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	USS Clueless - Non-Euclidean space
		Euclidean geometry is famously based on five axioms, but there's an unspoken axiom of uniformity, which assumes that the universe is geometrically the same at every location and at all scales.
		In Euclidean geometry, the fifth axiom was: if there is a line on a plane, and a point on that plane which is not on that line, then there is exactly one line on that plane passing through that point which is parallel to the other line.
		In Euclidean geometry, the sum of the angles of a triangle always add up to exactly 180 degrees, no matter where it is nor how large it is. But that sum is not a constant in spherical geometry.
	denbeste.nu /cd_log_entries/2003/10/Non-Euclideanspace.shtml (2230 words)

	Learn more about Euclidean geometry in the online encyclopedia. (Site not responding. Last check: 2007-11-06)
		Euclidean geometry, also called "flat" or "parabolic" geometry, is named after the Greek mathematician Euclid.
		Euclidean geometry is distinguished from other geometries by the parallel postulate, which is more easily phrased as follows
		In particular, this postulate separates Euclidean geometry from hyperbolic geometry, where many parallel lines could be drawn through the point, and from elliptic and projective geometry, where no parallel lines exist.
	www.onlineencyclopedia.org /e/eu/euclidean_geometry.html (719 words)

	Euclidean space (Site not responding. Last check: 2007-11-06)
		A Euclidean space is a particular metric space that enables the investigation of topological properties such as compactness.
		Euclidean space plays a part in the definition of a manifold which embraces the concepts of both Euclidean and non-Euclidean geometry.
		Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric.
	www.worldhistory.com /wiki/E/Euclidean-space.htm (816 words)

	Euclidean space (Site not responding. Last check: 2007-11-06)
		A Euclidean space is a particular metric space that enables the investigation of topologytopological properties such as compactness.
		Euclidean space plays a part in the definition of a manifold which embraces the concepts of both Euclidean geometryEuclidean and non-Euclidean geometry.
		Euclidean ''n''-space is the prototypical example of an ''n''-manifold, in fact, a smooth manifold.
	www.infothis.com /find/Euclidean_space (769 words)

	Non-Euclidean Geometry
		Euclidean geometry consists basically of the geometric rules and theorems taught to kids in today’s schools.
		In Euclidean geometry, we can show that parallel lines are always equidistant, but in hyperbolic geometries, of course, this is not the case.
		Newtonian physics, based upon Euclidean geometry, failed to consider the curvature of space, and that this constituted for major errors in the equations of planetary motion and gravity.
	www.geocities.com /CapeCanaveral/7997/noneuclid.html (2640 words)

	Euclidean
		Euclidean geometry In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensio...
		Euclidean space Euclidean space is the usual n-dimensional mathematical space, a generalization of the 2- and 3-dimensio...
		Extended Euclidean algorithm The extended Euclidean algorithm is a version of the greatest common divisor (gcd) as well...
	www.brainyencyclopedia.com /topics/euclidean.html (135 words)

	A Kuhnian Approach to the Non-Euclidean Revolution in Geometry
		Euclidean Geometry as "Normal Science" circa 1800 _________________________________________________ Thus, at the dawn of the nineteenth century, geometry was clearly in a state of what Kuhn calls "normal science." Euclid's _Elements_ was recognized as "the scientific achievement.
		In fact, Euclidean geometry -- with Hilbert's corrections -- was wholly contained within projective geometry, and the non-Euclidean sectors of the new paradigm retained most of the Euclidean structure and logic, despite their differing postulates.
		The pre-revolutionary Euclidean paradigm, and its practice in a world shaped by Kantian ideals, had settled to a state of "normal science" by the late eighteenth century.
	www.vivboard.net /doc/n0065.htm (8069 words)

	What is non-Euclidean geometry?
		Einstein realized that Newtonian physics, based upon Euclidean geometry, failed to consider the curvature of space, and that this simple matter constituted for major errors in the equations of planetary motion and gravity.
		Unlike a Euclidean geometry (where every point has a unique line parallel to another line), a Poincare geometry has an infinite number of lines parallel to another line and through a specific point.
		Euclidean distance is measured with the Pythagorean theorem; Taxi-Cab distance is measured as the sum of x and y movement.
	njnj.essortment.com /noneuclideange_risc.htm (963 words)

	ABSTRACT
		Euclidean Space –time is such that both relativistic transformation of the phase and relativistic composition law of velocities make use of Lorentz transformations in deriving the relativistic aberration and doppler formulas.
		This verification of equation (10) leads to the result that the statement, for w and k in equation (8) and (9), that are derived on the basis of Euclidean space-time concept, represent angular frequency and wave-vector of starlight photon in moving frame S and are in accordance with Lorentz transformation equation.
		In Euclidean space-time the path SP followed by photon on x-axis is moving length that is transversed by the star S to reach the intrepid traveller in the frame of traveller.
	www.rajandogra.freeservers.com /messagedecember21.htm (5738 words)

