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Topic: Euclidean plane

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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.
	planetmath.org /encyclopedia/EuclideanPlane.html (147 words)

	Euclidean space - Wikipedia, the free encyclopedia
		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.
		Euclidean n-space is the prototypical example of an n-manifold, in fact, a smooth manifold.
	en.wikipedia.org /wiki/Euclidean_space (740 words)

	Euclidean geometry - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-16)
		In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions.
		Plane geometry is the kind of geometry usually taught in high school.
		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.secaucus.us /project/wikipedia/index.php/Plane_geometry (710 words)

	Cabinet Magazine Online - Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina
		Like a Euclidean plane it is open and infinite, but it has a more complex and counterintuitive geometry.
		DH: The discovery of the hyperbolic plane came from the attempt to prove Euclid's fifth postulate, which is also known as the parallel postulate.
		On a Euclidean plane the internal angles of a triangle sum to 180 degrees, but on a hyperbolic plane they always sum to less than 180 degrees.
	www.cabinetmagazine.org /issues/16/crocheting.php (2537 words)

	Euclidean space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-16)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by (Greek geometer (3rd century BC)) Euclid.
		Euclidean space plays a part in the definition of a (A pipe that has several lateral outlets to or from other pipes) manifold which embraces the concepts of both (Click link for more info and facts about Euclidean) Euclidean and (Geometry based on axioms different from Euclid's) non-Euclidean geometry.
		Euclidean n-space is the prototypical example of an n- (A pipe that has several lateral outlets to or from other pipes) manifold, in fact, a (Click link for more info and facts about smooth manifold) smooth manifold.
	www.absoluteastronomy.com /encyclopedia/e/eu/euclidean_space.htm (1067 words)

	Circle - Wikipedia, the free encyclopedia
		In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center).
		Circles are simple closed curves, dividing the plane into an interior and exterior.
		Clifford discovered, in the ordinary Euclidean plane, a "sequence or chain of theorems" of increasing complexity, each building on the last in a natural progression.
	en.wikipedia.org /wiki/Circle (824 words)

	Tiling article - Tiling geometry tessellation Euclidean plane congruent topological - What-Means.com (Site not responding. Last check: 2007-10-16)
		In geometry, a tiling (also called tessellation, mosaic or dissection) of a given shape S consists of a collection of other shapes which precisely cover S. Often the shape S to be tiled is the Euclidean plane, but other shapes and three-dimensional objects are considered as well.
		Most topics in the area of tilings, patterns and packing problems are best known from examples in the two-dimensional Euclidean space, the Euclidean plane.
		It is still an unsolved problem, however, whether the plane can be tiled with a set of integral tiles such that each natural number is used exactly once as size of a square tile.
	www.what-means.com /encyclopedia/Tiling (2481 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.
		The "distance" between two points with coordinates z and w in the complex plane is: 2 arctanh{(z-w)/(1-wz)}, where z is the complex conjugate of z.
	www.geom.uiuc.edu /docs/forum/hype/model.html (519 words)

	CCSU Masters Thesis Abstract: The Euclidean and Hyperbolic Geometry underlying M.C. Escher's Regular Division Designs (Site not responding. Last check: 2007-10-16)
		The isometries of the Euclidean and the hyperbolic planes are discussed empirically.
		Tilings of the Euclidean plane and the hyperbolic plane may then be used to illustrate, visually, the differences between the two planes.
		Therefore a proof written to prove a theorem for the hyperbolic plane may give a student a clearer understanding of the importance of applying the rules of logic to a set of axioms and theorems to prove, or disprove, a conjecture they may have.
	wilson.ctstateu.edu /ccsu_theses/1491.html (663 words)

	H(+) plane and Euclidean plane (Site not responding. Last check: 2007-10-16)
		The hyperbolic length in the upper half-plane and the length in the Euclidean plane are the identical.
		The way of transforming a triangle is the same as we usually do with a pair of compasses in the Euclidean plane, except that the center of a circle in the upper half-plane shits up.
		In the hyperbolic plane, one end of the plane is at the intersection of the red and green hyperbolic parallel lines and the other end is perpendicular to the yellow base line.
	www1.kcn.ne.jp /~iittoo/us31_tool.htm (1017 words)

