61. Non-Euclidean Geometry: Elliptic Geometry encyclopediaEncyclopedia—noneuclidean geometry Elliptic Geometry.In elliptic geometry there are no parallels to a given line http://www.infoplease.com/ce6/sci/A0860016.html

Reduce your family medical costs by up to 60% now! Oral health..Save up to 60% on dental care. Easy Medical Diagnosis..stop smoking! All Infoplease All Almanacs General Entertainment Sports Biographies Dictionary Encyclopedia Infoplease Home Almanacs Atlas Dictionary ... Fact Monster Kids' reference Info:Daily Fun facts Homework Center Newsletter You've got info! Help Site Map Visit related sites from: Family Education Network Encyclopedia non-Euclidean geometry Elliptic Geometry In elliptic geometry there are no parallels to a given line L through an external point P, Sections in this article: Introduction Hyperbolic Geometry Elliptic Geometry Non-Euclidean Geometry and Curved Space Bibliography Hyperbolic Geometry non-Euclidean geometry Non-Euclidean Geometry and Curved Space Related content from on: non-Euclidean geometry: Elliptic Geometry Sommerville, Duncan MacLaren Young (1879-1934) (The Hutchinson Dictionary of Scientific Biography) MATHEMATICS (Stack of Lists) parallel lines and parallel planes (The Hutchinson Dictionary of Science) parallel lines and parallel planes (The Hutchinson Dictionary of Science) Cayley, Arthur (1821-1895)

62. Euclidean Geometry, Fractals And Adhesion euclidean geometry, Fractals and Adhesion. Sperling540 In EuclideanGeometry, a line is expressed as distance to the first power http://web.umr.edu/~wlf/Math/fractal.html

Euclidean Geometry, Fractals and Adhesion [Sperling-540] In Euclidean Geometry, a line is expressed as distance to the first power, and area is expressed as distance to the second power. Some irregular structures such as coastlines and interdiffusion boundaries (also called an interphase ) are better expressed as fractional powers between one and two. R. P. Wool and J. M. Long, Polym. Prepr., 31(2), 558 (1990) In two dimensions, D subscript f is 7/4. Last Update- January 5, 1995- wld

63. Trackstar: Euclidean Geometry euclidean geometry Track 3570 Annotations by Brett Cooper View Track •Grade(s)High School (912). •Subjects(s) Math. •Last Modified 07-OCT-02. http://trackstar.hprtec.org/main/display.php3?track_id=3570

64. Mathematics Online Compendium: Euclidean Geometry Fabrication and Illustration Tom Longtin http//www.sover.net/~tlongtin/ Catalogueof Algebraic Systems - Polygonal and Polyhedral Geometry http//www.math http://www.dei.unipd.it/~cuzzolin/MOCeuclidean.html

3D Modeling, Fabrication and Illustration - Tom Longtin http://www.sover.net/~tlongtin/ Catalogue of Algebraic Systems - Polygonal and Polyhedral Geometry http://www.math.niu.edu/~rusin/papers/known-math/polyhedral/ Faceted Objects: Tanks to Teapots - Paul Flavin http://www.frontiernet.net/~imaging/fo_arch.html Introduction to Quasicrystals - Steffen Weber http://www.nirim.go.jp/~weber/qc.html Imaging Raytracer: a raytracing applet - Paul Flavin http://www.frontiernet.net/~imaging/raytracing.html Imaging the Imagined - Paul Flavin http://www.frontiernet.net/~imaging/contents.html Geometry and the Imagination http://www.geom.umn.edu/docs/doyle/mpls/handouts/handouts.html Geometry and the Imagination in La Jolla http://math.ucsd.edu/~doyle/docs/gi/gi/gi.html The Geometry Junkyard - David Eppstein http://www.ics.uci.edu/~eppstein/junkyard/ Random Walk on the Speiser Graph of a Riemann Surface - Peter Doyle http://math.ucsd.edu/~doyle/docs/speiser/cover/cover.html Surfaces (Geometry and the Imagination) http://www.geom.umn.edu/docs/doyle/mpls/handouts/node14.html Modularity in Art and Mirror Curves - Slavik Jablan http://www.mi.sanu.ac.yu/~jablans/

