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Topic: Absolute geometry

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	Various Geometries
		Absolute Geometry is derived from the first four of Euclid's postulates.
		All theorems of Absolute Geometry are automatically true in the geometries of Euclid, Lobachevsky and Riemann since those three only differ in their treatment of the Fifth postulate.
		Affine Geometry is not concerned with the notions of circle, angle and distance.
	www.cut-the-knot.org /triangle/pythpar/Geometries.shtml (2183 words)

	Absolute geometry - Wikipedia, the free encyclopedia
		Absolute geometry is a geometry that does not assume the parallel postulate or any of its alternatives.
		It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate.
		Therefore this proposition is undecidable in absolute geometry.
	en.wikipedia.org /wiki/Absolute_geometry (226 words)

	Euclidean geometry - Wikipedia, the free encyclopedia
		In hyperbolic geometry the sum of the three angles are always less than 180° and can approach zero, while in elliptic geometry the sum is greater than 180°.
		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.
		For example, geometry on the surface of a sphere is a model of an elliptical geometry, carried out within a self-contained subset of a three-dimensional Euclidean space.
	en.wikipedia.org /wiki/Euclidean_geometry (2285 words)

	absolute - Wiktionary
		Absolute rights and duties are such as pertain to man in a state of nature as contradistinguished from relative rights and duties, or such as pertain to him in his social relations.
		absolute curvature (geometry): that curvature of a curve of double curvature, which is measured in the osculating plane of the curve.
		absolute temperature (physics): the temperature as measured on a scale determined by certain general thermo-dynamic principles, and reckoned from the absolute zero.
	en.wiktionary.org /wiki/absolute (489 words)

	The Craig Web Experience: Paper (Site not responding. Last check: 2007-10-28)
		We present Najm, a set of tools built on the axioms of absolute geometry for exploring the design space of Islamic star patterns.
		We describe a method for creating a parameterized set of motifs that can be used to fill the many regular polygons that comprise these tilings, as well as an algorithm to infer geometry for any irregular polygons that remain.
		Because Najm is built using only the axioms of absolute geometry, which makes no assumption about the behaviour of parallel lines, star patterns created by Najm can be designed equally well to fit the Euclidean plane, the hyperbolic plane, or the surface of a sphere.
	www.cgl.uwaterloo.ca /~csk/papers/tog2004.html (153 words)

	PlanetMath: neutral geometry
		A Euclidean geometry is a neutral geometry satisfying the Euclid's parallel axiom: for any given line and any given point not lying on the line, there is a unique line passing through the point and parallel to the given line.
		A hyperbolic geometry (or Bolyai-Lobachevsky geometry) is a neutral geometry satisfying the hyperbolic axiom: for any given line and any given point not lying on the line, there are at least two (distinct) lines passing through the point and parallel to the given line.
		This is version 5 of neutral geometry, born on 2005-11-04, modified 2005-11-05.
	planetmath.org /encyclopedia/NeutralGeometry.html (350 words)

	geometry. The Columbia Encyclopedia, Sixth Edition. 2001-05 (Site not responding. Last check: 2007-10-28)
		The general metric geometry consisting of all of Euclidean geometry except that part dependent on the parallel postulate is called absolute geometry; its propositions are valid for both Euclidean and non-Euclidean geometry.
		Another type of geometry, called affine geometry, includes Euclid’s parallel postulate but disregards two other postulates concerning circles and angle measurement; the propositions of affine geometry are also valid in the four-dimensional geometry of space-time used in the theory of relativity.
		For example, Euclidean geometry is the study of properties unchanged by similarity transformations, affine geometry is concerned with properties invariant under the linear transformations (affine collineations) that preserve parallelism, and projective geometry studies invariants under the more general projective transformations (collineations and correlations).
	www.bartleby.com /65/ge/geometry.html (681 words)

