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Topic: Euclidean geometry

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	PlanetMath: non-Euclidean geometry
		A non-Euclidean geometry is a geometry in which at least one of the axioms from Euclidean geometry fails.
		(This sum is not constant as in Euclidean geometry; it depends on the area of the triangle.
		Note also that, in spherical geometry, two distinct points do not necessarily determine a unique line; however, two distinct points that are not antipodal always determine a unique line.
	planetmath.org /encyclopedia/NonEuclideanGeometry.html (459 words)

	Euclidean geometry - Wikipedia, the free encyclopedia
		Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria.
		In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory, which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that probably cannot be constructed within the theory.
		Absolute geometry, formed by removing the parallel postulate, is also a consistent theory, as is non-Euclidean geometry, formed by alterations of the parallel postulate.
	en.wikipedia.org /wiki/Euclidean_geometry (2331 words)

	Encyclopedia :: encyclopedia : Non-euclidean geometry (Site not responding. Last check: )
		In Euclidean geometry, however, the lines remain at a constant distance, while in hyperbolic geometry they "curve away" from each other, increasing their distance as one moves farther from the point of intersection with the common perpendicular.
		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).
	www.hallencyclopedia.com /Non-euclidean_geometry (1057 words)

	non-Euclidean geometry – FREE non-Euclidean geometry Information \| Encyclopedia.com: Facts, Pictures, Information!
		The results of these two types of non-Euclidean geometry are identical with those of Euclidean geometry in every respect except those propositions involving parallel lines, either explicitly or implicitly (as in the theorem for the sum of the angles of a triangle).
		In hyperbolic geometry the two rays extending out in either direction from a point P and not meeting a line L are considered distinct parallels to L ; among the results of this geometry is the theorem that the sum of the angles of a triangle is less than 180°.
		An idea of the geometry on such a plane is obtained by considering the geometry on the surface of a sphere, which is a special case of an ellipsoid.
	www.encyclopedia.com /doc/1E1-nonEucli.html (1114 words)

	Learn more about Euclidean geometry in the online encyclopedia. (Site not responding. Last check: )
		This is the kind of geometry familiar to most people, since it is the kind usually taught in high school.
		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)

	Non-Euclidean geometry - Wikipedia, the free encyclopedia
		Another way to describe the differences between these geometries is as follows: Consider two lines in a two-dimensional plane that are both perpendicular to a third line.
		In Euclidean geometry the lines remain at a constant distance, intersecting only in the infinite; while in hyperbolic geometry they "curve away" from each other, increasing their distance as one moves further from the point of intersection with the common perpendicular.
		Lobachevsky termed Euclidean geometry, "ordinary geometry," and this new hyperbolic geometry, "imaginary geometry." However, the possibility still remained that the axioms for hyperbolic geometry were logically inconsistent.
	en.wikipedia.org /wiki/Non-Euclidean_geometry (1213 words)

	Online Encyclopedia and Dictionary - Euclidean geometry (Site not responding. Last check: )
		In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions.
		Euclidean geometry sometimes means geometry in the plane which is also called plane geometry.
		The traditional presentation of Euclidean geometry is as an axiomatic system, setting out to prove all the "true statements" as theorems in geometry from a set of finite number of axioms.
	www.fact-archive.com /encyclopedia/Euclidean_geometry (671 words)

	Non-Euclidean geometry
		Lambert noticed that, in this new geometry, the angle sum of a triangle increased as the area of the triangle decreased.
		It reduced the problem of consistency of the axioms of non-Euclidean geometry to that of the consistency of the axioms of Euclidean geometry.
		Euclidean geometry is a limiting case between the two where for each line there are two coincident infinitely distant points.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Non-Euclidean_geometry.html (1879 words)

	Geometry - MSN Encarta
		Euclidean geometry describes most aspects of the everyday world and was named after Euclid, the ancient Greek mathematician who developed it.
		While the postulates of Euclidean geometry do seem plausible when applied to physical space in our universe, there is evidence that Euclidean geometry is not the perfect system for describing space.
		Two-dimensional Euclidean geometry is often called plane geometry; three-dimensional Euclidean geometry is frequently referred to as solid geometry.
	encarta.msn.com /encyclopedia_761569706_2/Geometry.html (633 words)

	PlanetMath: geometry
		Geometry, or literally, the measurement of land, is among the oldest and largest areas of mathematics.
		Over the centuries, geometry has grown from its humble origins in land measurement to a study of the properties of space in the widest sense of the term.
		We start with the synthetic (or axiomatic) approach to Euclidean geometry not only because that is historically the oldest, but because it is the approach one is most likely to encounter first.
	planetmath.org /encyclopedia/EuclideanGeometry.html (1032 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)