65. Citations: Non-Euclidean Geometry - Coxeter (ResearchIndex) HSM Coxeter. Noneuclidean geometry. The hyperbolic plane is a non Euclideangeometry in which parallel lines diverge away from each other. http://citeseer.nj.nec.com/context/416235/0

13 citations found. Retrieving documents... Harold S.M. Coxeter. Non-Euclidean geometry . The University of Toronto Press, 1957. Home/Search Document Not in Database Summary Related Articles Check This paper is cited in the following contexts: Some New Randomized Approximation Algorithms - Andersson (2000) (1 citation) (Correct) ....for the volume function of spherical tetrahedra, and there is reason to believe that none exists as it is related to the dilogarithm function [63] One therefore has to resort to numerical integration to evaluate it. Several di#erent integrals representing the volume function have been proposed The most recent is also the most useful for the purposes of numerical integration. It is due to Hsiang [49] and reduces the problem of evaluating the volume function to calculating a complicated one dimensional integral. Let us begin by revisiting the approximation algorithm of Feige and .... Harold S.M. Coxeter. Non-Euclidean geometry . The University of Toronto Press, 1957. A Focus+Context Technique Based on Hyperbolic Geometry for .. - Lamping, Rao, Pirolli (1995) (14 citations) (Correct) ....browser can display up to 10 times as many nodes while providing more effective navigation around the hierarchy. The scale advantage is obtained by the dynamic distortion of the tree display according to the varying interest levels of its parts.

66. IAP 2002 Activity: Fun With Euclidean Geometry Fun With euclidean geometry John Gonzalez, Glenn Iba Mon, Wed, Fri, Jan 14, 16, 18,23, 25, 28, 30, 030400pm, 24-619 No enrollment limit, no advance sign up http://student.mit.edu/searchiap/iap-4207.html

IAP 2002 Activity Fun With Euclidean Geometry John Gonzalez, Glenn Iba Mon, Wed, Fri, Jan 14, 16, 18, 23, 25, 28, 30, 03-04:00pm, 24-619 No enrollment limit, no advance sign up Participants welcome at individual sessions (series) Prereq: Familiarity with high school geometry Euclidean geometry has become a forgotten subject in the midst of modern geometry such as topology and differential geometry. Come learn about this fascinating subject. In this seminar, we will explore some interesting topics connected mainly with triangles and circles (such as the nine point circle, pedal triangles, theorems of Ceva, Morley, and others). Seminar participation will be encouraged but not required. Web: http://web.mit.edu/esg/www/iap Contact: John Gonzalez, 24-612, x3-7786, johngonz@mit.edu Sponsor: Experimental Study Group Latest update: 14-Jan-2002 iap-www@mit.edu IAP Office, Room 7-104, 617.253.1668 Listing generated: 05-Feb-2002

67. Non-Euclidean Geometry Noneuclidean geometry. Article Nor is Bolyai's work diminished because Lobachevskypublished a work on non-euclidean geometry in 1829. Neither http://www.meta-religion.com/Mathematics/Articles/non-euclidean_geometry.htm

Non-Euclidean geometry Article by: J J O'Connor and E F Robertson From: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Non-Euclidean_geometry.html In about 300 BC Euclid wrote The Elements, a book which was to become one of the most famous books ever written. Euclid stated five postulates on which he based all his theorems: To draw a straight line from any point to any other. To produce a finite straight line continuously in a straight line. To describe a circle with any centre and distance. That all right angles are equal to each other. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles. It is clear that the fifth postulate is different from the other four. It did not satisfy Euclid and he tried to avoid its use as long as possible - in fact the first 28 propositions of The Elements are proved without using it. Another comment worth making at this point is that Euclid, and many that were to follow him, assumed that straight lines were infinite. Proclus (410-485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced a false 'proof'. Proclus then goes on to give a false proof of his own. However he did give the following postulate which is equivalent to the fifth postulate.