	Intro 1.1 (Site not responding. Last check: 2007-10-28)
		So that, the character of space (this is, its geometry) is determined by the choice of the initial elements and their mutual relations expressed by axioms.
		The axioms of the usual approaches to geometry can be divided into a number of groups: axioms of incidence, axioms of order, axioms of continuity, axioms of congruence and axioms of parallelism.
		Geometry based on the first three groups of axioms is called "ordered geometry ", while geometry based on the first four groups of axioms is called "absolute geometry "; to the latter corresponds the n-dimensional absolute space denoted by S
	www.emis.de /monographs/jablan/chap11.htm (337 words)

	GEOMETRY (Site not responding. Last check: 2007-10-28)
		Geometry uses problem situations, physical models, and appropriate technology to investigate and justify geometric concepts and relationships.
		Then they transfer the design to a piece of 8" X 11" pane of plexiglass and paint the pane to create a “stained glass.” Students construct one of the regular 3-dimensional solid and compute the volume and surface area.
		Draw two circles on one dartboard and three on the other. Throw randomly and count the throws that hit the board to determine which board yields the highest probability of a dart’s landing in a circle.
	www.state.tn.us /education/ci/cimathhighschool/cimathgeometry.htm (2064 words)

	Nineteenth Century Geometry
		Today projective geometry does not play a big role in mathematics, but in the late nineteenth century it came to be synonymous with modern geometry.
		one may say that the truth of the geometry of Euclid is not incompatible with the truth of the geometry of Lobachevsky, for the existence of a group is not incompatible with that of another group.
		Geometry distinguishes itself from other natural sciences because it obtains only very few concepts and laws directly from experience, and aims at obtaining from them the laws of more complex phenomena by purely deductive means.
	plato.stanford.edu /entries/geometry-19th (4782 words)

	Geometry :: Math
		Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts.
		Geometry and the Imagination in Minneapolis: Geometry exercises for a two-week summer workshop led by John Conway, Peter Doyle, Jane Gilman and Bill Thurston at the Geometry Center in Minneapolis, June 1991.
		Geometry from the Land of the Incas: Presents problems involving circles and triangles, with proofs, SAT practice quizzes and famous quotes.
	science.gourt.com /Math/Geometry.html (1235 words)

	geometry on Encyclopedia.com (Site not responding. Last check: 2007-10-28)
		GEOMETRY [geometry] [Gr.,=earth measuring], branch of mathematics concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts.
		Another type of non-Euclidean geometry was discovered by Bernhard Riemann (1854), who also showed how the various geometries could be generalized to any number of dimensions.
		Geometry in Space: this technology-rich project uses explorations of Mars to teach geometry and science to middle and high school students.
	www.encyclopedia.com /html/section/geometry_TheirRelationshiptoEachOther.asp (958 words)

	Search the Internet - InternetDJ.com (Site not responding. Last check: 2007-10-28)
		Geometry and the Imagination in Minneapolis - - Geometry exercises for a two-week summer workshop led by John Conway, Peter Doyle, Jane Gilman and Bill Thurston at the Geometry Center in Minneapolis, June 1991.
		Geometry Formulas and Facts - - Excerpts from the 30th Edition of the CRC Standard Mathematical Tables and Formulas (1995), namely, the geometry section minus differential geometry.
		Geometry from the Land of the Incas - - Presents problems involving circles and triangles, with proofs, SAT practice quizzes and famous quotes.
	www.internetdj.com /search/search.php?browse=/Science/Math/Geometry (980 words)

	[No title] (Site not responding. Last check: 2007-10-28)
		In the introduction to chapter 12, Ordered Geometry, Coxeter defines "absolute geometry" to be geometry without the parallel postulate, so theorems of absolute geometry are theorem of both Euclidean and of hyperbolic geometry.
		He also defines "affine geometry" as the part of euclidean geometry that is preserved by parallel projection from one plane to another.
		Ordered geometry lies in the theory that is common to both absolute geometry and affine geometry.
	www.math.niu.edu /~rusin/known-math/94/betweenness (308 words)

