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Topic: Hyperbolic space

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Hyperbolic plane

Hyperbolic geometry

Ultraparallel theorem

Elliptic geometry

Perpendicular

Ricci scalar

Einstein manifold

Metric tensor

Einstein summation convention

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	Hyperbolic Geometry for Visualizing Large Hierarchies
		Thus hierarchies---which tend to expand exponentially with depth---can be laid out in hyperbolic space in a uniform way, so that the distance (as measured in the hyperbolic geometry) between parents, children, and siblings is approximately the same everywhere in the hierarchy.
		Another novel tree browsing technique is treemaps (Johnson and Schneiderman, 1991) which allocates the entire space of a display area to the nodes of the tree by dividing the space of a node among itself and its descendants according to properties of the node.
		This technique utilizes space efficiently and can be used to look for values and patterns amongst a large collection of values which agglomerate hierarchically, however it tends to obscure the hierarchical structure of the values and provides no way of focusing on one part of a hierarchy without losing the context.
	members.tripod.com /enaction/id180.htm (4720 words)

	Hyperbolic Geometry
		Each of the triangles is the same shape in hyperbolic space with angles of 90°, 45° and 30°, and each has the same area in hyperbolic space.
		Hyperbolic geometry is not restricted to two-dimensional space.
		In three-dimensional hyperbolic space it is possible to construct regular dodecahedra with dihedral angles, the angles between two faces, equal to 90°.
	www.btinternet.com /~connectionsinspace/Patterns_and_Space_Filling/Hyperbolic_Geometry/body_hyperbolic_geometry.html (565 words)

	Hyperbolic Metric Spaces (Site not responding. Last check: 2007-10-18)
		Using this property of hyperbolic space, it is possible to generalize the notion of hyperbolicity to abstract geodesic metric spaces; and thanks to some deeply insightful definitions of M. Gromov in the 1980's to metric spaces in general.
		Hyperbolic metric spaces arise naturally as the boundaries at infinity of hyperbolic manifolds (the boundary at infinity of n-dimensional hyperbolic space is the n-1 sphere).
		Hyperbolic groups have lots of nice properties from a combinatorial viewpoint; they are finitely generated, and the word and conjugacy problems are solveable for these groups.
	www.math.lsa.umich.edu /~kwildric/Research/Hyperbolic.html (306 words)

	Hyperbolic Space (Site not responding. Last check: 2007-10-18)
		Hyperbolic space is an unbounded infinite space with constant negative curvature.
		If you were inside this space, it would seem more cramped than euclidean space, as all of space would appear to be inside a finite sphere.
		Another interesting thing that happens in hyperbolic space is that the apparent size of an object decreases exponentially as you move away from it.
	www.cs.brown.edu /people/tor/vr/doc/node5.html (212 words)

	The Geometry Junkyard: Hyperbolic Geometry
		Embedding the hyperbolic plane in higher dimensional Euclidean spaces.
		Packing circles in the hyperbolic plane, Java animation by Kevin Pilgrim illustrating the effects of changing radii in the hyperbolic plane.
		The tractrix and the pseudosphere, hyperbolic surfaces modeled in Cabri.
	www.ics.uci.edu /~eppstein/junkyard/hyper.html (376 words)

	Hyperbolic Space-Time, and Hyperbolic Numerical Manifolds
		As a subset of a hyperbolic manifold, the numerical logic of space-time likewise is hyperbolic.
		It attests to the pervasiveness of hyperbolic and tetrahedral effects attaching to dimensional (and polymensional) considerations.
		It is the hyperbolic nature of both the superior numerical manifold, and the lesser manifold subtending space-time, which is responsible for the phenomenon of dimensionality.
	members.aol.com /YuzhnoeMore/arg1.html (1141 words)

	Cabinet Magazine Online - Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina
		The discovery of hyperbolic space in the 1820s and 1830s by the Hungarian mathematician Janos Bolyai and the Russian mathematician Nicholay Lobatchevsky marked a turning point in mathematics and initiated the formal field of non-Euclidean geometry.
		In the 1970s the American geometer William Thurston had described a model of hyperbolic space that could be made by taping together a series of paper annuli, or thin circular strips.
		One potential geometry is a dodecahedral space, in which the basic symmetry of the universe is that of the dodecahedron, one of the five Platonic solids.
	www.cabinetmagazine.org /issues/16/crocheting.php (2537 words)