68. Euclidean/Non-Euclidean Geometry Euclidean/Noneuclidean geometry A Source of Ideas for Mathematics Teachers. Thenames are hyperbolic geometry and elliptic geometry . euclidean geometry. http://westview.tdsb.on.ca/Mathematics/non-euclidean.html

This page will be under constant revision so please use the reload button on your browser to refresh the page. Last revised December 21, 2000 Euclidean/Non-Euclidean Geometry A Source of Ideas for Mathematics Teachers Introduction The primary difference has to do with the parallel postulate. In Euclidian geometry, given a line with a point outside (and not on said line), you can draw only one line thru the point which will be parallel to the aforementioned line. Non-Euclidian geometries can allow no parallel lines thru the point or an infinite number of parallel lines. The names are "hyperbolic geometry" and "elliptic geometry". Euclidean Geometry Euclid's presentation of plane geometry is based on a number of theorems that can all be derived from five postulates (axioms) and five common notions. The axioms are: Exactly one straight line can be drawn between any two points. A straight line can be continued indefinitely. With any point as center, a circle with any radius may be described. All right angles are equal. Through a given point outside a given straight line, there passes only one line parallel to the given line; that is, such a line does not intersect the given line.

69. Non-Euclidean Geometry Noneuclidean geometry. (80561) Main source David W. Henderson, ExperiencingGeometry in Euclidean, Spherical, and Hyperbolic Spaces, Second Edition. http://www.ma.huji.ac.il/~karshon/teaching/2001-02/nonEuc/

Non-Euclidean Geometry Undergraduate student seminar 2001-02, First semester Instructor: Yael Karshon , math building room 314, Tel. 6584123, karshon@math.huji.ac.il , Office hours: by appointment. Course meetings: Sundays 10:00-11:45, Shprintzak 114 Grading: Student presentations, participation, and problem sets. Student presentations: Every two students will prepare a topic together and will present it in one course meeting. Problem sets will be given on most weeks. On the following week you are to hand in your written solution and be ready to discuss it in class. You are encouraged to collaborate; however, you must write up the answers by yourself. Goal: The seminar provides an informal "hands-on" introduction to geometry, with emphasis on the Euclidean plane, the sphere, and the hyperbolic plane. Main source: David W. Henderson, Experiencing Geometry in Euclidean, Spherical, and Hyperbolic Spaces , Second Edition. Syllabus: We aim to cover Polking's "Geometry of the Sphere" , and most of chapters 1, 2, 3, 5, 6, 7, 8, 10, 14, 15, and 16 of Henderson's book. However, this plan is flexible. Source for projective geometry: http://www.cs.elte.hu/geometry/csikos/proj/proj.html

70. Non-Euclidean Geometry. The Columbia Encyclopedia, Sixth Edition. 2001 The Columbia Encyclopedia, Sixth Edition. 2001. noneuclidean geometry.branch of 3. Non-euclidean geometry and Curved Space. What distinguishes http://www.bartleby.com/65/no/nonEucli.html

Select Search All Bartleby.com All Reference Columbia Encyclopedia World History Encyclopedia World Factbook Columbia Gazetteer American Heritage Coll. Dictionary Roget's Thesauri Roget's II: Thesaurus Roget's Int'l Thesaurus Quotations Bartlett's Quotations Columbia Quotations Simpson's Quotations English Usage Modern Usage American English Fowler's King's English Strunk's Style Mencken's Language Cambridge History The King James Bible Oxford Shakespeare Gray's Anatomy Farmer's Cookbook Post's Etiquette Bulfinch's Mythology Frazer's Golden Bough All Verse Anthologies Dickinson, E. Eliot, T.S. Frost, R. Hopkins, G.M. Keats, J. Lawrence, D.H. Masters, E.L. Sandburg, C. Sassoon, S. Whitman, W. Wordsworth, W. Yeats, W.B. All Nonfiction Harvard Classics American Essays Einstein's Relativity Grant, U.S. Roosevelt, T. Wells's History Presidential Inaugurals All Fiction Shelf of Fiction Ghost Stories Short Stories Shaw, G.B. Stein, G. Stevenson, R.L. Wells, H.G. Reference Columbia Encyclopedia PREVIOUS NEXT ... BIBLIOGRAPHIC RECORD The Columbia Encyclopedia, Sixth Edition. non-Euclidean geometry branch of geometry in which the fifth postulate of Euclidean geometry, which allows one and only one line parallel to a given line through a given external point, is replaced by one of two alternative postulates. Allowing two parallels through any external point, the first alternative to