	[No title] (Site not responding. Last check: 2007-10-28)
		Description of the Course This course is an axiomatic study of various geometries including finite geometry, absolute (neutral) geometry, Euclidean geometry, Lobachevskian geometry, and Riemannian geometry.
		Apply concepts of a particular geometry to the analysis, solution, and coherent presentation of problems within that geometry.
		Absolute (Neutral) Geometry (6 hours) Introduction to geometry without the Euclidean parallel postulate culminating in the Saccheri-Legendre Theorem 3.
	www.lhup.edu /UCC/mathematics/MATH307_rev.doc (520 words)

	Math 123 Course Information (Site not responding. Last check: 2007-10-28)
		The purpose of Math 123 is to study the axiom sets and models for various geometries, with particular attention paid to Euclid's parallel postulate and to models for geometries that violate the parallel postulate (noneuclidean geometries).
		One such geometry is the hyperbolic geometry of the unit disk in the complex plane, in which the "lines" are arcs of circles perpendicular to the unit circle.
		One of the goals of Math 123 is to study noneuclidean geometries, and in particular to study the hyperbolic geometry of the unit disk.
	www.math.ucla.edu /undergrad/courses/math123 (402 words)

	[No title]
		Explain what absolute geometry is and be able to state the elliptic, Euclidean and hyperbolic parallel postulate.
		Explain the relationship between sum of the angles of a triangle, the area of a triangle and congruence in hyperbolic and elliptical.
		Describe the relationship between Absolute Geometry, Euclidean, Hyperbolic, and Spherical Geometry in terms of the axioms studied.
	www.math.ohiou.edu /~moss/StudyGuide_2.doc (494 words)

	Open Directory - Science: Math: Geometry
		Geometry Formulas and Facts - Excerpts from the 30th Edition of the CRC Standard Mathematical Tables and Formulas (1995), namely, the geometry section minus differential geometry.
		Geometry from the Land of the Incas - Presents problems involving circles and triangles, with proofs, SAT practice quizzes and famous quotes.
		Geometry in Action - Includes collections from various areas in which ideas from discrete and computational geometry meet real world applications.
	www.findthelinks.com /dmozurl/science/Math/Geometry (884 words)

	PlanetMath:
		absolute norm (in norm and trace of algebraic number) owned by pahio
		absolute trace (in norm and trace of algebraic number) owned by pahio
		absolute value of complex number (=modulus of complex number) owned by matte
	planetmath.org /encyclopedia/A (2174 words)

	Foundations of Plane Geometry (Site not responding. Last check: 2007-10-28)
		Unique in approach, it combines an extended theme--the study of a generalized absolute plane from axioms through classification into the three fundamental classical planes--with a leisurely development that allows ample time for mathematical growth.
		It is purposefully structured to facilitate the development of analytic and reasoning skills and to promote an awareness of the depth, power, and subtlety of the axiomatic method in general, and of Euclidean and non-Euclidean plane geometry in particular.
		Focus on one main topic--The axiomatic development of the absolute plane--which is pursued through a classification into Euclidean, hyperbolic, and spherical planes.
	www.indiaplaza.com /books/pd.aspx?sku=0130479543 (182 words)

	How and Why Hyperbolic Geometry Came to Be
		When talking about geometry, most people that you meet on the street would assume that Euclidean Geometry is the only form of geometry accepted in the mathematical world.
		Euclidean Geometry is the oldest form of geometry, but it is by far not the only accepted form.
		Euclidean Geometry was created when Euclid wrote The Elements around 300 BC as a collection of the mathematics known to the Greeks at that time.
	filebox.vt.edu /users/jtoffene/HowandWhyHyperbolicGeometryCametoBe.htm (1887 words)

	Citebase - New Path Equations in Absolute Parallelism Geometry
		Authors: Wanas, M. Melek, M. Kahil, M. The Bazanski approach, for deriving the geodesic equations in Riemannian geometry, is generalized in the absolute parallelism geometry.
		Paths in an appropriate geometry are usually used as trajectories of test particles in geometric theories of gravity.
		Absolute parallelism geometry is frequently used for physical applications.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:gr-qc/0207113 (922 words)

