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Topic: Hyperbolic geometry

Related Topics

Hyperbolic space

Kleinian model

Fuchsian model

Fuchsian group

Ultraparallel theorem

Elliptic geometry

Upper half plane

Angle of parallelism

Mathematical model

Parallel Postulate

Spherical geometry

Uniformization theorem

Euclidean geometry

Modular group

Riemann sphere

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	JÁNOS BOLYAI CONFERENCE ON HYPERBOLIC GEOMETRY
		After he had recognized the impossibility of this task, he developed absolute geometry that is independent of the fifth postulate and also hyperbolic geometry where this postulate is negated.
		In remembrance of his brilliant mind a Conference on Hyperbolic Geometry will be held in the capital of Hungary, Budapest from July 8th to 12th, 2002.
		The Conference is intended to highlight the historical significance of the invention of hyperbolic geometry and to present new ideas and developments related to hyperbolic geometry.
	www.conferences.hu /Bolyai (0 words)

	Cabinet Magazine Online - Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina
		A hyperbolic plane is a surface in which the space curves away from itself at every point.
		The hyperbolic plane is sometimes described as a surface in which the space expands.
		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)

	PlanetMath: non-Euclidean geometry
		A non-Euclidean geometry is a geometry in which at least one of the axioms from Euclidean geometry fails.
		It is relatively easy to see that, in this geometry, given a line and a point not on the line, there are infinitely many lines passing through the point that are parallel to the given line.
		One final example of a non-Euclidean geometry is semi-Euclidean geometry, in which the axiom of Archimedes fails.
	planetmath.org /encyclopedia/NonEuclideanGeometry.html (420 words)

	Non-Euclidean Geometry - Wasil Intsar Mohar
		This is the geometry on a sphere such as the surface of the earth.
		The significance of the discovery of non-Euclidean Geometry is immense.
		One of the branches of non-Euclidean geometry is the theory of Elliptic Curves.
	community.middlebury.edu /~wmohar/Non-EuclideanGeometry-WasilMohar.htm (2291 words)

	Hyperbolic geometry Summary
		In the hyperbolic disk, the shortest path between two points is not an ordinary straight line, but rather is an arc of a semicircle whose center is on the boundary of the disk; diameters of the disk are also shortest paths, but other straight lines are not.
		Hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is rejected.
		Hyperbolic geometry was initially explored by Omar Khayyám and later Giovanni Gerolamo Saccheri, in an attempt to prove it inconsistent and thereby prove the parallel postulate.
	www.bookrags.com /Hyperbolic_geometry (3334 words)

	Euclidean and Non-Euclidean Geometry
		Hyperbolic geometry does, however, have applications to certain areas of science such as the orbit prediction of objects within intense gradational fields, space travel and astronomy.
		In hyperbolic geometry, the sum of the angles of a triangle is less than 180°.
		In hyperbolic space, the concept of perpendicular to a line can be illustrated as seen in the picture at the right.
	www.regentsprep.org /regents/math/geometry/GG1/Euclidean.htm (845 words)

	Hyperbolic geometry - ExampleProblems.com
		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.
		Hyperbolic geometry has many properties foreign to Euclidean geometry, all of which are consequences of the hyperbolic postulate.
	www.exampleproblems.com /wiki/index.php?title=Hyperbolic_geometry&redirect=no (1154 words)

	Hyperbolic Geometry
		Hyperbolic Geometry refutes Euclid's fifth postulate by saying that given line l and point p, there are an infinite number of lines through p parallel to l.
		This type of geometry is very tough to visualize, and that is one of the reasons that it took so long to be discovered.
		In the Hyperbolic Geometry Exhibit there is a nice Java simulation of the Poincare Disk.
	members.tripod.com /~noneuclidean/hyperbolic.html (463 words)

	non-Euclidean geometry
		When a body revolves around another body, it appears to move in a curved path due to some force exerted by the central body, but it is actually moving along a geodesic, without any force acting on it.
		Another consequence of non-Euclidean geometry is the possibility of the existence of a fourth dimension.
		Non-Euclidean geometry has applications in other areas of mathematics, including the theory of elliptic curves, which was important in the proof of Fermat’s last theorem.
	www.daviddarling.info /encyclopedia/N/non-Euclidean_geometry.html (713 words)

	Hyperbolic Geometry
		In hyperbolic geometry the sum of the angles of a triangle is always less than 180°.
		Hyperbolic geometry opens up new possibilities for producing decorative tessellations, some of which were explored by the master of tessellation M. Escher.
		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)