	Crystallographic Topology - Orbifold 2
		There are 17 plane groups (wallpaper groups) defining the symmetry in all patterns that repeat by 2-dimensional lattice translations in Euclidean 2-space.
		The notation under the crystallographic drawing is the standard plane group name and that under the orbifold drawing is our notation for the Euclidean 2-orbifold with S, D, and RP denoting sphere, disk, and real projective plane, respectively.
		For the projective plane orbifold, RP22, 1/4 of the unit cell is required for the asymmetric unit.
	www.ornl.gov /sci/ortep/topology/orbfld2.html (1387 words)

	The Institute For Figuring // An Interview with David Henderson and Daina Taimina
		The soccer model of the hyperbolic plane was conceived by Keith Henderson.
		On a Euclidean plane the internal angles of a triangle sum to 180, but on a hyperbolic plane they always sum to less than 180.
		This is one of the exercises we give the students, and it’s very mysterious because the area of any ideal triangle on a hyperbolic plane will always be πr2, where r is the radius of curvature of the plane.
	www.theiff.org /lectures/05a.html (2350 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.
		The dual tessellation {4,5} of the hyperbolic plane
	aleph0.clarku.edu /~djoyce/poincare/poincare.html (784 words)

	non-Euclidean (hyperbolic) geometry applet (Site not responding. Last check: 2007-10-16)
		This is analogous to ordinary "sliding" of objects in Euclidean space; however, in this non-Euclidean geometry the Euclidean picture of it makes things appear to become smaller as they move toward the edge.
		The preservation of angles should be detectable if one keeps in mind that the angles are angles between the arcs of circles at their point of intersection.
		Since the bounding circle is "infinitely far away", the motion of the picture does not exactly parallel the mouse drag motion, but instead moves about the same non-Euclidean distance as the Euclidean distance moved by the mouse.
	www.math.umn.edu /~garrett/a02/H2.html (214 words)

	Body
		Note that in the construction of a hyperbolic plane is dependent on the r (the radius of the annuli) which is often called the radius of the hyperbolic plane.
		The upper half plane model is a convenient way to study the hyperbolic plane -- think of it as a map of the hyperbolic plane in the same way that we use planar maps of the spherical surface of the earth.
		(0) to vectors in the tangent plane at z(p).
	www.math.cornell.edu /~dwh/books/eg99/Ch05/Ch05.html (4021 words)

	Modern Algebra: transformation groups
		One way to study the Euclidean plane is to concentrate on distance and find enough characteristics of this concept of distance to be able to describe of of Euclidean plane geometry in terms of distance.
		The group of isometries of the Euclidean plane is an example of a transformation group.
		The Euclidean plane could be replaced by a hyperbolic plane, or a sphere, or a cylinder.
	aleph0.clarku.edu /~djoyce/modalg/transgroups.html (1374 words)

	Citations: Computing the constrained relative neighborhood graphs and constrained Gabriel graphs in Euclidean plane - ...
		Computing the constrained relative neighborhood graphs and gabriel graphs in euclidean plane.
		SU AND R. Computing the constrained relative neighborhood graphs and constrained Gabriel graphs in Euclidean plane.
		Clearly a simple polygon is a special case of a set of line segments and hence under their visibility constraint the RND of a simple polygon can be computed in O(n log n) time.
	citeseer.ist.psu.edu /context/167411/0 (1206 words)

	Search Results for Euclidean geometry
		Euclidean geometry] but thoroughly consistent, which I have developed to my entire satisfaction, so that I can solve every problem in it excepting the determination of a constant, which cannot be fixed a priori.
		The proof is very typical of Motzkin in that the Euclidean algorithm is given a new formulation, which at first seems to be leading away from the problem at hand, but is suddenly seen to be the decisive key to its solution.
		Euclidean geometry is a limiting case between the two where for each line there are two coincident infinitely distant points.
	www-groups.dcs.st-andrews.ac.uk /history/Search/historysearch.cgi?SUGGESTION=Euclidean+geometry&CONTEXT=1 (15645 words)

	Euclidean geometry
		Euclidean geometry usually refers to geometry in the plane which is also called plane geometry.
		It is plane geometry which is the topic of this article.
		Many revised systems of axioms were constructed, the most standard ones are Hilbert's axioms and Birkhoff's axioms.
	www.sciencedaily.com /encyclopedia/euclidean_geometry (611 words)

