hyperbolic viewer has been developed as a contribution to the bioWidget project. More about the bioWidget Consortium and hyperbolic viewers can be found on the ViBE pages. ...

• 1-Hyperbolic Graphs.
• 1-Hyperbolic Graphs: SIAM Journal on Discrete Mathematics Vol. ... 2The shortest-path metric d of a graph G=(V,E) is called $\delta$-{\it hyperbolic} if for any four vertices $u,v,w,x\in X$ the two larger of the three sums d(u,v)+d(w,x),d(u,w)+d(v,x),d(u,x)+d(v,w) differ by at most $\delta. $ In this paper, we characterize the graphs with 1-hyperbolic metrics in terms of a convexity condition and forbidden isometric subgraphs. ...

• Special Topics:Hyperbolic Geometry.
• In the other, there are many parallel lines through the given point, and this is called hyperbolic geometry. ...
• 237 Tiling of the Hyperbolic Plane .
• 4 Dodecahedra in Hyperbolic Space .
• Hyperbolic Dirichlet Domains .
• Hyperbolic Dodecahedron.
• Hyperbolic Kaleidoscope: 2-3-5 .
• Hyperbolic Kaleidoscope: 3-4-4 .
• Hyperbolic Kaleidoscope: S2 .
• Hyperbolic Space Tiled with Dodecahedra, 1.
• Hyperbolic Space Tiled with Dodecahedra, 2 .

• Poincaré Disk and the Upper Half-Plane Models of Hyperbolic Geometry.
• Hyperbolic Geometry is a geometry of Einstein's General Theory of Relativity and Curved Hyperspace. ...
• What is Non-Euclidean Geometry: - Euclidean Geometry, Spherical Geometry, Hyperbolic Geometry, and others. ...
• Parallel Lines: - In Hyperbolic Geometry, a pair of intersecting lines can both be parallel to a third line. ...
• Axioms and Theorems: - Euclid's Postulates, Hyperbolic Parallel Postulate, SAS Postulate, Hyperbolic Geometry Proofs. ...
• Area: - Exaimation of A=½bh and A=s² in Hyperbolic Geometry, Properties Necessary for an Area Function, Altitudes of a Hyperbolic Triangle, Defect of a Triangle, Defect of a Polygon, and an Upper Bound to Area. ...
• X-Y Coordinate System: - A description of how an x-y coordinate system can be set up in Hyperbolic Geometry. ...
• Disk and Upper Half-Plane Models: - An informal development of these two models of Hyperbolic Geometry. ...
• For The Teacher:Why is it Important for Students to Study Hyperbolic Geometry? .

• Nonlinear Hyperbolic Waves in Phase Dynamics and Astrophysics.
• Quasilinear hyperbolic systems in divergence form, commonly called 'hyperbolic conservation laws', govern a broad spectrum of physical phenomena in compressible fluid dynamics, nonlinear material science, etc. ...
• Hyperbolic problems in astrophysics and cosmology (relativistic compressible fluid models, the Einstein Field equations of general relativity) are particularly challenging for the applied mathematician; they are essential in order to uncover the structure and formation of the Universe. Hyperbolic problems arising in nonlinear material science (propagating phase boundaries in solids undergoing phase transitions of the austenite-martensite type, for instance) face serious mathematical difficulties (failure of strict hyperbolicity, elliptic regions in phase space); these models are an essential component for understanding the dynamics of shape memory alloys (also called smart materials). ... Recent developments on the theory of systems of conservation laws will be covered; L1 well-posedness, singular limits based on diffusion-dispersion or relaxation, multidimensional hyperbolic problems, degenerate shock wave structure, failure of strict hyperbolicity, etc.
• January to February: Kinetic Relations for Nonclassical Shocks and Phase Boundaries February to March: General Theory of Hyperbolic Conservation Laws March to April: Shock Waves having Anomalous Structure and Multi-dimensional Problems April to May: Multiphase Flows in Fluids June to July: Hyperbolic Problems from Astrophysics and Cosmology. ...

