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Topic: Inversive ring geometry

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	Projective Geometry Encyclopedia Article @ Infinitely.org (Site not responding. Last check: 2007-11-04)
		Projective geometry is a non-metrical form of geometry that emerged in the early 19th century.
		Projective geometry is a non-Euclidean geometry that formalizes one of the central principles of perspective art: that parallel lines meet at infinity and therefore are to be drawn that way.
		In essence, a projective geometry may be thought of as an extension of Euclidean geometry in which the "direction" of each line is subsumed within the line as an extra "point", and in which a "horizon" of directions corresponding to coplanar lines is regarded as a "line".
	www.infinitely.org /encyclopedia/Projective_geometry (2800 words)

	Semple and Kneebone's geometry
		Geometry is commonly regarded as having had its origins in ancient Egypt and Babylonia, where much empirical knowledge was acquired through the experience of surveyors, architects, and builders; but it was in the Greek world that this knowledge took on the characteristic form with which we are now familiar.
		Projective geometry is more symmetrical than euclidean, by virtue both of the existence of a principle of duality and also of the fact that it may be handled by means of homogeneous coordinates.
		Since the discussion of projective geometry which follows in Part II is to be analytical, we shall conclude this chapter by touching upon the use of coordinates; but it should be realized, nevertheless ' that we are under no logical compulsion to introduce a coordinate system at all.
	www-history.mcs.st-andrews.ac.uk /Extras/Semple_Kneebone_geometry.html (2801 words)

	Projective geometry (Site not responding. Last check: 2007-11-04)
		In a historical perspective on mathematics, the field of geometry that developed in the first half of the nineteenth century under the name projective geometry was a stepping stone from analytic geometry to algebraic geometry.
		This period in geometry was rather overtaken by the research on the general algebraic curve by Clebsch, Riemann, Max Noether and others, which stretched existing techniques, and then by invariant theory.
		Some important work was done in enumerative geometry in particular, by Schubert, that is now considered an anticipation of the theory of Chern classes in their guise as representing the algebraic topology of Grassmannians.
	projective-geometry.iqnaut.net (626 words)

	Inversive ring geometry - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-04)
		Two elements of a ring are relatively prime if the ideal in A that they generate is the whole of A.
		In case A is the ring of biquaternions, the mappings include both the ordinary and hyperbolic rotations of the Lorentz group.
		The ring of dual numbers D gave Joseph Grunbaum opportunity to exhibit P(D) in 1906.
	en.wikipedia.org /wiki/Inversive_ring_geometry (772 words)

	Vassar College: Mathematics Department
		Topics to be chosen from: conic sections, transformational geometry, Euclidean geometry, affine geometry, projective geometry, inversive geometry, nonEuclidean geometry, spherical geometry, convexity, fractal geometry, solid geometry, foundations of geometry.
		Aspects of the elementary geometry and topology of differentiable manifolds.
		Rings and fields, with a particular emphasis on Galois theory.
	catalogue.vassar.edu /oldcatalogue00_01/math.html (1240 words)

	The Geometry Junkyard: All Topics
		This Geometry Forum problem of the week asks for the number of different hexominoes, and for how many of them can be folded into a cube.
		The Geometry Center's collection includes programs for generating Penrose tilings, making periodic drawings a la Escher in the Euclidean and hyperbolic planes, playing pinball in negatively curved spaces, viewing 3d objects, exploring the space of angle geometries, and visualizing Riemann surfaces.
		Geometry forum discussion on the Reuleux triangle and its ability to drill out (most of) a square hole.
	www.math.ntnu.edu.tw /~jcchuan/all.html (7425 words)