	How and Why Hyperbolic Geometry Came to Be
		When talking about geometry, most people that you meet on the street would assume that Euclidean Geometry is the only form of geometry accepted in the mathematical world.
		Euclidean Geometry is the oldest form of geometry, but it is by far not the only accepted form.
		Hyperbolic or non-Euclidean Geometry has its roots in Euclidean Geometry even though they are based on opposite postulates.
	filebox.vt.edu /users/jtoffene/HowandWhyHyperbolicGeometryCametoBe.htm (1887 words)

	Hyperbolic Geometry (Site not responding. Last check: )
		This is a quick glance at some of the tools the geometry program Cinderella offers to someone starting to study hyperbolic geometry.
		In this complex geometry, the conics that represent circles flip from being ellipses to being hyperbolas once their centers move outside the gray circle.
		In hyperbolic geometry, while EF is perpendicular to AB, EF is not perpendicular to ED or to FC.
	www.cs.earlham.edu /~timm/Geometry/hyperbolicGeom.html (806 words)

	Knit Theory - - science news articles online technology magazine articles Knit Theory (Site not responding. Last check: )
		Hyperbolic geometry describes a world that is curving away from itself at every point, making it the precise opposite of a sphere, whatever that might look like.
		They declared that all the normal rules of euclidean geometry would apply to this geometry except for Euclid's parallel postulate, which states that if you have a straight line and a point not on that line, there exists at most one straight line that passes through the point and is parallel to the line.
		When she was assigned to teach his class on hyperbolic geometry at Cornell, where she had an appointment as a visiting professor, she was forced to confront it.
	www.discover.com /issues/mar-06/features/knit-theory (1899 words)

	Warwick Mathematics Institute Hyperbolic Geometry Seminars
		We investigate the relationship between $\sumb_n$ and $\mathbf{K}(1b_n)$ with hyperbolic geometry and use this geometry to construct a sequence $b_n$ of real numbers for which both $\sum b_n$ and $\mathbf{K}(1b_n)$ converge, thereby answering Wall's question.
		Abstract: In broad terms, the programme of quantum geometry is applying the usual rules of quantum nechanics to geometry or geometric structure.
		Unlike non-commutative geometry, quantum geometry applies quantum mechanics not to the space of functions, but to the geometry of the space itself.
	www.maths.warwick.ac.uk /~jaram/seminar0405.html (1456 words)

	A Unified Algebraic Framework for Classical Geometry
		Though fundamental ideas of classical geometry are permanently imbedded and broadly applied in mathematics and physics, the subject itself has practically disappeared from the modern mathematics curriculum.
		Because the three geometries are obtained by interpreting null vectors of the same Minkowski space differently, natural correspondences exist among geometric entities and constraints of these geometries.
		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 Tessellations
		The hyperbolic plane can not be metrically represented in the Euclidean plane, but Poincaré described ways that it can be conformally represented in the Euclidean plane.
		For this representation, a straight line in the hyperbolic plane is represented as the part (in the disk) of a circle that meets the boundary of the disk at right angles.
		For instance, here is a representation of the tessellation of the hyperbolic plane by pentagons where four pentagons meet at each vertex, that is, the {5,4}-tessellation.
	aleph0.clarku.edu /~djoyce/poincare/poincare.html (784 words)

	MathDL \| Hyperbolic Geometry
		Hyperbolic geometry is a geometry for which we accept the first four axioms of Euclidean geometry but negate the fifth postulate, i.e., we assume that there exists a line and a point not on the line with at least two parallels to the given line passing through the given point.
		The Poincare half-plane model is conformal, which means that hyperbolic angles in the Poincare half-plane model are exactly the same as the Euclidean angles -- with the angles between two intersecting circles being the angle between their tangent lines at the point of intersection.
		The points are connected by a (hyperbolic) line segment, the radius, in red, and the (hyperbolic) circle itself is drawn in blue.
	mathdl.maa.org /mathDL/4/?pa=content&sa=viewDocument&nodeId=455&pf=1 (1951 words)

	The Institute For Figuring // An Interview with David Henderson and Daina Taimina
		The discovery of hyperbolic space by the Hungarian mathematician Janos Bolyai and the Russian mathematician Nicholay Lobatchevsky in the 1820’s and 1830’s marked a turning point in mathematics and initiated the formal study of non-Euclidean geometry.
		In higher-level mathematics courses it is often defined as the geometry that is described by the “upper half-plane model.”; One way of understanding it is that it’s the geometric opposite of the sphere.
		The hyperbolic plane is sometimes described as a surface in which the space expands.
	www.theiff.org /lectures/05a.html (2372 words)