	PlanetMath: Euclidean distance
		The resulting (topological and vectorial) space is known as Euclidean space.
		Euclidean metric, standard metric, standard topology, Euclidean, canonical topology
		This is version 10 of Euclidean distance, born on 2002-01-05, modified 2005-03-04.
	planetmath.org /encyclopedia/EuclideanDistance.html (199 words)

	Euclidean Distance (Site not responding. Last check: 2007-11-06)
		Euclidean Distance is the most common use of distance.
		Euclidean distance or simply 'distance' examines the root of square differences between coordinates of a pair of objects.
		Euclidean distance is a special case of Minkowski distance with
	www.people.revoledu.com /kardi/tutorial/Similarity/EuclideanDistance.html (77 words)

	Non-Euclidean Pictures?
		It should be interesting to know whether she is aware of the works carried out with the help of Fractint, Tierazon, Flarium and so on, closer to my idea of a popular fractal art, and if so, how she reacts to them.
		Notice that a spherical surface would be Euclidean in these two senses while mathematicians regard it as a non-Euclidian surface.
		The geometry of a spherical surface is not Euclidean.
	perso.wanadoo.fr /charles.vassallo/en/art/non_euclide.html (1080 words)

	PlanetMath: Euclidean space
		Alternatively, we can consider Euclidean space as an inner product space that has forgotten which point is its origin.
		It is common to refer to 2-dimensional Euclidean space as the Euclidean plane.
		This is version 9 of Euclidean space, born on 2004-04-08, modified 2005-09-04.
	planetmath.org /encyclopedia/EuclideanPlane.html (138 words)

	Models of the Hyperbolic Plane
		In the Klein model of the hyperbolic plane, the "plane" is the unit disk; in other words, the interior of the Euclidean unit circle.
		The upper half plane model takes the Euclidean upper half plane as the "plane." Now the "lines" are portions of circles with their center on the boundary, as shown in Figure 1.
		Thus as in the Klein model, the "distance" to the boundary of the disk is infinite, and postulate 2 holds.
	www.geom.uiuc.edu /docs/forum/hype/model.html (519 words)

	Non-Euclidean Geometry - Wasil Intsar Mohar
		Engineering and architecture indicate that Euclidean geometry is extremely useful for ordinary distances, but not so for large scale distances.
		It may seem that Euclidean geometry may be most convenient – it is for ordinary engineering but not for the theory of relativity.
		It is proposed that if space-time does happen to have an overall positive curvature, then the universe will stop expanding after a fixed amount of time and then start to shrink resulting in a big crunch as opposed to the big bang that resulted in its creation.
	community.middlebury.edu /~wmohar/Non-EuclideanGeometry-WasilMohar.htm (2291 words)

	Non-Euclidean Geometries, Models
		Gauss expressed his conviction in consistency of the theory he had in mind in a letter in 1824.
		However, hesitant of the public reaction to the idea that, by the side of Euclidean, there is another geometry, he never published anything on the subject.
		Euclidean segments of constant length grow in the non-Euclidean length when moved towards that circle.
	www.cut-the-knot.org /triangle/pythpar/Model.shtml (763 words)

	Non-Euclidean geometry
		It is important to realise that neither Bolyai's nor Lobachevsky's description of their new geometry had been proved to be consistent.
		In fact Beltrami's model was incomplete but it certainly gave a final decision on the fifth postulate of Euclid since the model provided a setting in which Euclid's first four postulates held but the fifth did not hold.
		Euclidean geometry is a limiting case between the two where for each line there are two coincident infinitely distant points.
	www-gap.dcs.st-and.ac.uk /~history/HistTopics/Non-Euclidean_geometry.html (1843 words)

	Euclidean & Non Euclidean Geometry (Site not responding. Last check: 2007-11-06)
		Before I address the topic of Euclidean geometry, I'd like to try to clarify exactly what Euclidean geometry is. Euclidean geometry consists of all the known rules, definitions, propositions, and thereoms before and up to the time of the Greek scholar Euclid.
		I think the best way to describe Euclidean geometry is to say that it is all based on the daily human perception of the world, and with the relationship between objects.
		In Euclidean geometry, the shortest distance between two points is a straight line.
	dsdk12.net /project/euclid/GEOEUC~1.HTM (713 words)