	NonEuclid: 1: Non-Euclidean Geometry
		The essential difference between Euclidean geometry and these two non-Euclidean geometries is the nature of parallel lines: In Euclidean geometry, given a point and a line, there is exactly one line through the point that is in the same plane as the given line and never intersects it.
		In hyperbolic geometry there are at least two distinct lines that pass through the point and are parallel to (in the same plane as and do not intersect) the given line.
		Spherical geometry is a plane geometry on the surface of a sphere.
	cs.unm.edu /~joel/NonEuclid/noneuclidean.html (557 words)

	What really is Geometry?
		In this geometry, the most important theorem is the Exterior Angle Theorem that the measure of the exterior angle of a triangle is greater than either of the non-adjacent interior angles.
		In Euclidean Geometry, the measure of the exterior angle is equal to the sum of the measures of the nonadjacent interior angles.
		A non-Euclidean Geometry is a Geometry that has the negation of the Parallel Postulate as one of its axioms.
	www.mathpath.org /concepts/geometries.htm (1888 words)

	Applications Of Non-Euclidean Geometry
		It has been argued that Euclidean Geometry, while good for architecture and to survey land, when it is moved into the third dimension, the postulates do not hold up as well as those of hyperbolical and spherical geometry.
		While Euclidean geometry provides an excellent representation for the part of the universe that we inhabit, like Newton's Laws of physics, they break down when placed in situations that their originators could not have imagined.
		This is analagous to one of the quirks of each geometry; in hyperbolic geometry the sum of the angles of a triangle is greater than 180 degrees, while Euclidean has the sum of a triangles angles to be 180 degrees exactly.
	members.tripod.com /~noneuclidean/applications.html (663 words)

	History of Non-euclidean Geometry
		Lambert noticed that, in this new geometry, the angle sum of a triangle increased as the area of the triangle decreased.
		Elementary geometry was by this time engulfed in the problems of the parallel postulate.
		Euclidean geometry is a limiting case between the two where for each line there are two coincident infinitely distant points.
	members.tripod.com /~noneuclidean/history.html (1813 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 geometry - Search Results - MSN Encarta
		Plane Geometry, branch of elementary geometry dealing with the properties of flat surfaces and of planar figures, such as the triangle or the...
		Perhaps the most familiar and intuitive geometry is called Euclidean geometry.
		Euclidean geometry describes most aspects of the everyday world and...
	encarta.msn.com /Euclidean+geometry.html (156 words)

	Euclidean and Non-Euclidean Geometry
		Riemannian Geometry is named for the German mathematician, Bernhard Riemann, who in 1889 rediscovered the work of Girolamo Saccheri (Italian) showing certain flaws in 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°.
	www.regentsprep.org /Regents/math/geometry/GG1/Euclidean.htm (845 words)

	Non-Euclidean Geometry - Search Results - MSN Encarta
		Non-Euclidean Geometry, branch of geometry based on axioms different from those enumerated by Euclid in The Elements.
		Geometry (Greek geō, “Earth”; metrein, “to measure”), branch of mathematics that deals with the properties of space.
		A further 19th-century discovery that was considered apparently abstract and useless at the time was non-Euclidean geometry.
	uk.encarta.msn.com /Non-Euclidean_Geometry.html (151 words)

	Euclidean Geometry at the Library of Math (Free Online Mathematics)
		The word geometry comes from the Greek geometrein (geo meaning earth, and metrein meaning to measure); geometry was originally the science of measuring the land.
		It wasn't until after the discovery of non-Euclidean geometry that mathematicians began examining the foundations of Euclidean geometry and formulating precise sets of axioms for it.
		These axioms, which do give rise to all theorems in Euclidean geometry, are not minimal in nature and are meant to move the student almost immediately to more interesting and less intuitively obvious results.
	libraryofmath.com /Euclidean_Geometry.html (2264 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 (711 words)

	NonEuclid: Non-Euclidean Geometery (Site not responding. Last check: )
		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.
		Hyperbolic Geometry is a "curved" space, and plays an important role in Einstein's General theory of Relativity.
	www.cs.unm.edu /~joel/NonEuclid/noneuclidean.html (333 words)

	Euclidean & Non Euclidean Geometry (Site not responding. Last check: )
		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.
	www.dsdk12.net /project/euclid/GEOEUC~1.HTM (713 words)

	A Unified Algebraic Framework for Classical Geometry
		With roots in ancient times, the great flowering of classical geometry was in the 19th century, when Euclidean, non-Euclidean and projective geometries were given precise mathematical formulations and the rich properties of geometric objects were explored.
		All three of the above models are built in Euclidean space, and the latter two are conformal in the sense that the metric is a point-to-point scaling of the Euclidean metric.
		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.
	modelingnts.la.asu.edu /html/UAFCG.html (2056 words)

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