Non Euclidean geometries

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Topic: Non Euclidean geometries

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	Learn more about Euclidean geometry in the online encyclopedia.
		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.
		Euclidean geometry is distinguished from other geometries by the parallel postulate, which is more easily phrased as follows
		(Euclidean geometry does, however, share the parallel postulate with some other geometries, such as certain finite geometries and affine geometry.)
	www.onlineencyclopedia.org /e/eu/euclidean_geometry.html (719 words)

	51: Geometry
		This large area includes classical Euclidean geometry and synthetic (non-Euclidean) geometries; analytic geometry; incidence geometries (including projective planes); metric properties (lengths and angles); and combinatorial geometries such as those arising in finite group theory.
		Cabri-geometry is used for teaching secondary school geometry, but, equally important, is its use for university level instruction and as a tool by mathematicians in their research work.
		Solid geometry is placed here (actually in 51M05) because it mirrors elementary plane geometry, but spherical geometry is primarily on the page for general convex geometry.
	www.math.niu.edu /~rusin/known-math/index/51-XX.html (828 words)

	Non-Euclidean Geometry
		Although the formulas for computing distance and angles in these geometries differ from Euclidean geometry, they can be built into mathematical visualization systems by hand.
		Mathematicians in the nineteenth century showed that it was possible to create consistent geometries in which Euclid's Parallel Postulate was no longer true- Absence of parallels leads to spherical, or elliptic, geometry; abundance of parallels leads to hyperbolic geometry.
		By mid-century the English mathematician Arthur Cayley had constructed analytic models of these three geometries that had a common descent from projective geometry, which one may think of as the formalization of the renaissance theory of perspective.
	www.geom.uiuc.edu /docs/research/ieee94/node12.html (652 words)

	"GEOMETRY" related terms, short phrases and links (Archive 2002)
		Euclidean and Non-Euclidean Geometries : Development and History von Marvin Jay Greenberg Computational Geometry.
		Non-Euclidean geometries are those geometries obtained by relaxing the fifth postulate of Euclid (see Euclidean geometry).
		There is a special bargain pack: Geometry and Topology Monographs bargain pack at $64 for individual purchasers.
	keywen.com /Science/Math/Geometry (652 words)

	Non-Euclidean Geometry
		Although the formulas for computing distance and angles in these geometries differ from Euclidean geometry, they can be built into mathematical visualization systems by hand.
		Mathematicians in the nineteenth century showed that it was possible to create consistent geometries in which Euclid's Parallel Postulate was no longer true- Absence of parallels leads to spherical, or elliptic, geometry; abundance of parallels leads to hyperbolic geometry.
		By mid-century the English mathematician Arthur Cayley had constructed analytic models of these three geometries that had a common descent from projective geometry, which one may think of as the formalization of the renaissance theory of perspective.
	www.geom.uiuc.edu /docs/research/ieee94/node12.html (652 words)

	Amazon.com: Books: Non-Euclidean Geometry (Spectrum)
		After this the Euclidean and hyperbolic geometries are built up axiomatically as special cases of a more general 'descriptive geometry'.
		Euclidean and Non-Euclidean Geometry by Patrick J. Ryan
		In so many ways, Euclidean geometry is but the middle way between the two other geometries.
	www.amazon.com /exec/obidos/tg/detail/-/0883855224?v=glance (1186 words)

	Learn more about Euclidean geometry in the online encyclopedia.
		(Euclidean geometry does, however, share the parallel postulate with some other geometries, such as certain finite geometries and affine geometry.)
		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.
		Euclidean geometry, also called "flat" or "parabolic" geometry, is named after the Greek mathematician Euclid.
	www.onlineencyclopedia.org /e/eu/euclidean_geometry.html (719 words)

	Non-Euclidean geometry - Wikipedia, the free encyclopedia
		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).
		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.
	en.wikipedia.org /wiki/Non-Euclidean_geometry (1060 words)

	Non-Euclidean Geometry Seminar
		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).
		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.
	www.math.columbia.edu /~pinkham/teaching/seminars/NonEuclidean.html (446 words)

	non-Euclidean geometry -- Encyclopædia Britannica
		The entire scope of mathematics was enriched by the discovery of controversial areas of study such as non-Euclidean geometries and transfinite set theory.
		Although the term is frequently used to refer only to hyperbolic geometry, common usage includes those few geometries (hyperbolic and spherical) that differ from but are very close to Euclidean geometry (see table).
		Brief introduction to the life and works of this New Zealand mathematician known for his contributions to non-euclidean geometry and the history of mathematics.
	www.britannica.com /eb/article-9111073 (809 words)