71. The Historical Importance Of Non-Euclidean Geometry What is the historical importance of noneuclidean geometry? I intend towrite in more detail on this topic. For now, here is a brief summary. http://www.dpmms.cam.ac.uk/~wtg10/historyetc.html

What is the historical importance of non-Euclidean geometry? I intend to write in more detail on this topic. For now, here is a brief summary. The development of non-Euclidean geometry caused a profound revolution, not just in mathematics, but in science and philosophy as well. The philosophical importance of non-Euclidean geometry was that it greatly clarified the relationship between mathematics, science and observation. Before hyperbolic geometry was discovered, it was thought to be completely obvious that Euclidean geometry correctly described physical space, and attempts were even made, by Kant and others, to show that this was necessarily true. Gauss was one of the first to understand that the truth or otherwise of Euclidean geometry was a matter to be determined by experiment, and he even went so far as to measure the angles of the triangle formed by three mountain peaks to see whether they added to 180. (Because of experimental error, the result was inconclusive.) Our present-day understanding of models of axioms, relative consistency and so on can all be traced back to this development, as can the separation of mathematics from science. The scientific importance is that it paved the way for Riemannian geometry, which in turn paved the way for Einstein's General Theory of Relativity. After Gauss, it was still reasonable to think that, although Euclidean geometry was not

72. PHY 209 - Euclidean Geometry PHY 209 euclidean geometry Angles, Lengths, Areas, Volumes. 209 209 RobSalgado (salgado@physics.syr.edu) Last modified Sun Feb 23 170008 1997. http://physics.syr.edu/courses/PHY209.95Fall/notes/angles.html

PHY 209 - Euclidean Geometry: Angles, Lengths, Areas, Volumes Rob Salgado (salgado@physics.syr.edu) Last modified: Sun Feb 23 17:00:08 1997

73. Non-Euclidean Geometry Noneuclidean geometry. Taxicab Geometry An Adventure in Non-euclidean geometry;The Nature and Power of Mathematics; Non-Euclideon Geometry. Back to Mathematics. http://www.arkanar.com.by/69/Geometry_Non_Euclidean_index.htm

Non-Euclidean Geometry Compact Riemann Surfaces : An Introduction to Contemporary Mathematics (Universitext) Generalized Cauchy-Riemann Systems With a Singular Point (Pitman Monographs and Surveys in Pure and Applied Mathematics) Taxicab Geometry : An Adventure in Non-Euclidean Geometry The Nature and Power of Mathematics ... Mathematics In Association with Amazon.com Amazon.co.uk Amazon.de Advertise at this Site ... Eugene Kisly and Victor Kisly

74. Non-Euclidean Geometry ->Topological Formation Noneuclidean geometry - Topological Formation. Follow Ups Post Followup Geometry III FAQ Posted by sol on December 14, 2002 at 103056 http://superstringtheory.com/forum/geomboard/messages3/48.html

String Theory Discussion Forum String Theory Home Forum Index Non-Euclidean Geometry ->Topological Formation Follow Ups Post Followup Geometry III FAQ Posted by sol on December 14, 2002 at 10:30:56: At the Heart of Science The very idea that Non Euclidean geometry could expressed different values in what could have been determined of the triangle, on the sphere, has move perception beyond what is currently understood, and paved the way for movement in the topological understandings. The sphere now becomes the model of apprehension about a world that can move beyonsuch coordinates and getting to this realm time was added and in it, changes in position relative too, and we see in such relation how the world on the background of strings, demonstrates such movements. It incoporates the very nature of such movement, and holds within it, the very understanding of maths and physics so far to date, that has allowed such theoretical formulation, to move to the higher dimensional understanding. I would like to see if any would like to expand slowly here, and the Jordan Theorom as described by DickT, is a very important piece of what could have defined such movements topologically. Such truth of the logic, of mathematical orientatation, has been placed at the basis of Descartes ontological causality. Such reductionism, had become the understanding of Einsteins relation with Bohr, that such probing would have move Bohr to consider more deeply the Quantum world?

