21. The Math Forum - Math Library - Non-Euclidean Geom. KSEG is a Linux program for dynamically exploring euclidean geometry. Create a construction, such as a triangle with a circumcenter; drag its vertices, http://mathforum.org/library/topics/noneuclid_g/

Browse and Search the Library Home Math Topics Geometry : Non-Euclidean Geom. Library Home Search Full Table of Contents Suggest a Link ... Library Help Subcategories (see also All Sites in this category Hyperbolic Geometry Projective Geometry Selected Sites (see also All Sites in this category Non-Euclidean Geometry - MacTutor Math History Archives Covers Euclid's Elements through the end of the 1800's, with 23 references (books/articles). more>> All Sites - 73 items found, showing 1 to 50 abraCAdaBRI - Yves Martin ...more>> Advanced Geometry - Math Forum Links to some of the best Internet resources for advanced geometry: Web sites, software, Internet projects, publications, and public forums for discussion. ...more>> An Artist's Timely Riddles - Ivars Peterson - Science News Online The late 19th and early 20th centuries were a time of great popular interest in visualizing a fourth spatial dimension - a concept that appeared to offer painters and sculptors, in particular, an avenue of escape from conventional representation. Moreover, ...more>> ...more>> C26: Geometry - Scott Thatcher; Dept. of Mathematics, Northwestern Univ., Evanston, IL

22. The Triangle Figure Introduction to a Euclidean Model of Hyperbolic Geometry In euclidean geometry we can use translation, rotation, and reflection to make congruent copies http://www.math.clemson.edu/~simms/triangle/hyperbolic.html

Introduction to a Euclidean Model of Hyperbolic Geometry Hyperbolic geometry is the result of replacing the parallel axiom of Euclidean Geometry with the alternative of there being, through a given point, at least two lines parallel to a given line. It's one thing to develop the theorems in such a geometry, but it's another to do constructions a euclidean model where the compass and straight edge may be used would be advantageous. The angle formed by two intersecting h-lines is determined by substituting the tangent to any type-2 h-lines at the point of intersection and measuring the angle as we would between two intersecting e-lines. Here's a Java version of a construction of a circle inversion from the inside of a circle to the outside and vice-versa. I-O circle inversion The point inside the circle may be dragged and the figure adjusts accordingly. O-I circle inversion The point outside the circle may be dragged and the figure adjusts accordingly. Another use for circle inversions is in the fact that a circle passing through both points of a circle inversion pair will be orthogonal to the circle of inversion. So circle inversion may be used to aid in the construction of a type-2 h-line through a given point that is perpendicular to a given type-2 h-line. Links to other hyperbolic geometry pages Models of Hyperbolic Geometry NonEuclid Models of the Hyperbolic Plane , at the Geometry Center , U. of Minnesota

23. Non-Euclidean Geometry And Indra's Pearls In our ordinary geometry, called euclidean geometry after the ancient Greek mathematician Euclid, these points form a circle, namely the circle that has http://plus.maths.org/issue43/features/serieswright/

about Plus support Plus ... Plus search plus with google home latest issue news ... Plus Permission is granted to print and copy this page on paper for non-commercial use. For other uses, including electronic redistribution, please contact us. Issue 43 June 2007 Contents Features Non-Euclidean geometry and Indra's pearls What is the Area of a Circle? Quantum geometry Euler's polyhedron formula ... Maths goes public Career interview Teacher packages Teacher package: Differential equations Regulars Plus puzzle Editorial Outer space Reviews 'Mathematics and common sense' 'Why beauty is truth' 'A beautiful math' All the latest news ... posters! June 2007 Features Non-Euclidean geometry and Indra's pearls by Caroline Series and David Wright Many people will have seen and been amazed by the beauty and intricacy of fractals like the one shown on the right. This particular fractal is known as the Apollonian gasket and consists of a complicated arrangement of tangent circles. [Click on the image to see this fractal evolve in a movie created by David Wright.] Few people know, however, that fractal pictures like this one are intimately related to tilings of what mathematicians call

24. Bob Gardner's "Non-Euclidean Geometry" Webpage INTRODUCTION This part of your Independent Study class will be a one week introduction to noneuclidean geometry with a review of euclidean geometry. http://www.etsu.edu/math/gardner/noneuclidean/syllabus.htm

25. Non-Euclidean Geometry Despite the fact that noneuclidean geometry has found its use in numerous applications (the most striking example being 3-dimensional topology), http://www.mccme.ru/mathinmoscow/courses/view.php?name=Non-Euclidean geometry.ht

26. Non-Euclidean Geometry Seminar In this course we traced the development of hyperbolic (nonEuclidean) geometry from ancient Greece up to the turn of the century. This was accomplished by http://www.math.columbia.edu/~pinkham/teaching/seminars/NonEuclidean.html

