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Topic: Synthetic geometry

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	Synthetic geometry - Wikipedia, the free encyclopedia
		Synthetic geometry is the branch of geometry which makes use of theorems and synthetic observations to draw conclusions, as opposed to analytic geometry which uses algebra to perform geometric computations and solve problems.
		The heyday of synthetic geometry can be considered to have been the nineteenth century; when methods based on coordinates and calculus were ignored by some geometers such as Jakob Steiner, in favour of a synthetic development of projective geometry.
		Synthetic differential geometry is an application of topos theory to the foundations of smooth manifold theory.
	en.wikipedia.org /wiki/Synthetic_geometry (283 words)

	Differential Synthetic Geometry
		An interest in developing a differential version of synthetic geometry is motivated by its potential for gravitational theory, especially a version which might gracefully incorporate spin.
		Synthetic geometry is that kind of geometry which deals purely with geometric objects directly endowed with geometrical properties by abstract axioms.
		Synthetic geometry is the kind of geometry for which Euclid is famous and that we all learned in high school.
	enlightenment.supersaturated.com /essays/text/craigspencer/thesis (626 words)

	Synthetic geometry (Site not responding. Last check: 2007-11-06)
		Synthetic geometry is a descriptive term that describes a methodology of geometry which makes use of theorems and synthetic observations tocreate theorems or solve problems, as opposed to analyticgeometry which uses algebra, numbers, computations to draw theorems or solve problems.
		The heyday of synthetic geometry can be considered to have been the nineteenth century ; when methods based on coordinates and calculus were ignored by some geometers suchas Jakob Steiner, in favour of a synthetic development of projective geometry.
		The close axiomatic studyof Euclidean geometry led to the discovery of non-Euclidean geometry.
	www.therfcc.org /synthetic-geometry-69262.html (246 words)

	A Critique of the Kantian View of Geometry (Site not responding. Last check: 2007-11-06)
		Synthetic truths, on the other hand, require that a concept to be "synthesized" or combined with some other information, perhaps another concept or some sensory data, to produce something truly new.
		Geometry is literally an empirical science, a branch of physics, not just a way of visualizing analytic geometry, since it is a posteriori, and thus constrained by the results of physics, which tells us that space is Einsteinian.
		I believe the issue of the syntheticity of arithmetic is a more subtle issue, requiring a full understanding of recursion theory, and in particular the work of Kurt Gödel [Gödel 1931], and thus beyond the scope of the current paper.
	home.ican.net /~arandall/Kant/Geometry (9289 words)

	Synthetic geometry (Site not responding. Last check: 2007-11-06)
		The heyday of synthetic geometry can considered to have been the nineteenth century ; when methods based on coordinates and calculus were ignored by some geometers such Jakob Steiner in favour of a synthetic development projective geometry.
		The close axiomatic of Euclidean geometry led to the discovery of non-Euclidean geometry.
		Basic Concepts of Synthetic Differential Geometry (Kluwer Texts in the Mathematical Sciences, V. This book is by far the most readable introduction to Synthetic Differential Geometry that there currently is. The book concentrates on building up axiomatic SDG with hardly a reference to ways of modelling it (ie: using topos theory), leaving the complic...
	www.freeglossary.com /Synthetic_geometry (454 words)

	KEGP
		He explains that geometry can be viewed as relating either to a set of phenomenal (and thus unfalsifiable), pictures, or to the logic behind them, or to their application to objects in physical space.
		Kant's silence on the subject of physical geometry need not be interpreted as an identification of phenomenal with physical, but may simply reflect his recognition that physical geometry (as a posteriori) is a subject which need not be addressed by the transcendental philosopher.
		Moreover, in the geometry of curved space, the perspective-lessness of the observer (or the unobservability of the perspective) is of utmost importance.
	www.hkbu.edu.hk /~ppp/srp/arts/KEGP.html (9752 words)

	Peter Suber, "Geometry and Arithmetic are Synthetic"
		Hence the advent of non-Euclidean geometry shortly after the Critique appeared, and especially Hilbert's proof in 1899 that it is as consistent as Euclidean geometry, seem to falsify Kant's account of geometry.
		Hence when Kant asserts that the propositions of geometry are synthetic, he is making a claim about the relation of these propositions to contradiction, not to physical space.
		Wiredu, J.E., "Kant's Synthetic A Priori in Geometry and the Rise of Non-Euclidean Geometries," 61 (1970) 5-27.
	www.earlham.edu /~peters/writing/synth.htm (6077 words)