	List of Publications In Physics Encyclopedia Article @ Resounded.net (Site not responding. Last check: 2007-11-04)
		Gibbs: Vector Analysis and Bonola: Non-Euclidean Geometry) to provide entry into mathematical physics including a vector-based introduction to quaternions and a primer on matrix notation for linear transformations of 4-vectors.
		The actual technique in geometric arithmetic comes about with inversive ring geometry applied to biquaternions.
		This text, with its ambitious development of pseudo-Riemannian geometry for gravitational theory, set an austere standard with relativity enthusiasts.
	www.resounded.net /encyclopedia/List_of_publications_in_physics (1969 words)

	TITLE OF THE ARTICLE:
		A cuboctahedron, for example, is constructed from 4 hexagonal rings, which hold together through their springiness.
		After removing 6 of the edges of a stiff icosahedron in a cubically symmetric way, the structure can be made to expand into a cuboctahedron or to collapse into an octahedron or even further into a flat triangle.
		Further, one can construct a flexible ring of 6 or more tetrahedra which can be used to demonstrate a generalization of Euler’s theorem on the number of faces, edges and vertices of any convex polyhedron.
	www.mi.sanu.ac.yu /vismath/visbook/higgins (972 words)

	Course Catalog ~ Colorado College Mathematics (Site not responding. Last check: 2007-11-04)
		After the introduction of Euclidean geometry, the course then considers one of the most historically and philosophically important developments in all of mathematics, the advent of non-Euclidean geometry.
		In plane Euclidean geometry, the parallel postulate says that through each point not on a line, there is exactly one parallel line.
		Geometry is an excellent course to take soon after Number Theory; it provides good practice for reading and writing proofs about some fascinating mathematics.
	www2.coloradocollege.edu /dept/ma/Courses/catalog.asp (4829 words)

	ECTS Guide - Department of Mathematics
		This course continues the course Linear Algebra and Analytic Geometry I and its objective is to supply the students with a basic and sound education in Linear Algebra and Analytic Geometry.
		The goal of the course is to provide students with essential tools for the study of the Differential Geometry of arbitrary dimension submanifolds of Euclidean space, making a bridge with classical curves and surfaces and with an accent on the coordinate-free viewpoint.
		The purpose of the course is to pursue the study of Euclidean Geometry with a first glance at models of other geometries.
	www.fc.ul.pt /en/mathematics2.html (5423 words)

	Orðasafn: I
		2 (of a ring or field) eining, einingarstak, margföldunarhlutleysa, = identity 4, = one 2, = unit element, = unit 2, = unity 2, = unity element.
		integral ring heilbaugur, = domain 2, = domain of integrity, = entire ring, = integral domain.
		inversive geometry Möbíusar-rúmfræði, hringarúmfræði, = conformal geometry 1, = Möbius geometry.
	www.hi.is /~mmh/ord/safn/safnI.html (2408 words)

	Topics for Mathematics Honours Teaching (Site not responding. Last check: 2007-11-04)
		The course covers a selection of some basic topics involved in the axiomatic foundations of various elementary type of ``geometry'', several of which have particular relevance to the study and development of geometrical concepts studied at school and in early university courses.
		Amongst the types of ``geometries'' which may be considered in this respect are affine, projective, Euclidean, hyperbolic and elliptic planes, as well as finite and higher-dimensional versions of these.
		Various important results of Euclidean geometry, some of which were known over 2000 years ago, and some of which were discovered during the Renaissance and thereafter, were recognized by the French mathematicians of the Napoleonic era not to depend on distance and angle.
	www.wits.ac.za /science/maths/postgrad/Topics_Mathematics_Honours_.html (1681 words)

	York University Faculty of Graduate Studies 2002-2004 Calendar Graduate Programme in Mathematics & Statistics
		The intent of this course is to give the student an appreciation of mathematical structure through the study of fields, rings and groups, with examples from, and applications to, number theory and geometry.
		Various geometries, including Euclidean, affine, projective, inversive, non-Euclidean, and finite geometries, and the transformations associated with these geometries, are studied from the unifying point of view of affine and metric affine geometry.
		Topics include the geometry of the classical groups over a field, the construction of the finite simple groups, discontinuous groups of motion of the Euclidean and non-Euclidean planes, geometry of linear fractional transformations, Fuchsian groups, groups generated by reflections (Coxeter groups), surface, knot, and braid groups.
	www.yorku.ca /grads/cal/mat.htm (5849 words)