75. Non Euclidean Geometry ->Topological Transformation Non euclidean geometry Topological Transformation. Follow Ups Post 47Non euclidean geometry - Topological Transformation. This will http://superstringtheory.com/forum/metaboard/messages14/99.html

String Theory Discussion Forum String Theory Home Forum Index Non Euclidean Geometry ->Topological Transformation Follow Ups Post Followup Mystics and Metaphysics XIV FAQ Posted by sol on December 14, 2002 at 12:05:56: In Reply to: Logic, as Basis of Mathematcial Truth posted by sol on December 14, 2002 at 11:59:47: Non Euclidean Geometry ->Topological Transformation This will give one a complete picture of what any Pattern might do, that we will have understood the math, as well as the need to apply the Physics for confirmnation and concretization of such knowledge accumulation. Why Paradigm models are important. They become a model of the world of understanding. Sol (Report this post to the moderator) Follow Ups: (Reload page to see most recent) Post a Followup Follow Ups Post Followup Mystics and Metaphysics XIV FAQ

76. Non-Euclidian Geometry NON euclidean geometry RESOURCES. Non-euclidean geometry links. Video Non-euclidean geometry . When Further Non-euclidean geometry. Before http://www.scit.wlv.ac.uk/university/scit/modules/mm2217/neg.htm

NON - EUCLIDEAN GEOMETRY RESOURCES Non-Euclidean Geometry links Non-Euclidean Geometry Non-Euclidean geometry Video "Non-Euclidean Geometry" When the initial reading for Non-Euclidean geometry has been completed. You should study the above video produced by the "Open University". This video can be viewed in the University library when is open 9 am - 9 pm Monday to Thursday, 9 am - 5.15 pm Friday and 9 am - 12.30 pm Saturday. It is just necessary to ask for the video by name at the library enquiries. Further Non-Euclidean Geometry Before attempting the assessment, it may be necessary to read about Non-Euclidean Geometry in more detail. It is suggested that you look at the Non-Euclidean Geometry links and read the section on Non-Euclidean geometry from "The History of Mathematics - A Reader". (See reading list). D Thompson has a copy of this book which may be borrowed, if copies are unavailable in libraries. Assessment New Geometries, New Worlds History of Mathematics Module Links to other History of Mathematics sites ... Module Leader These pages are maintained by M.I.Woodcock.

77. Spherical And Non-Euclidean Geometry next up previous Next Spatial Geometry Up Math 170 Possible FinalPrevious Euclid's Geometry. Spherical and Noneuclidean geometry. http://www.math.uga.edu/~cantarel/teaching/math170/projects/node2.html

Next: Spatial Geometry Up: Math 170 Possible Final Previous: Euclid's Geometry Spherical and Non-Euclidean Geometry We have proved in class that there are no similar triangles on the sphere which are not congruent. But this is puzzling! Construct an equilateral triangle on the sphere- and copy the triangle 4 times to form a big triangle, as below: All the small triangles are congruent, equilateral and equiangular. Why doesn't the big one have the same vertex angles as the small ones? Perform this construction on a real ball, using rubber bands to form your lines. (You must turn in the ball.) Tilings. A tiling is a way to cover a space without gaps using identical copies of a single geometrical figure. The equilateral triangle tiles the plane, in a tiling where six triangles meet at each vertex (see below) Find a tiling of the sphere using 20 equilateral triangles where five triangles meet at each vertex. Draw your tiling (using rubber bands for lines) on a real sphere, and turn it in. Read Riemann's "On the Foundations underlying the study of space" and write a 5-10 page essay on the philosophical implications of non-Euclidean geometry. Can we tell what the geometry of our universe is? Does it matter? Write a 5-10 page paper on historical figures who attempted to prove the 5th postulate, and their (failed) proofs. Some mathematicians to include might be Proclus, John Wallis, Johann Lambert, Saccheri, Playfair, Legendre, al-Tusi, and Khayyam. Look at Jeremy Gray's book

