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Topic: Euclidean space

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	PlanetMath: Euclidean space
		The difference between Euclidean space and a Euclidean vector space is one of loss of structure.
		Euclidean space is a Euclidean vector space that has “forgotten” its origin.
		This is version 13 of Euclidean space, born on 2004-04-08, modified 2006-01-22.
	www.planetmath.org /encyclopedia/EuclideanPlane.html (148 words)

	Encyclopedia article: Euclidean space (Site not responding. Last check: )
		In mathematics (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement), Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid (Greek geometer (3rd century BC)).
		Euclidean space plays a part in the definition of a manifold (A pipe that has several lateral outlets to or from other pipes) which embraces the concepts of both Euclidean (additional info and facts about Euclidean) and non-Euclidean geometry (Geometry based on axioms different from Euclid's).
		Euclidean n-space is the prototypical example of an n-manifold (A pipe that has several lateral outlets to or from other pipes), in fact, a smooth manifold (additional info and facts about smooth manifold).
	www.absoluteastronomy.com /encyclopedia/e/eu/euclidean_space.htm (1021 words)

	Encyclopedia: Euclidean space (Site not responding. Last check: )
		In mathematics, the dimension of a vector space V is the cardinality (i.
		A Euclidean space is a particular metric space that enables the investigation of topological properties such as compactness.
		Euclidean space is the usual n -dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid.
	www.nationmaster.com /encyclopedia/Euclidean-space (868 words)

	Causality, Measurement and Space
		Space is often supposed to be a sort of box in which existence is placed, or a sort of insubstantial stage on which the drama of reality unfolds.
		Space is a grid of reference lines which we imagine to be constructed according to geometrical method.
		Space is not a cause because space is not an entity.
	www.quackgrass.com /space.html (5250 words)

	USS Clueless - Non-Euclidean space
		I have often seen Einstein's notion of curved space illustrated as a sheet where the sun lies in a big dent and the earth rolls around in that dent or in a trough.
		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)

	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.
		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).
		In case of Euclidean space with usual metric the two notions of similarity are equivalent.
	en.wikipedia.org /wiki/Metric_space (1832 words)

	Summary
		Affine space is obtained from projective space by fixing an arbitrary hyperplane to serve as the hyperplane of points `at infinity': requiring that this be fixed puts n constraints on the allowable projective transformations, reducing them to the affine subgroup.
		Euclidean space is obtained from affine space by designating a conic in the hyperplane at infinity to serve as the absolute conic.
		Scaled Euclidean space has all the familiar properties of conventional 3D space, except that there is no notion of scale or absolute length.
	homepages.inf.ed.ac.uk /rbf/CVonline/LOCAL_COPIES/MOHR_TRIGGS/node45.html (326 words)

	Euclidean space -- Britannica Concise Encyclopedia - Your gateway to all Britannica has to offer!
		In geometry, a two- or three-dimensional space in which the axioms and postulates of Euclidean geometry apply; also, a space in any finite number of dimensions, in which points are designated by coordinates (one for each dimension) and the distance between two points is given by a distance formula.
		Though non-Euclidean spaces, such as those that emerge from elliptic geometry and hyperbolic geometry, have led scientists to a better understanding of the universe and of mathematics itself, Euclidean space remains the point of departure for their study.
		In its rough outline, Euclidean geometry is the plane and solid geometry commonly taught in secondary schools.
	concise.britannica.com /ebc/article-9363979 (806 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.
		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.
		But in Euclidean space - time the path followed by light signal in moving frame C' is the moving length and therefore (x - vt) denotes the moving length that is transversed by the light signal with speed c.
	www.rajandogra.freeservers.com /messagedecember21.htm (5738 words)

	natural religion > glossary > euclidean space (Site not responding. Last check: )
		Euclidean (or Cartesian) space is the mathematical abstraction and extension of the 'ordinary' three dimensional space of everyday life.
		Spaces may be devised in which there are no lines fulfilling definition 23, or in which lines described by Postulate 5 never meet.
		An important feature of Euclidean space is that Pythagoras' theorem ('the square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other two sides') holds within it.
	www.naturaltheology.net /Glossary/euclideanSpace.html (666 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/EuclideanVectorSpace.html (138 words)

	PlanetMath: Euclidean distance
		The resulting (topological and vectorial) space is known as Euclidean space.
		Cross-references: Euclidean space, canonical, absolute value, formula, open balls, basis, induced, topology, metric, distance, vector space, vector, difference, norm, segment, length, plane
		This is version 10 of Euclidean distance, born on 2002-01-05, modified 2005-03-04.
	planetmath.org /encyclopedia/Euclidean.html (199 words)

	Non-Euclidean Geometry
		To say our space is Euclidean, is to say our space is not “curved”, which seems to make a lot of sense regarding our drawings on paper, however non-Euclidean geometry is an example of curved space.
		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.
		In layman’s terms, this explains that the phrase “curved space” is not a curvature in the usual sense but a curve that exists of spacetime itself and that this “curve” is in the direction of the fourth dimension.
	www.geocities.com /capecanaveral/7997/noneuclid.html (2640 words)

	Euclidean space - Art History Online Reference and Guide (Site not responding. Last check: )
		Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric.
		Euclidean n-space is the prototypical example of an n-manifold, in fact, a smooth manifold.
		is a Euclidean space of m dimensions embedded in it (as an affine subspace).
	www.arthistoryclub.com /art_history/Euclidean_space (753 words)