• : Hyperbolic Geom. ...
• Hyperbolic Geometry (The Geometry Junkyard) - David Eppstein, Theory Group, ICS, UC Irvine.
• An extensive annotated list of links to material in hyperbolic geometry, in which infinitely many parallels to a line can go through the same point. ...
• Course notes that include: Origins of geometry; spherical geometry; logic and the axiomatic method; proof; Euclid's mathematical system; incidence geometry; betweenness axioms; congruence theorems; axioms of continuity; neutral geometry; theorems of continuity; the work of Saccheri and Gauss; hyperbolic geometry; classification of parallels; the pseudosphere; hyperbolic trigonometry and hyperbolic analytic geometry; and more. ...
• A Java program offering straightedge and compass constructions in hyperbolic geometry for high school and college students. ... The site includes an introduction to hyperbolic geometry, plus activities for getting started and exploring with NonEuclid. ...
• Contents include: Introduction and Historical Background; Notation and Definitions; General Theorems; Hyperbolic Results; and. ...
• An list of suggestions for projects in geometry, topology, symmetry, making geometric solids, calendars, spherical and hyperbolic trigonometry, puzzles, models, etc. ...
• Mandelbrot and Julia sets, with a generation form and an Explorer on which you can point and click to get more refined images; Newton Basins and a form to generate them; a page describing tiling the hyperbolic plane; some wallpaper groups; a gallery. ...
• Hyperbolic tessellations shown in various stages of truncation, and represented by their Schlafli symbols. Java applets explore hyperbolic tessellations, a magic Rubik's-like cube in four dimensions, polygon reciprocation (dual), modern portfolio theory. ...
• A collection of interactive geometric objects represented graphically including 3D-graphics, space curves, trigonometric functions, hyperbolic functions, and the trajectory of projectiles. ...
• Hyperbolic Geometry (Geometry and the Imagination) - Conway, Doyle, Gilman, Thurston; The Geometry Center.
• Hyperbolic Geometry (Graphics Archive, Special Topics) - The Geometry Center.
• A definition of hyperbolic geometry and illustrations of hyperbolic structures, with a link to Not Knot, a guided tour through computer-animated hyperbolic space. ...
• Hyperbolic Geometry Using Cabri - Tim Lister.

• Hyperbolic geometry.
• Hyperbolic geometry, also called saddle geometry or Lobachevskian geometry, is the non-Euclidean geometry obtained by replacing the parallel postulate with the hyperbolic postulate, which states: "Given a line L and any point A not on L, at least two distinct lines exist which pass through A and are parallel to L. ...
• In hyperbolic geometry, the term parallel only applies to lines that don't intersect in the hyperbolic plane but intersect at the circle at infinity. Lines that neither intersect in the hyperbolic plane nor the circle at infinity are called ultraparallel. One remarkable property of the hyperbolic plane is that there is a unique common perpendicular for each pair of ultraparallel lines (see Ultraparallel theorem).
• Hyperbolic geometry was initially explored by Saccheri in the 1700s, who nevertheless believed that it was inconsistent, and later by Bolyai, Gauss, and Lobachevsky, after whom it is sometimes named. ...
• There are four models commonly used for hyperbolic geometry: the Klein model, the Poincaré disc model, the Poincaré half-plane model and the Lorentz model.
• The Klein model, also known as the projective disc model and Beltrami-Klein model, uses the interior of a circle for the hyperbolic plane, and chords of the circle as lines. This model has the advantage of simplicity, but the disadvantage that angles in the hyperbolic plane are distorted.
• The Poincaré half-plane model takes one-half of the Euclidean plane, as determined by a Euclidean line B, to be the hyperbolic plane (B itself is not included). Hyperbolic lines are then either half-circles orthogonal to B or rays perpendicular to B.
• Both Poincaré models preserve hyperbolic angles, and are thereby conformal. ...
• Hyperbolic geometry has many properties foreign to Euclidean geometry, all of which are consequences of the hyperbolic postulate.
• Visualizing hyperbolic geometry.
• It is also possible to see quite plainly the negative curvature of the hyperbolic plane, via its effect on the sum of angles in triangles and squares.
• Hence we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is the fact that the hyperbolic plane has exponential growth. ... The angels have equal hyperbolic area, so the area of a ball of radius n must rise exponentially in n.

• Hyperbolic geometry.
• Figure 1: A hyperbolic triangle in the unit disk. ...
• Hyperbolic geometry is an example of non-Euclidean geometry, where the parallel axiom is discarded: .
• In figure 1 an example of a hyperbolic triangle is shown. ...

• sinh-Hyperbolic Sine .
• sinh() computes the hyperbolic sine of x. ...

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