	Encyclopaedia of Design Theory: Glossary
		A flag in an incidence geometry is a set of mutually incident objects or varieties.
		An identity element of a ring is an element 1 satisfying 1a=a1=a for all a in R.
		An incidence geometry consists of a set V of varieties, a set T of types, a type map t from V to T and symmetric incidence relation on V such that two varieties of the same type are incident if and only if they are equal.
	designtheory.org /library/encyc/glossary (13994 words)

	The Geometry Junkyard: All Topics
		Geometry problems involving circles and triangles, with proofs.
		A Geometrical Picturebook of finite and combinatorial geometries, B.
		Meru Foundation appears to be another sacred geometry site, with animated gifs of torus knots and other geometric visualizations and articles.
	www.ics.uci.edu /~eppstein/junkyard/all.html (9768 words)

	list_of_publications_in_physics (Site not responding. Last check: 2007-11-04)
		The expressionshould have the form of an inner automorphism but Silberstein inexplicably uses theexpression Q Q, failing to supply one of the Q's with a − 1 exponent.
		Theactual technique in geometric arithmetic comes about with inversive ring geometryapplied to biquaternions.
		Gone is anymention of quaternions or hyperbolic geometry since tensor calculus subsumes them.Thus for learning the mechanics of modern relativity this text still serves, but formotivation and context of the special theory, Silberstein is better.
	www.freesimslotsgames.com /wiki/?title=List_of_publications_in_physics (1513 words)

	Wits University Postgrad studies in the School of Mathematics
		Ring localizations, integral elements, prime and maximal ideals, Dedekind domains, unique factorization of ideals, algebraic number fields, integral bases, discriminants, norms, class number.
		A synthesis of differential geometry, geometric topology, and algebraic geometry.
		The interplay between the geometry, algebra and analysis is highlighted.
	www.maths.wits.ac.za /postgradinfo.html (6190 words)

	News \| Gainesville.com \| The Gainesville Sun \| Gainesville, Fla. (Site not responding. Last check: 2007-11-04)
		Passing to projective space is a common tool in algebraic geometry because the added points at infinity lead to simpler formulations of many theorems.
		The Bohr compactification of a topological group arises from the consideration of almost periodic functions.
		One can compactify a topological ring by forming a projective line with inversive ring geometry.
	www.gainesville.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=compactification_(mathematics) (1082 words)

	Inversion
		Among all circles inscribed in the segment, the one that is tangent to MT' on the side of T is invariant under the inversion with center S and radius SM.
		It is customary to complement the definitions by assigning the point at infinity to the center of inversion (and vice versa, of course.) This is not the same infinity as that shared by all parallel lines, rather every straight line closes on itself at the "new" infinity.
		The straight lines thus may be (and are in inversive geometry) looked at as circles with center at infinity and an infinite radius.
	www.cut-the-knot.org /ctk/Circle.shtml (2218 words)

	METU MATHEMATICS DEPARTMENT
		Goals: The aim of this course is to introduce the student to some algebraic and differential topological ideas at an early stage emphasizing unity with geometry and more generally introduce the student to the relation of the modern axiomatic approach in mathematics to geometric intuition.
		Geometry from a Differentiate Viewpoint, J. McCIeary, Cambridge Univ., Cambridge, 1994.
		Projective Algebraic Geometry: Projective plane, projective space and projective varieties, the Projective Algebra-geometry dictionary, The projective closure of an Affine variety, Projective elimination theory, The geometry of Quadric hypersurfaces.
	www.math.metu.edu.tr /courses/elective.shtml (6325 words)