	Math 371 Geometry Notes on Line
		In the distinction between synthetic and analytic geometry the key connecting concept is the use of measurements.
		Euclid's division algorithm is stated in geometry that of ON is a segement and OD is a segment that is contained as a subsegment of ON, then ON is congruent to q*OD with possibly a remaining segment RN which is congruent to a subsegment of OD.
		The impact of this on geometry was that one could not presume that all of geometry could be handled by using simple ratios of whole numbers for measurements.
	www.humboldt.edu /~mef2/Courses/m371notes04.html (8713 words)

	A Unified Algebraic Framework for Classical Geometry (Site not responding. Last check: 2007-11-06)
		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.
		Though fundamental ideas of classical geometry are permanently imbedded and broadly applied in mathematics and physics, the subject itself has practically disappeared from the modern mathematics curriculum.
		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)

	Learn more about Pythagorean theorem in the online encyclopedia. (Site not responding. Last check: 2007-11-06)
		Note that this proof does not work in non-Euclidean geometries, since, say, on a sphere, the angles of a triangle don't add up to 180 degrees, and the above "square" cannot be formed.
		If one erects similar figures (see geometry) on the sides of a right triangle, then the sum of the areas of the two smaller ones equals the area of the larger one.
		Since the Pythagorean theorem is derived from the axioms of Euclidean geometry, and physical space may not always be Euclidean, it need not be true of triangles in physical space.
	www.onlineencyclopedia.org /p/py/pythagorean_theorem.html (926 words)

	Triangle
		A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments.
		In Euclidean geometry, the sum of the angles Î± + Î² + Î³ is equal to two right angles (180Â° or Ï radians).
		Geometers also study non-planar triangles in noneuclidean geometries, such as spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry.
	www.algebra.com /algebra/homework/formulas/Triangle.wikipedia (2176 words)

	Mathematics Books (Site not responding. Last check: 2007-11-06)
		Geometry algorithms and computer graphics uses a lot of math, and many algorithms books assume the reader has some knowledge of basic math (geometry, algebra, trig, etc).
		This is a thorough summary of Euclidean synthetic geometry.
		Even though it predates widespread use of computers, it is a fundamental reference for modern geometry, and its republication in 1989 attests to this fact.
	www.geometryalgorithms.com /books_mathematics.shtml (1057 words)

	Fashion and Mathematics
		However, the explanation was in terms of algebra, and not in terms of Synthetic Geometry.
		Geometry was largely neglected, and branches of geometry such as Descriptive Geometry and Synthetic Geometry were completely ignored.
		In my book on Synthetic Geometry there is no mention of congruence or some other words missing in English language literature.
	ca.geocities.com /ingsaler6/fashionandmathematics.html (524 words)

	51: Geometry
		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.
		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.
		A useful collection of Geometry Formulas and Facts is taken from the CRC Standard Mathematical Tables and Formulas, and available at the The Geometry Center.
	www.math.niu.edu /~rusin/known-math/index/51-XX.html (828 words)

	MATH5450.06 Geometry for Teachers.
		Geometry has an important classical side: Euclidean Geometry from the Greeks moving, in the last two centuries, to non-Euclidean geometries (which differ by their assumptions about parallel lines), including spherical, hyperbolic and projective geometries.
		In modern geometry, the interplay of abstraction, axiomatics, synthetic geometry, analytic methods, and groups of transformations presents a rich mix of mathematical methods and problems to be explored.
		Geometry is also a good subject to explore the role of the visual in the practice of mathematics: generating insights, problem solving, communication, remembering, etc.
	www.math.yorku.ca /Who/Faculty/Whiteley/5450_outline.html (758 words)

	math lessons - Category:Geometry
		Geometry is the branch of mathematics dealing with spatial relationships.
		From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry.
		Such axioms are insusceptible to proof, but can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions.
	www.mathdaily.com /lessons/Category:Geometry (72 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Synthetic geometry is a branch of geometry that relies on the visual observation of figures and can be developed independently of number concepts and of algebraic notions.
		In the late nineteenth century synthetic geometry was developed to full perfection.
		In what follows we shall show how synthetic geometry can be investigated profitably under the environment of one dynamic geometry software, the Geometer's Sketchpad.
	www.atcminc.com /mPublications/EP/EPATCM97/ATCMP021/abstract.html (130 words)