	Compact space - Wikipedia, the free encyclopedia (Site not responding. Last check: )
		In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space R
		If a topological space has a sub-base such that every cover of the space by members of the sub-base has a finite subcover, then the space is compact.
		At one time, when primarily metric spaces were studied, compact was taken to mean the weaker sequentially compact, that every sequence has a convergent subsequence.
	xahlee.org /_p/wiki/Compact.html (1402 words)

	Euclidean space (Site not responding. Last check: )
		Euclidean space is the usual n-dimensional mathematical space, a generalization of the 2- and3-dimensional spaces studied by Euclid.
		The term "n-dimensional Euclidean space" is usually abbreviated to "Euclidean n-space", or even just"n-space".
		Euclidean n-space can also be considered as an n-dimensional real vector space, in fact a Hilbert space, in a naturalway.
	www.therfcc.org /euclidean-space-3886.html (266 words)

	The Geometric Algebra of 3D Euclidean Space
		The axioms for a vector space are designed to encode the intuitive idea of addition of arrows and multiplication of an arrow by a number.
		Again the space is oriented so that there is a sense of unrolling and rolling space that corresponds to positive and negative volumes.
		The objects in the geometric algebra are particularly useful for representing the isometries of the space.
	omega.albany.edu:8008 /mat220dir/ga3d-dir/GA3d.html (2218 words)

	The Shape of Our Universe by Dr. Sarah
		In this quarter-turn space, unmarked walls are glued to one another in the simple, straight-across way while the marked side shows that we should glue that side and its opposite side with a rotation by 90 degrees (a quarter of a turn and hence the quarter-turn universe).
		Even though this Euclidean space is finite, we have the illusion of flying in an infinite space because we never reach an edge.
		The brightness of a shining object in Euclidean space is inversely proportional to the square of the distance to the object.
	www.mathsci.appstate.edu /~sjg/class/1010/wc/geom/universe2a.html (3337 words)

	Embedding Non-Euclidean Spaces in Euclidean Spaces (Site not responding. Last check: )
		On the other hand, it isn't clear that a formally Euclidean space with imaginary distances is any more intuitive than a curved space with strictly real distances.
		A metric space, in the strict sense of the term, is a manifold that satisfies the triangle inequality, which is the property that leads to our intuitive impressions of "locality".
		In particular, locality is transitive in a metric space, meaning that if A is close to B, and B is close to C, then A can't be too far from C. Spacetime doesn't satisfy this condition.
	www.mathpages.com /home/kmath342.htm (407 words)

	Body
		The vast empty space appears to locally (on a medium scale much larger than the scale of the distortions near stars) be a geometry the local symmetries of Euclidean 3-space.
		Euclidean 3-space, 3-spheres, and hyperbolic 3-spaces are the only simply-connected (every loop can be continuously shrunk to a point in the space) 3-dimensional geometries that locally have the same symmetries as Euclidean 3-space.
		Consider a cube in Euclidean 3-space with the opposite faces glued through a reflection in the plane that is midway between the opposite faces.
	www.math.cornell.edu /~dwh/books/eg99/Ch20/Ch20.html (4024 words)

	The Ontology and Cosmology of Non-Euclidean Geometry
		Thus the surface of a sphere is the classic model of a two-dimensional, positively curved Riemannian space; but while great circles are the straight lines (geodesics) according to the intrinsic properties of that surface, we see the surface as itself curved into the third dimension of Euclidean space.
		That is not true in terms of astronomical space, where the lines drawn by freefalling bodies in gravitational fields are most evidently curved to our three dimensional imaginations, even while they are understood to be geodesics only in terms of their form in the higher dimension of spacetime.
		The argument that, in empty space, with no "distant galaxies," there would be no centrifugal force in the bucket and the water in one would be just as flat as in the other is not a necessary conclusion, but only a theory.
	www.friesian.com /curved-1.htm (6358 words)

	Euclidean space (Site not responding. Last check: )
		him, he would argue that what is valid in non- Euclidean planning isn't new, and what is new isn't valid...
		What is new about non- Euclidean planning is that by putting together a...
		facilities such as the sailing club room and a general purpose space complete with a kitchen.
	hallencyclopedia.com /Euclidean_space (556 words)

	Models of the Hyperbolic Plane
		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.
		In this case the "space" is the unit sphere, "lines" are portions of circles intersecting the boundary of the unit sphere at right angles, and "planes" are portions of spheres which meet the unit sphere at right angles.
	www.geom.uiuc.edu /docs/forum/hype/model.html (519 words)

	Math Forum Discussions - Re: Embedding of Riemann manifold in Euclidean space of higher dimension
		Re: Embedding of Riemann manifold in Euclidean space of higher dimension
		Embedding of Riemann manifold in Euclidean space of higher dimension
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.com /kb/thread.jspa?forumID=13&threadID=1125265&messageID=3690033 (274 words)

	Euclidean Tensors
		This paper explains the concept of tensor under rotations in a Euclidean space, and the methods of calculating with indices.
		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)

	Euclidean n-dimensional space (from topology) -- Encyclopædia Britannica (Site not responding. Last check: )
		More results on "Euclidean n-dimensional space (from topology)" when you join.
		set of regulations governing international conduct in space beyond the Earth's atmosphere; concept introduced by U.S. president Dwight D. Eisenhower in 1957 in conjunction with disarmament talks; established that traditional laws allowing nations to claim any uninhabited lands are not valid in outer space; Outer Space Committee formed in 1959 in United Nations to promote...
		The exploration of space is among the most fascinating ventures of modern times.
	www.britannica.com /eb/article-69101 (791 words)

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