	Syllabus for Math 511 (Site not responding. Last check: 2007-10-28)
		It is a rigorous course focusing on absolute geometry where each step of the argument is justified by previously proven axioms and theorems.
		In the undergraduate course, a model of the theorem is not required; in some cases a model is undesirable since students tend to rely on the model as justification of the theorem.
		In the graduate course, students, who usually teach geometry in high school, are taught the transition of proof-writing, how to start with proofs that are largely intuitive and go through the process of developing a more formal proof.
	www.uncp.edu /home/truman/mat511/511syl.htm (282 words)

	Selected topics in modern geometry
		This course is prefaced by a careful examination of the foundations of geometry.
		Considerable attention is given to the modern alliance of geometry with linear and abstract algebra.
		To enrich the students appreciation of the geometries used in the modern world of mathematics by studying elementary and advanced Euclidean geometry and then continuing the investigation by studying the non-Euclidean geometries of Riemann and Lobachevski.
	www.njcu.edu /dept/math/grad05sylb/MA624.htm (210 words)

	Synergistic Research (Site not responding. Last check: 2007-10-28)
		Absolute Reference (X2) interconnects are carefully crafted by hand from the highest quality Silver Matrix conductors.
		Absolute Reference builds on Designers' Reference dimension and detail and adds a significant level of refinement and musicality through it's use of the all active four way geometry.
		Absolute Reference possesses a unique blend of state-of-the-art resolution and three dimensional realism with musicality and smoothness.
	www.synergisticresearch.com /abs-ref-a-ic.html (217 words)

	[No title]
		School geometry is traditionally approached either by the synthetic method (Euclid, Steiner, Hilbert) or analytically (Riemann, Klein, Birkhoff).
		Geometry and the Axiomatic Method Let me say first that I was thoroughly delighted to see this chapter here.
		Perhaps it could be called Jackiwian geometry, after the iventor of the Geometry Sketchpad, which is (I think, but may be mistaken) the first of the genre.
	www.math.uiuc.edu /~gfrancis/public/cggeall (6831 words)

	The Math Forum - Math Library - Geometry (Site not responding. Last check: 2007-10-28)
		A collection of handouts for a two-week summer workshop entitled 'Geometry and the Imagination', led by John Conway, Peter Doyle, Jane Gilman and Bill Thurston at the Geometry Center in Minneapolis, June 17-28, 1991.
		Some notes on a most general definition of "geometry," first elucidated by Felix Klein, which is based on a set of geometric invariants under a group of transformations.
		The FoCM's primary aim is to further the understanding of the deep relationships between mathematical analysis, topology, geometry and algebra and the computational process as they are evolving together with the modern computer.
	mathforum.org /library/topics/geometry (2313 words)

	SOAR Winter 2003 Homework Two (Site not responding. Last check: 2007-10-28)
		That is, assuming ``absolute geometry'' (Euclid's first four postulates), prove that...
		the pdf version in the context of absolute geometry.
		Using the previous problem and the fact that the angle of parallelism is acute (under 90°), show that, in hyperbolic geometry, the distance between parallel lines is decreasing.
	www.math.toronto.edu /mathnet/soar/Winter/HTML/homework02.html (341 words)

	geometry: Their Relationship to Each Other
		Another type of geometry, called affine geometry, includes Euclid's parallel postulate but disregards two other postulates concerning circles and angle measurement; the propositions of affine geometry are also valid in the four-dimensional geometry of space-time used in the theory of
		An important step in recognizing the connections between the different types of geometry was the Erlangen program, proposed by the German Felix Klein in his inaugural address at the Univ. of Erlangen (1872), according to which geometries are classified with respect to the geometrical properties that are left unchanged (invariant) under a given
		, perhaps the most general type of geometry although often considered a separate branch of mathematics, is concerned with properties invariant under continuous transformations, which carry neighborhoods of points into neighborhoods of their images.
	www.factmonster.com /ce6/sci/A0858361.html (344 words)

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