	The Geometry Junkyard: Geometric Topology
		Similar questions in three dimensions have more complicated answers; Thurston showed that there are eight possible geometries, and conjectured that all 3-manifolds can be split into pieces having these geometries.
		For instance, compact two dimensional surfaces can have a local geometry based on the sphere (the sphere itself, and the projective plane), based on the Euclidean plane (the torus and the Klein bottle), or based on the hyperbolic plane (all other surfaces).
		This area of mathematics is about the assignment of geometric structures to topological spaces, so that they "look like" geometric spaces.
	www.ics.uci.edu /~eppstein/junkyard/topo.html (809 words)

	hyperbolic_tris
		One might also consider scaling down a putative hyperbolic trisection to the infinitesimal scale to obtain a Euclidean trisection (which we know to be impossible), but the hyperbolic plane admits no scaling transformations (unlike the Euclidean plane), and anyway the limit of a valid construction may not be a valid construction (degeneracies may arise).
		Because of this, drawing a hyperbolic circle through a given point with a given center is different from drawing a Euclidean circle through a given point with a given center, and it is not clear (to me) that the former construction can be reduced to the latter (or to any other Euclidean construction).
		Since this interpretation fails for other geometries such as the hyperbolic plane or the sphere, it would be nice to understand the problem in a way that would extend naturally to other geometries.
	www.math.niu.edu /~rusin/known-math/98/hyperbolic_tris (457 words)

	Non-Euclidean Geometries Books - Non-Euclidean Geometries - Geometry & Topology - Mathematics - Professional Science - Professional & Technical - Find the Lowest Price with Buyer's Wiz Price Comparison Shopping
		Non-Euclidean Geometries Books - Non-Euclidean Geometries - Geometry & Topology - Mathematics- Professional Science - Professional and Technical - Find the Lowest Price with Buyer's Wiz Price Comparison Shopping
	www.buyerswiz.com /browse/books/6467.html (457 words)

	non-Euclidean geometry -- Encyclopædia Britannica
		The entire scope of mathematics was enriched by the discovery of controversial areas of study such as non-Euclidean geometries and transfinite set theory.
		Although the term is frequently used to refer only to hyperbolic geometry, common usage includes those few geometries (hyperbolic and spherical) that differ from but are very close to Euclidean geometry (see table).
		In its rough outline, Euclidean geometry is the plane and solid geometry commonly taught in secondary schools.
	www.britannica.com /eb/article-9111073 (798 words)

	Non-Euclidean Geometry Seminar
		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).
		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.
	www.math.columbia.edu /~pinkham/teaching/seminars/NonEuclidean.html (446 words)

	USS Clueless - Non-Euclidean space
		It's impossible to describe curved space in Euclidean geometry, but it is definitely possible in some non-Euclidean geometries.
		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.
	denbeste.nu /cd_log_entries/2003/10/Non-Euclideanspace.shtml (2230 words)

	Non-Euclidean Geometry
		In Euclidean geometry, we can show that parallel lines are always equidistant, but in hyperbolic geometries, of course, this is not the case.
		Euclidean geometry consists basically of the geometric rules and theorems taught to kids in today’s schools.
		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)

	Articles - Non-Euclidean geometry
		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.
		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.
		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.gaple.com /articles/Non-Euclidean_geometry (962 words)

	Question Corner -- Euclidean Geometry
		Another reason it is given the special name "Euclidean geometry" is to distinguish it from non-Euclidean geometries (described in the answer to another question).
		Euclidean geometry is just another name for the familiar geometry which is typically taught in grade school: the theory of points, lines, angles, etc. on a flat plane.
		The difference is that Euclidean geometry satisfies the Parallel Postulate (sometimes known as the Fifth Postulate).
	www.math.toronto.edu /mathnet/questionCorner/euclidgeom.html (230 words)

	NonEuclid: Non-Euclidean Geometery
		One of the most useful non-Euclidean geometries is Spherical Geometry which describes the surface of a sphere.
		non-Euclidean Geometry is any geometry that is different from Euclidean Geometry.
		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.
	www.cs.unm.edu /~joel/NonEuclid/noneuclidean.html (333 words)