	LEXIS-NEXIS® Academic Universe-Document
		It may be that the space we live in is negatively curved on the largest scales.
		The hyperbolic plane is often described as being like a saddle, curved downwards in one direction and upwards in the other.
		Instead of designing fancy tiles to cover the space, Radin and Bowen were interested in packing in the simplest shapes of all--circles and spheres.
	www.ma.utexas.edu /users/radin/reviews/newscientist2.html (2109 words)

	Hyperbolic Interactive Visualization - HMDS, HSOM, H2DV
		The hyperbolic space is a non-euclidean space with negative curvature.
		Hyperbolic Tree Layout (HTL) is a recursive partitioning of the space by the nodes (and their children).
		The Hyperbolic Self-Organizing Maps (HSOM) is the transfer of Kohonen's SOM algorithm operating on a finite grid structure in H2.
	www.techfak.uni-bielefeld.de /~walter/h2vis (864 words)

	Hyperbolic Space
		Further explanation of the ramifications of the hyperbolic metric can be found in one of the many mathematical textbooks which cover hyperbolic geometry [Mar75] [Wol45].
		Although we could simply place euclidean objects into hyperbolic 3-space and move them around according to the rules of hyperbolic geometry, we would not be exploiting the exponential amount of room available in hyperbolic space.
		Distinct hyperbolic coordinates which are too far from the origin will be projected so close to the surface of the unit ball that there are not enough bits to distinguish between their euclidean coordinates.
	graphics.stanford.edu /papers/h3/html/node4.htm (968 words)

	A drawing program for hyperbolic space
		Description of the project: Hyperbolic space is a variety of non-Euclidean space (or "curved space") in which the angles of a triangle add up to less than 180 degrees.
		Hyperbolic space plays a role in many areas of mathematics and physics, but it is not entirely easy to develop a good intuition for its unfamiliar geometry.
		Several rudimentary applets of this kind exist already (do a Google search on "hyperbolic space java applet"), but the present project will seek to go further in the means it provides for applying transformations to existing designs.
	homepages.inf.ed.ac.uk /mcryan/projs0607/project.php?number=P140 (286 words)

	Models of Hyperbolic Geometry (Site not responding. Last check: 2007-10-18)
		Possibly the single most fascinating aspect of hyperbolic geometry is the parallel axiom, which states: Given a line L and a point p not on the line, there are an infinite number of lines through p and parallel to L.
		Given a model of a hyperbolic geometry, that does not include its boundary, two lines that have no common points within the model are said to be asymptotically parallel if they intersect on the boundary.
		Given the slightly disorientating nature of the hyperbolic parallel axiom, models of hyperbolic geometries are invaluable in our visualization of hyperbolic space.
	www.geom.uiuc.edu /~crobles/hyperbolic/hypr/modl (253 words)

	Chapter 5
		However, to skip studying hyperbolic planes would be to skip an important notion in the history of geometry and to skip the geometry which may be the basis of the geometry of our physical universe.
		Hyperbolic geometry, discovered more than 170 years ago by C.F. Gauss (1777-1855, German), János Bolyai (1802-1860, Hungarian), and N.I. Lobatchevsky (1792-1856, Russian), is special from a formal axiomatic point of view because it satisfies all the postulates (axioms) of Euclidean geometry except for the parallel postulate.
		Hyperbolic geometry and non-Euclidean geometry are considered in many books as being synonymous, but as we have seen there are other non-Euclidean geometries, particularly spherical geometry.
	www.math.cornell.edu /~dwh/books/eg00/00EG-05 (3493 words)

	[No title]
		Poincaré extensions from sphere to ball, begin: hyperbolic geometry and the hyperbolic metric on the unit ball..
		Hyperbolic isometries of the unit ball are Möbius transformations, and lengths of paths in metric spaces.
		Gromov hyperbolic groups have boundaries at infinity which admit visual metrics, and these are independent of choice of generators for their associated Cayley graphs.
	www.math.lsa.umich.edu /~jgong/qfractals/index.html (1075 words)