	Beyond3D - Which was nice. (Site not responding. Last check: 2007-11-06)
		In the case of geometry test, resolution made virtually no impact on the results and hence the software rendering path was always limited by the CPU’s geometry throughput.
		Unlike the synthetic tests, within a game the CPU will be handling a lot more than just the geometry, such as collision detection, physics, sound, etc., and offloading the geometry processing to the 3D chip should provide a performance increase.
		Increasing the flexibility in the number of textures allowed per pass will reduce the number of times the card is forced into ‘multi-pass rendering’, which increases both the geometry required for the scene (by needing to recalculate the entire scene for the number of times it is passed) and the external memory bandwidth used.
	www.beyond3d.com /reviews/ati/radeon8500/index10.php (876 words)

	Undefined: Points, Lines, and Planes
		The geometry of a saddle shaped surface is known as hyperbolic geometry (from the Greek to exceed).
		The geometry of a sphere required additional changes to the usual axioms because betweenness is no longer meaningful and must be replaced with separation.
		This geometry was popularized by Albert Einstein when he developed his theory of General Relativity with the notion that space is curved by the presence of mass.
	www.andrews.edu /~calkins/math/webtexts/geom01.htm (2268 words)

	20th WCP: Knowledge by Invention: Extending a Kantian Dichotomy to a Poincaréan Trichotomy
		For Hilbert geometry was mostly analytic à priori because of his emphasis on deductive inference, whereas the axioms could be synthetic à priori.
		I suggest a Poincaréan reconstruction in which the postulates of geometry have truth value and are known by their very invention of them.
		Are the postulates of geometry synthetic or analytic?
	www.bu.edu /wcp/Papers/TKno/TKnoJetl.htm (2685 words)

	Mailgate: sci.stat.math: Re: Synthetic Geometry (Site not responding. Last check: 2007-11-06)
		Euclid's Elements is the canonical model for synthetic geometry (so-called because proofs of theorems are synthesized from postulates and other theorems already proved--same word origin as "synthesis" in chemistry).
		This is sometimes called "coordinate-free" geometry because, as Rich said, it does not rely on any particular choice of coordinate system or origin.
		To give a more practical answer to the original question, the synthetic geometry items on a teacher certification exam are likely to involve things such as using triangle congruence postulates and theorems (SSS, SAS, etc) to determine whether given triangles are congruent, using theorems about angles inscribed in circles to determine angle measures, etc.
	mailgate.supereva.it /sci/sci.stat.math/msg10088.html (450 words)

	Sturm_Rudolf (Site not responding. Last check: 2007-11-06)
		There he was taught by Schroeter who encouraged him to study synthetic geometry.
		Sturm wrote extensively on geometry and, other than the teaching textbook on descriptive geometry and graphical statics which we mentioned above and one other teaching text Maxima und Minima in der elementaren Geometrie which he published in 1910, all his work was on synthetic geometry.
		and Schubert's enumerative geometry the first two volumes of which he published in 1908 and the second two volumes in 1909.
	www-history.mcs.st-and.ac.uk /history/Mathematicians/Sturm_Rudolf.html (689 words)

	The Math Forum - Math Library - Geometry (Site not responding. Last check: 2007-11-06)
		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.
		A short article designed to provide an introduction to geometry, including 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.
		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.
	mathforum.org /library/topics/geometry (2304 words)

	History of Trigonometry Outline
		Trigonometry is, of course, a branch of geometry, but it differs from the synthetic geometry of Euclid and the ancient Greeks by being computational in nature.
		The geometry of the sphere was called "spherics" and formed one part of the quadrivium of study.
		The current name for the subject is "elliptic geometry." Trigonometry apparently arose to solve problems posed in spherics rather than problems posed in plane geometry.
	aleph0.clarku.edu /~djoyce/ma105/trighist.html (1546 words)

	Synthetic Projective Geometry and Poincaré's Theorem on Automorphisms of the Ball (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Synthetic Projective Geometry and Poincaré's Theorem on Automorphisms of the Ball (ResearchIndex)
		Synthetic Projective Geometry and Poincaré's Theorem on Automorphisms of the Ball
		7 On pseudo-conformal geometry of hypersurfaces of the space o..
	citeseer.ist.psu.edu /85035.html (318 words)

	Geometric areas of Mathematics
		At one extreme, geometry includes the very precise study of rigid structures first seen in Euclid's Elements; at the other extreme, general topology focuses on the very fundamental kinships among shapes.
		53: Differential geometry is the language of modern physics as well as an area of mathematical delight.
		The geometric areas share with the fields of algebra the tendency to distill their inquiry to the study of certain axioms and their consequences; during the last half-century the ties between these broad areas have increased.
	www.math.niu.edu /~rusin/known-math/index/tour_geo.html (668 words)

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