	Algebra, Geometry and Topology - Cambridge University Press
		Two of the key subjects in mathematics Algebra and Geometry have much in common, and Cambridge's portfolio in this area is an impressive array of modern cutting edge theory and exciting textbooks.
		This monograph is a bridge between the classical theory and modern approach via arithmetic geometry.
		Circle packing has an experimental and visual character that is unique in pure mathematics, and the book exploits that character to carry the reader from the very beginnings to links with complex analysis and Riemann surfaces.
	www.cambridge.org /uk/browse/browse_highlights.asp?subjectid=1010168 (534 words)

	San Antonio Special Session (10.I.99)
		We describe an algorithm for numerical uniformization of piecewise flat surfaces, ie, for constructing fundamental domains for such surfaces in the standard geometries as well as the corresponding conformal uniformizing maps.
		The crucial ingredient there was the ability to perform "conformal subdivisions" that allowed us to construct an iterative scheme based on circle packings and prove convergence of the scheme to a uniformizing map.
		We remedy this by preserving the subdivision process while modifying the circle packing geometry by allowing "circle packings" involving pairwise disjoint collections of circles to model the conformal geometry of the surfaces.
	www.emilvolcheck.com /sanantonio.html (2302 words)

	The CTK Exchange Forums
		From my classes in descriptive geometry, I remembered that although we cannot construct an ellipse with a ruler and a compass, we certainly can construct as many points of the ellipse as we wish.
		Moreover, descriptive geometry deals with projections of objects, including cylinders, cones, etc. Even students not familiar with descriptive geometry should realize that the projection of a circle is generally an ellipse.
		The true joy of geometry is in solving the problem, not just in seeing the solution (like in everything else).
	www.cut-the-knot.org /htdocs/dcforum/DCForumID3/205.shtml (5192 words)

	Inversive ring geometry (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-11-04)
		In mathematics, inversive ring geometry is the extension to the context of associative rings, of the concepts of projective line, homogeneous coordinates, projective transformations, and cross-ratio, concepts usually built upon rings that happen to be fields.
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	www.mispedia.org.cob-web.org:8888 /Inversive_ring_geometry.html (115 words)

	Amazon.ca: Introduction to Circle Packing : The Theory of Discrete Analytic Functions: Books: Kenneth Stephenson (Site not responding. Last check: 2007-11-04)
		The topic can be enjoyed for the visual appeal of the packing images - over 200 in the book - and the elegance of circle geometry, for the clean line of theory, for the deep connections to classical topics, or for the emerging applications.
		The topic of circle packing was born of the computer age but takes its inspiration and themes from core areas of classical mathematics.
		Circle packing has an experimental and visual character which is unique in pure mathematics, and the book exploits that to carry the reader from the very beginnings to links with complex analysis and Riemann surfaces.
	www.amazon.ca /Introduction-Circle-Packing-Analytic-Functions/dp/0521823560 (582 words)

	March 18 (Site not responding. Last check: 2007-11-04)
		Steiner discovered inversive geometry and is considered one of the greatest geometers of modern times.
		The chair of geometry was established for him at the University of Berlin and he occupied that chair from 1834 until his death in 1863.
		Koffka worked with Wolfgang Köhler and Max Wertheimer at the University of Giessen to develop a holistic approach to psychology which is known as "gestalt psychology".
	courseweb.stthomas.edu /paschons/language_http/calendar/march18.html (583 words)

	Biquaternion (via CobWeb/3.1 planetlab-3.cs.princeton.edu) (Site not responding. Last check: 2007-11-04)
		Given any 2x2 complex matrix, there are complex values u, v, w, and x to put it in this form so that the matrix ring is isomorphic to the biquaternion ring.
		Then {a + b Î¹ j : a, b â R } is a subring of biquaternions isomorphic to the split-complex number ring.
		The actual exhibition of individual Lorentz transformations involves extensions of inner automorphisms of the group of units of biquaternions to the singular elements through inversive ring geometry.
	www.e-tv.co.za.cob-web.org:8888 /b/i/q/Biquaternion.html (609 words)

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