	A Unified Algebraic Framework for Classical Geometry
		This deficiency in the vector space model was corrected early in the 19th century by removing the origin from the plane and placing it one dimension higher.
		Because stereographic projections are conformal maps, the conformal groups of n-dimensional Euclidean, spherical, and hyperbolic spaces are isometric to each other, and are all isometric to the group of isometries of hyperbolic (n+1)-space, according to observations of Klein [K1872], [K1872].
		Geometric Algebra was applied to hyperbolic geometry by H. Li in [L97], stimulated by Iversen's book [I92] on the algebraic treatment of hyperbolic geometry and by the paper of Hestenes and Ziegler [HZ91] on projective geometry with Geometric Algebra.
	modelingnts.la.asu.edu /html/UAFCG.html (2056 words)

	Hyperbolic and Spherical Tiling Gallery
		In the Klein projection, straight lines in hyperbolic space are seen as straight lines, but angles are distorted by the projection.
		This is the hyperbolic equivalent of an icosidodecahedron.
		This is supposed to be a hyperbolic tiling where each vertex is surrounded by a 7-gon and two hexagons.
	bork.hampshire.edu /~bernie/hyper (292 words)

	11011110: Hyperbolic
		The geometry of gaussians is a hyperbolic space.
		Here cat(0) seems to mean that when you look at a triangle abc in the cat(0) space, and a triangle ABC in Euclidean space with the same edge lengths, the distance between two edge midpoints in abc is less than the distance between endpoints in ABC (and similarly for other corresponding points on edges).
		There seems to be the claim that, in such spaces, there is a constant k such that, if all the k-hop subpaths of a path are shortest, then the whole path is shortest.
	11011110.livejournal.com /47479.html (889 words)

	The Math Forum - Math Library - Hyperbolic Geom. (Site not responding. Last check: 2007-10-18)
		Hyperbolic tessellations shown in various stages of truncation, and represented by their Schlafli symbols.
		Krickl's diploma thesis: a list of all pentahedra that tessellate hyperbolic 3-space in the sense that they are a fundamental domain to their discrete reflection group.
		One illustrates that hyperbolic reflection in the Poincare disk corresponds to Euclidean inversion.
	mathforum.org /library/topics/hyperbolic_g (1487 words)

	NonEuclid: The Shape of Space (Site not responding. Last check: 2007-10-18)
		Large-scale curvature is the overall shape of space, and is the cumulative result of the totality of all the mater and energy in space.
		Hyperbolic Space has its own geometry, Hyperbolic Geometry, which has many similarities to our usual Euclidean Geometry, but also has some striking differences.
		All geometric theorems of Euclidean Geometry, therefore, appear to be true in a small chunk of Hyperbolic Space.
	www.cs.unm.edu /~joel/NonEuclid/space.html (789 words)

	Is Universe expanding
		This is how author concludes that the redshift of Galaxies are cause by the nature of Hyperbolic space, in which the space stretches the light spherical front.
		However, if we assume the Universe is in Hyperbolic space, very logically, we must derive its rules and formulas from Hyperbolic rules and Hyperbolic formulas.
		If he believed that the Universe was in Hyperbolic space, then there is a space curvature which is the cosmological constant.
	www.angelfire.com /ca6/aliou/expanduniverse.htm (1518 words)

	OpenGL Demos by Bernie Freidin
		Hyperbolic space can be thought of as "the opposite of spherical space".
		Hyperbolic space has many interesting geometrical properties: a good introduction to the subject is Geometry of Surfaces by John Stilwell.
		Essentially, the hyperbolic space would function as a very real representation of the "alternate dimensions" often referred to in science fiction.
	bork.hampshire.edu /~bernie (1287 words)

	Body
		This second paper and tape hyperbolic surface (used in classes and workshops for the next 11 years) was the one that Daina witnessed in use.
		This is the usual upper half plane model of the hyperbolic plane thought of as a map of the hyperbolic plane in the same way that we use planar maps of the spherical surface of the earth.
		Thus, we have established that the annular hyperbolic plane is the same as the usual upper half plane model of the hyperbolic plane.
	www.math.cornell.edu /~dwh/papers/crochet/crochet.html (3801 words)