	Additional Reading (from non-Euclidean geometry) -- Encyclopædia Britannica
		Although the term is frequently used to refer only to hyperbolic geometry, common usage includes those few geometries (hyperbolic and spherical) that differ from but are very close to Euclidean geometry (seetable).
		, Experiencing Geometry: Euclidean and Non-Euclidean with History, 3rd ed.
		In its rough outline, Euclidean geometry is the plane and solid geometry commonly taught in secondary schools.
	www.britannica.com /eb/article-70472?tocId=70472 (956 words)

	Non-Euclidean Geometry Seminar
		Four general references were used throughout this course: Bonola's Non-Euclidean Geometry, Jeremy Gray's Ideas of Space, Greenberg's Euclidean and Non-Euclidean Geometries, and McCleary's Geometry from a Differential Viewpoint.
		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).
		Almost all criticisms of Euclid up to the 19th century were centered on his fifth postulate, the so-called Parallel Postulate.The first half of the course dealt with various attempts by ancient, medieval, and (relatively) modern mathematicians to prove this postulate from Euclid's others.
	www.math.columbia.edu /~pinkham/teaching/seminars/NonEuclidean.html (446 words)

	Non-Euclidean Geometry
		In all cases, however, the standard built-in illumination computations are implicitly Euclidean; correct rendering of surface shading requires custom software shaders that use alternative inner products for computing distances and angles in non-Euclidean geometry.
		The other two models are from the outsider's point of view, and we can see the position of the insider's camera marked by the blue X. In the projective (Beltrami-Klein) model of hyperbolic space, shown in Figure 8b, geodesics (the paths of light rays) are Euclidean straight lines, while angles are non-Euclidean in character.
		Although the formulas for computing distance and angles in these geometries differ from Euclidean geometry, they can be built into mathematical visualization systems by hand.
	www.geom.uiuc.edu /docs/research/ieee94/node12.html (652 words)

	Non-Euclidean geometry - Wikipedia, the free encyclopedia
		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.
		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.
		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 (1101 words)

	Distance - Wikipedia, the free encyclopedia
		In the study of complicated geometries, we call the most common type of distance Euclidean distance, as we define it from the Pythagorean theorem.
		In the case of two locations on Earth, usually the distance along the surface is meant: either "as the crow flies" (along a great circle) or by road, railroad, etc. Distance is sometimes expressed in terms of the time to cover it, for example walking or by car.
		Alternatively, the distance between sets may indicate "how different they are", by taking the supremum over one set of the distance from a point in that set to the other set, and conversely, and taking the larger of the two values (Hausdorff distance).
	en.wikipedia.org /wiki/Distance (678 words)

	USS Clueless - Non-Euclidean space
		It's impossible to describe curved space in Euclidean geometry, but it is definitely possible in some non-Euclidean geometries.
		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.
	denbeste.nu /cd_log_entries/2003/10/Non-Euclideanspace.shtml (2230 words)

	Metric space - Wikipedia, the free encyclopedia
		The geometry of the space depends on the metric chosen, and by using a different metric we can construct interesting non-Euclidean geometries such as those used in the theory of general relativity.
		In case of Euclidean space with usual metric the two notions of similarity are equivalent.
		An important consequence is that every metric space admits partitions of unity and that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietze extension theorem).
	en.wikipedia.org /wiki/Metric_space (1832 words)

	Causality, Measurement and Space
		He found that these measurements characterized the many-parallel class of non-Euclidean geometries as an infinity of spaces of constant negative curvature, the no-parallel class as an infinity of spaces of constant positive curvature, and Euclidean geometry as a unique space of constant zero curvature.
		In the Euclidean case, the metric is independent of position and yields ordinary length.
		Space is not a cause because space is not an entity.
	www.quackgrass.com /space.html (5250 words)

	Group theory
		Möbius in 1827, although he was completely unaware of the group concept, began to classify geometries using the fact that a particular geometry studies properties invariant under a particular group.
		At that time the only known groups were groups of permutations and even this was a radically new area, yet Cayley defines an abstract group and gives a table to display the group multiplication.
		Hölder was to prove it in the context of abstract groups in 1889.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Development_group_theory.html (5250 words)

	Non-Euclidean Geometry
		In Euclidean geometry, we can show that parallel lines are always equidistant, but in hyperbolic geometries, of course, this is not the case.
		Euclidean geometry consists basically of the geometric rules and theorems taught to kids in today’s schools.
		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)

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