	Semi-Regular Tilings of the Plane Part 1: Introduction and Historical Background
		In the Euclidean plane, tilings using only regular polygonal tiles were known and used in antiquity, but were not completely and systematically classified.
		The hyperbolic plane is amenable to an infinite number of different tilings by regular polygons.
		In fact, the author was able to locate only a single such illustration [R, Figure 11] which shows a tiling of the hyperbolic plane using two regular octagons and a single hexagon at each vertex.
	people.hws.edu /mitchell/tilings/part1.html (871 words)

	Euclidean space (Site not responding. Last check: 2007-10-16)
		Equivalent primal and dual differentiable reformulations of the Euclidean multifacility location problem.
		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...
	hallencyclopedia.com /Euclidean_space (533 words)

	The Imperfect Plane
		The Euclidean plane is a strange and unnatural beast.
		The flatness is guaranteed by the parallel postulate (a curved plane would have non-Euclidean regions–if the axiom is universal, the plane is flat) the ungrainy, uninterrupted evenness by the continuity axioms, the symmetry by the congruence axioms, the infinitude and simplicity by betweeness and continuity.
		Points and lines were redefined and then "undefined" in that first giddy frenzy, as it became clear that the crippled versions of the plane which had horrified earlier geometers, violating their monotheistic faith in that perfect form, were in fact merely new constructs.
	indagabo.orcon.net.nz /non-Euclidean.htm (1837 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations (Site not responding. Last check: 2007-10-16)
		On the hyperbolic plane a set of two damped harmonic oscillators, each time-reversed from the other, is shown to be equivalent to a single undamped harmonic oscillator.
		The equations for the damped oscillators are proven to be the same as the ones for the Lorentz force acting on two particles carrying opposite charge in a constant magnetic field and in the electric harmonic potential.
		The symplectic structure of the reduced theory is finally discussed in the Dirac constrained canonical formalism and in the Faddeev{endash}Jackiw symplectic formalism.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=486548 (297 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		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).
		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).
		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)

	Euclidean space : Euclidean plane (Site not responding. Last check: 2007-10-16)
		terms defined : Euclidean space : Euclidean plane
		All is still licensed under the GNU FDL.
		The only official was that of steward to one of the "powerful families" then engaged undermining the ducal authority; this appointment was a kind of measures of grain and checking the heads of cattle.
	www.termsdefined.net /eu/euclidean-plane.html (501 words)

	Encyclopedia: Euclidean plane
		In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid.
		In its turn, a Euclidean space plays a part in the definition of a manifold which embraces the concepts of both Euclidean and Non-Euclidean geometry.
		The (interior) angle θ; between x and y is then given by
	www.nationmaster.com /encyclopedia/Euclidean-plane (793 words)

	Finite Fields as Models for Euclidean Plane Geometry: A Supplement to Finitism in Geometry (Site not responding. Last check: 2007-10-16)
		The second condition that is needed to guarantee the existence of the Euclidean "kernel" is a non-trivial statement.
		Note: As Ernst Welti points out, it would be a rather annoying situation for a finitist if the proof that shows that there are an infinite number of primes of the right form were not finitistically acceptable.
		Although the original proof of Dirichlet was in fact unacceptable, fortunately there does now exist a finitistically acceptable proof of the theorem.
	plato.stanford.edu /entries/geometry-finitism/supplement.html (631 words)

	Dieter Ruoff - Proportionality in the non-Euclidean plane
		The Euclidean proportionality theorems involving an angle that is intersected by a pair of parallel lines do not extend to the hyperbolic plane; on the the one hand the uniqueness of the parallel line and on the other a meaningful concept of similarity are missing.
		In fact, the proportionality theorems also fail when parallelism is interpreted in the narrower sense of being boundary parallel.
		As will be shown the proof that this is so is not straightforward but requires some interesting lemmas concerning the hyperbolic plane.
	www.camel.math.ca /Events/winter99/abstracts/node187.f?nomenu=1 (154 words)

	M470
		Write and Submit a proof of the theorem that in euclidean geometry, the sum of the interior angles in a triangle is 180 degrees.
		For points and lines of a projective plane: Given a point p and a line L not containing p, prove that the function q -> (the line joining p and q) gives a bijection between the set of points on L and the set of lines through p.
		Prove that in a projective plane, there is a bijection between the points of any two lines.
	www.math.colostate.edu /courses/spring00/m470sp00.html (768 words)

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