	Professor Lets Her Fingers Do the Talking - New York Times (Site not responding. Last check: 2007-10-18)
		Hyperbolic space is useful to many professionals - engineers, architects and landscapers, among others.
		Math professors have been teaching about hyperbolic space for decades, but did not think it was possible to create an exact physical model.
		Because of editing errors, a headline Monday about a mathematician at Cornell University who crochets objects to illustrate hyperbolic space, an advanced geometric shape, misstated her title, and a picture caption misspelled her given name and the surname of her husband.
	www.nytimes.com /2005/07/11/nyregion/11cornell.html?ex=1278734400&en=4f34f654e57655d2&ei=5090&partner=rssuserland&emc=rss (1078 words)

	InfoVis CyberInfrastructure- Hyperbolic Trees
		Hyperbolic graph layout uses a context + focus technique to represent and manipulate large tree hierarchies on limited screen size.
		Hyperbolic trees are based on Poincare's model of the (hyperbolic) non-Euclidean plane.
		Distortion Technique: Hyperbolic layout uses a nonlinear (distortion) technique to accommodate focus and context for a large number of nodes.
	iv.slis.indiana.edu /sw/hyptree.html (650 words)

	The Institute For Figuring // Online Exhibit: Hyperbolic Space
		Noting that one of the qualities of hyperbolic space is that as you move away from a point the space around it expands exponentially, Thurston designed a paper model made up of thin cresent-shaped annuli taped together.
		The beauty of Taimina’s method is that many of the intrinsic properties of hyperbolic space now become visible to the eye and can be directly experienced by playing with the models.
		The radius of any given hyperbolic plane is the radius of a circle that would sit flatly on an a tabletop.
	theiff.org /oexhibits/oe1e.html (931 words)

	The Hyperbolic Surface Activity Page
		Hyperbolic space, which is three-dimensional, has more volume than ordinary Euclidean space!
		So, to make a hyperbolic plane surface, we need to arrange for there to be, in some sense, more surface around a point than usual.
		The Geometry Center at the University of Minnesota has a number of exhibits devoted to Hyperbolic Geometry, including a Java Applet for drawing Hyperbolic Triangles and some pictures from a Computer Generated Fly-through of Hyperbolic Space.
	members.tripod.com /professor_tom/hyperbolic/index.html (441 words)

	[No title]
		In fact, there is "exponential room" in 3-D hyperbolic space, in the sense that a circle's circumference is an exponential growth function of its radius, and similarly a sphere's area is an exponential growth function of its radius.
		This is in contrast to the "polynomial room" of 3-D Euclidean space: in 3-D Euclidean space, a circle's circumference is a polynomial function of its radius, and similarly a sphere's area is a polynomial function of its radius.
		The infinite 3-D hyperbolic space can be converted to a finite unit ball in 3-D Euclidean space using exactly the same "shooting eye rays" procedure, similar to a standard perspective projection.
	www.cs.utexas.edu /~kliu/graphics/project4 (3982 words)

	Graphics Archive - Special Topics:Hyperbolic Geometry (Site not responding. Last check: 2007-10-18)
		In the other, there are many parallel lines through the given point, and this is called hyperbolic geometry.
		It has several representations within the unit circle, or in the upper half-plane of 2-dimensional space.
		There are also higher-dimensional analogs, just as there are higher-dimensional Euclidean spaces.
	www.geom.uiuc.edu /graphics/pix/Special_Topics/Hyperbolic_Geometry (219 words)

	NonEuclid - Hyperbolic Geometry Article & Applet (Site not responding. Last check: 2007-10-18)
		Hyperbolic Geometry also has practical aspects such as orbit prediction of objects within intense gradational fields.
		Hyperbolic Geometry is used in Einstein's General Theory of Relativity and Curved Hyperspace.
		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.
	cs.unm.edu /~joel/NonEuclid/NonEuclid.html (538 words)

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