	Articles - Non-Euclidean geometry (Site not responding. Last check: 2007-11-06)
		While Euclidean geometry (named for the Greek mathematician Euclid) includes some of the oldest known mathematics, non-Euclidean geometries were not widely accepted as legitimate until the 19th century.
		Euclidean geometry is modelled by our notion of a "flat plane." The simplest model for elliptic geometry is a sphere, where lines are "great circles" (such as the equator or the meridians on a globe), and points opposite each other are identified (considered to be the same).
		Einstein's Theory of Relativity describes space as generally flat (i.e., Euclidean), but elliptically curved (i.e., non-Euclidean) in regions near where matter is present.
	www.gaple.com /articles/Non-Euclidean_geometry (962 words)

	KEGP (Site not responding. Last check: 2007-11-06)
		Kant clearly regards the established (Euclidean, Aristotelian and Newtonian) models as having attained a kind of absolute certainty, but this does not mean, as is generally assumed, that he therefore rejects the possibility of other, equally valid models being developed.
		When this is understood, the question of whether or not physical space is Euclidean can be seen in its proper perspective, as a side issue relating not to the validity of transcendental philosophy, but only to the question of the significance of Euclidean geometry for empirical science.
		Moreover, in the geometry of curved space, the perspective-lessness of the observer (or the unobservability of the perspective) is of utmost importance.
	www.hkbu.edu.hk /~ppp/srp/arts/KEGP.html (9752 words)

	Euclidism and Theory of Architecture by Michele Sbacchi for the Nexus Network Journal vol.3 no.3 Summer 2001 (Site not responding. Last check: 2007-11-06)
		In fact, Euclidean geometry is still an essential part of the curriculum in high schools worldwide, as it was in the quadrivium during the Middle Ages.
		Hence we can assume that an 'Euclidean culture associated with architecture,' existed for a long time and that it was probably the preeminent one among the masses and the workers.
		Traces of Euclidean studies can be found also in Leonardo: the M and I nanuscripts, the Foster, Madrid II and Atlantic codices contain Euclidean constructions and even the literal transcription of the first page of the Elements [Lorber 1985:114; Veltman 1986].
	www.nexusjournal.com /Sbacchi.html (5592 words)

	Dimensions in Special Relativity - a Euclidean interpretation (Site not responding. Last check: 2007-11-06)
		Euclidean relativity, both special and general, is steadily gaining attention as a viable alternative to the Minkowski framework.
		The links section gives a short overview of its main characteristics together with a number of references to articles on Euclidean relativity published by other authors.
		The Euclidean interpretation of special relativity that you will find on this site provides arguments for a geometrical unification of gravity and electromagnetism, which is worked out in another article.
	www.rfjvanlinden171.freeler.nl (457 words)

	Math 413 Lecture 2 - Divisibility & Euclidean Algorithm
		This is useful both because the Euclidean algorithm is the primary computational tool in number theory and because the existence of a Euclidean algorithm has strong consequences for the structure of the ring.
		The existence of a Euclidean algorithm is sufficiently important that we call a ring with such an algorithm a Euclidean ring.
		Careful analyses of the Euclidean algorithm and faster variants of the algorithm can be found in [Knuth2] and [BS96].
	www.math.umbc.edu /~campbell/Math413Spr01/Lectures/lecture2.html (455 words)

	NonEuclid: Non-Euclidean Geometery (Site not responding. Last check: 2007-11-06)
		Euclidean Geometry was named after Euclid, a Greek mathematician who lived in 300 BC.
		Euclidean Geometry was of great practical value to the ancient Greeks as they used it (and we still use it today) to design buildings and survey land.
		One of the most useful non-Euclidean geometries is Spherical Geometry which describes the surface of a sphere.
	www.cs.unm.edu /~joel/NonEuclid/noneuclidean.html (333 words)

	References on Euclidean Distance Transforms
		We analyze the error patterns for approximate Euclidean DT using finite masks, and we derive a rule defining, for any pixel location, the size of the neighborhood that guarantees the exactness of the DT.
		In Chapter 7, the 3D Euclidean DT is applied to the registration of MR images of the brain where the matching criterion is the distance between the surfaces of similar objects (skin, cortex, ventricular system,...) in both images.
		The parallel algorithms discussed are for the computation of exact Euclidean distance transform for all pixels with respect to fl pixels in an N × N binary image.
	www.lems.brown.edu /vision/people/leymarie/Refs/CompVision/DT/EDT.html (2076 words)

	Euclidean Tensors
		Euclidean tensors restore the clarity and generality of presentation, and make it easy to find general relations.
		Euclidean tensors are of special help in describing crystal properties; here, they are practically essential, since vector methods are of little aid.
		Its ordinary properties express the Euclidean nature of space, meaning that a vector is not changed by parallel displacement or by rotation, as revealed by ordinary experience.
	www.du.edu /~jcalvert/math/eucltens.htm (5403 words)

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