78. Non-Euclidean Geometry Ideas Ask A Scientist. Mathematics Archive. Noneuclidean geometry ideas.Author reilly a etc. on non-euclidean geometry. Response 1 http://newton.dep.anl.gov/newton/askasci/1995/math/MATH092.HTM

Ask A Scientist Mathematics Archive Non-Euclidean geometry ideas Author: reilly a conway If anyone is up to helping me, I would like your insight, ideas, etc. on non-Euclidean geometry. Response #: 1 of 1 Author: hawley Here is something to get you started: draw a triangle on the surface of a sphere. Show that the sum of the interior angles is now NEVER equal to 180 but is, in fact, 180 < sum <= 540 depending on the relative sizes of the sphere and the triangle. The surface of a sphere is a simple example of a non-Euclidean, two-dimensional space, that is easy to visualize. Things get more difficult with higher-dimensional spaces! Back to Mathematics Ask A Scientist Index NEWTON Homepage Ask A Question ... NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators. Argonne National Laboratory, Division of Educational Programs, Harold Myron, Ph.D., Division Director.

79. Deductive Euclidean Geometry Activity sheets Deductive euclidean geometry. Contents Activity 1 Problemsolving warmup Homework 1 Definitions Line segment; Activity http://plato.acadiau.ca/courses/educ/reid/Geometry/DeductiveUnit/Act-TOC.html

Activity sheets Deductive Euclidean Geometry Contents: Activity 1 Problem solving warmup Homework 1 Definitions: Line segment Activity 2 Class discussion of HW Activity 3 Two lines, parallel only + class discussion Homework 2 Text work Activity 4 Two lines, intersecting + class discussion Homework 3 Text work Activity 5 Three lines through three points + class discussion Homework 4 Text work Activity 6 Three lines, two parallel + class discussion Homework 5 Text work Activity 7 Four lines, two parallel, trapezoid + class discussion Homework 6 Text work Activity 8 Four lines, two parallel, triangle + class discussion Activity 9 Polygons Activity 10 Congruent Triangles Homework 8 Hypotenuse Length Theorem Activity 11 Midpoint problem Homework 9 Study for Geopardy Activity 12 Geopardy Additional activities [Added during unit] Homework 8.5 Notes on Homework Homework Tests Individual test Group test

80. Hyperbolic Geometry Cabri constructions for the demonstration of the basic concepts of hyperbolic geometry in the Poincare Category Science Math Geometry...... of a small pearl of a book Advanced euclidean geometry (Modern Geometry) An elementaryTreatise on the Geometry of the triangle and the Circle (to give its http://mcs.open.ac.uk/tcl2/nonE/nonE.html

Hyperbolic Geometry using Cabri This page and links maintained by Tim Lister, t.c.lister@open.ac.uk Last updated: A tessellation of the hyperbolic plane H Full screen version of diagram During the summer of 97 I had great fun playing with some marvelous software, Cabri Geometry , and devising constructions for use in teaching the basic ideas of a geometry course put on by the Open University. These started with some figures to demonstrate the transformations of Inversive Geometry, and progressed to figures for the Arbelos, the inversors of Peucellier and Hart, Coaxial Circles and so on, much of which was driven by the discovery of a Dover edition of a small pearl of a book Advanced Euclidean Geometry (Modern Geometry) An elementary Treatise on the Geometry of the triangle and the Circle (to give its full title) written by Roger A. Johnson and first published in 1929. It had languished on my bookshelves, having been bought years ago for 20 cents (South African) in some sale or other. I can recommend it as a fascinating read, or just for taking in the breathtaking complexity of the many hand crafted diagrams to be found on its pages.

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