	Amazon.com: Sources of Hyperbolic Geometry (History of Mathematics, V. 10): Books: John Stillwell (Site not responding. Last check: )
		Hyperbolic geometry is mathematics at its best: deep classical roots; stunning intrinsic beauty and conceptual simplicity; diverse and profound applications.
		The geodesic geometry of surfaces of constant negative curvature such as the pseudosphere capture much of the essence of hyperbolic geometry.
		Indeed, Beltrami's projective disc metric begs to be interpreted in terms of projective geometry: the distance between two points in the circle is easily expressed in terms of the cross-ratio of these two points and the two colinear points on the circle.
	www.amazon.com /Sources-Hyperbolic-Geometry-History-Mathematics/dp/0821809229 (1605 words)

	Crafty Geometry: Science News Online, Dec. 23, 2006 (Site not responding. Last check: )
		One mathematician's crocheted models of a counterintuitive shape called a hyperbolic plane are enabling her students and fellow mathematicians to gain new insight into startling properties.
		She knew that everyone can use intuition to conceive of the first two geometries, which are the realms of, say, sheets of paper and basketballs.
		Taimina realized that she could crochet a durable model of the hyperbolic plane using a simple rule: Increase the number of stitches in each row by a fixed factor, by adding a new stitch after, for instance, every two (or three or four or n) stitches.
	www.sciencenews.org /articles/20061223/bob10.asp (2423 words)

	The Hyperbolic Geometry Exhibit (Site not responding. Last check: )
		Hyperbolic geometry is one of the most important examples of a "non-Euclidean" geometry, with far reaching applications in math and science, including special relativity.
		Historically, hyperbolic geometry was discovered as a consequence of questions about the parallel postulate.
		As in Euclidean geometry, the isometries of Hyperbolic 2-Space are at the heart of the geometry.
	www.geom.uiuc.edu /~crobles/hyperbolic (391 words)

	The Hyperbolic Geometry Exhibit (Site not responding. Last check: )
		Historically, hyperbolic geometry was discovered as a consequence of questions about the parallel postulate.
		As in Euclidean geometry, the isometries of Hyperbolic 2-Space are at the heart of the geometry.
		In a way, hyperbolic geometry can be thought of as the geometry of a universe in which things travel faster than the speed of light.
	www.geom.umn.edu /~crobles/hyperbolic (391 words)

	Non-Euclidean Geometry Seminar (Site not responding. Last check: )
		We began with an exposition of Euclidean geometry, first from Euclid's perspective (as given in his Elements) and then from a modern perspective due to Hilbert (in his Foundations of Geometry).
		In this way properties of hyperbolic geometry were discovered, even though no one believed such a geometry to be possible.
		He expanded the class of non-Euclidean geometries to include elliptic geometry (which was then called Riemannian geometry) and also geometries whose properties may vary from point to point (which is now what is meant by Riemannian geometry).
	www.math.columbia.edu /~pinkham/teaching/seminars/NonEuclidean.html (446 words)

	Hyperbolic Geometry -- Mudd Math Fun Facts
		In the fun fact on Spherical Geometry, we saw an example of a space which is curved in such a way that the sum of angles in a triangle is greater than 180 degrees, where the sides of the triangle are "intrinsically" straight lines, or geodesics.
		In Euclidean geometry, given a line L there is exactly one line through any given point P that is parallel to L (the parallel postulate).
		However in hyperbolic geometry, there are infinitely many lines parallel to L passing through P. Mathematicians sometimes work with strange geometries by defining them in terms of a Riemannian metric, which gives a local notion of how to measure "distance" and "angles" on an arbitrary set.
	www.math.hmc.edu /funfacts/ffiles/30001.2.shtml (443 words)

	Hyperbolic Geometry (Site not responding. Last check: )
		Hyperbolic Geometry and here is a terrific site for it.
		Hyperbolic geometry is one of the most important examples of a "non-Euclidean" geometry, with far reaching applications in math and science, including special relativity.
		Here are a few visualizations and animations of hyperbolic geometry in the unit disk made by Juha Haataja at CSC.
	www.mccallie.org /myates/hyperbolic_geometry.htm (190 words)

	Hyperbolic geometry course announcement (Site not responding. Last check: )
		In 1868 Beltrami used differential geometry to present a Euclidean model of hyperbolic geometry; he proved that hyperbolic geometry is at least as consistent as Euclidean geometry.
		In 1872, Klein suggested that geometries should be studied by the analysis of their isometry groups.
		We shall see, for example, that the isometry group for the hyperbolic plane is closely related to the group SL(2,R) of 2 by 2 real matrices with determinant equal to 1.
	www.public.iastate.edu /~driessel/hyperbolic-geometry.html (350 words)

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