Seminar on the History of Hyperbolic Geometry Greg Schreiber In this course we traced the development of hyperbolic (non-Euclidean) geometry from ancient Greece up to the turn of the century. This was accomplished by focusing chronologically on those mathematicians who made the most significant contributions to the subject. We began with an exposition of Euclidean geometry, first from Euclid's perspective (as given in his Elements) and then from a modern perspective due to Hilbert (in his Foundations of Geometry). Almost all criticisms of Euclid up to the 19th century were centered on his fifth postulate, the so-called Parallel Postulate.The first half of the course dealt with various attempts by ancient, medieval, and (relatively) modern mathematicians to prove this postulate from Euclid's others. Some of the most noteworthy efforts were by the Roman mathematician Proclus, the Islamic mathematicians Omar Khayyam and Nasir al-Din al-Tusi, the Jesuit priest Girolamo Sacchieri, the Englishman John Wallis, and the Frenchmen Lambert and Legendre. Each one gave a flawed proof of the parallel postulate, containing some hidden assumption equivalent to that postulate. In this way properties of hyperbolic geometry were discovered, even though no one believed such a geometry to be possible. References: Four general references were used throughout this course: Bonola's Non-Euclidean Geometry, Jeremy Gray's Ideas of Space, Greenberg's Euclidean and Non-Euclidean Geometries, and McCleary's Geometry from a Differential Viewpoint. In addition, original works of these mathematicians were used whenever possible, as well as biographies of them. These books included Euclid's Elements, Hilbert's Foundations of Geometry, Proclus's A Commentary on the First Book of Euclid's Elements, Saccheri's Euclid Vindicated, Bolyai's Science of Absolute Space, Lobachevskii's Geometrical Researches in the Theory of Parallels, and Riemann's "On the Hypotheses Which Lie at the Foundations of Geometry," among others.

27. Non-Euclidean Geometries, Discovery Gauss invented the term Noneuclidean geometry but never published anything on the subject. On the other hand, he introduced the idea of surface curvature http://www.cut-the-knot.org/triangle/pythpar/Drama.shtml

var MyPageLoc = document.location; var MyPageTitle = document.title; G o o g ... e Web CTK Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Non-Euclidean Geometries Drama of the Discovery Four names - C. F. Gauss N. Lobachevsky J. Bolyai (1802-1860), and B. Riemann (1826-1866) - are traditionally associated with the discovery of non-Euclidean geometries. In non-Euclidean geometries, the fifth postulate is replaced with one of its negations : through a point not on a line, either there is none (B) or more than 1 (C) line parallel to the given one. Carl Friedrich Gauss was apparently the first to arrive at the conclusion that no contradiction may be obtained this way. In a private letter of 1824 Gauss wrote: The assumption that (in a triangle) the sum of the three angles is less than 180 o leads to a curious geometry, quite different from ours, but thoroughly consistent, which I have developed to my entire satisfaction. From another letter of 1829, it appears that Gauss was hesitant to publish his research because he suspected the mediocre mathematical community would not be able to accept a revolutionary denial of Euclid's geometry. Gauss invented the term "Non-Euclidean Geometry" but never published anything on the subject. On the other hand, he introduced the idea of surface curvature on the basis of which Riemann later developed Differential Geometry that served as a foundation for Einstein's General Theory of Relativity.

28. Advanced Euclidean Geometry Excursions For Secondary Teachers And This book reviews the essentials of high school geometry, then expands those concepts to advanced euclidean geometry to give teachers more confidence in http://www.keypress.com/x5760.xml

Customer Support Contact Us About Key Home Educators Parents, Mentors, Students Educational Retailers State Resources Professional Development View Cart Checkout Advanced Euclidean Geometry Overview Ordering Home Educators ... Geometry Advanced Euclidean Geometry Advanced Euclidean Geometry Excursions for Secondary Teachers and Students Alfred S. Posamentier Build Confidence Guiding Student Explorations This book reviews the essentials of high school geometry , then expands those concepts to advanced Euclidean geometry to give teachers more confidence in guiding student explorations and questions. The text includes hundreds of illustrations created with The Geometer's Sketchpad and a CD with more than 100 interactive sketches . Access to Sketchpad Explore Supplementary Books and Materials Algebra 1 and Algebra 2 Geometry Precalculus and Calculus Statistics ... Bay Area Green Business

29. The Geometer's Sketchpad® - Euclidean And Non-Euclidean Geometry Conference talk by Scott Steketee with downloadable sketches. http://www.dynamicgeometry.com/general_resources/recent_talks/euclidean_geom_wie

Getting Started Product Information How to Order Curriculum Modules ... recent talks euclidean geom wien98 Euclidean and Non-Euclidean Geometry with The Geometer's Sketchpad by Scott Steketee stek@keypress.com Key Curriculum Press Downloadable Sketches: Geometry.sea.hqx Geometry.exe : Download the sketches in PC format

30. Applications Of Non-Euclidean Geometry It has been argued that euclidean geometry, while good for architecture and to survey land, when it is moved into the third dimension, the postulates do not http://members.tripod.com/noneuclidean/applications.html

The Applications Of Non-Euclidean Geometry Table of Contents Where Euclidean Geometry Is Wrong The Theory of General Relativity Spherical Geometry Celestial Mechanics Where Euclidean Geometry Is Wrong Since Euclid first published his book Elements in 300 B.C. it has remained remarkably correct and accurate to real world situations faced on Earth. The one problem that some find with it is that it is not accurate enough to represent the three dimensional universe that we live in. It has been argued that Euclidean Geometry, while good for architecture and to survey land, when it is moved into the third dimension, the postulates do not hold up as well as those of hyperbolical and spherical geometry. Both of those geometries hold up to a two dimensional world, as well as the third dimension. Back To Top Cosmology - Cosmology is the study of the origin, constitution, structure, and evolution of the universe. Back To Top The Theory of General Relativity Einstein's Theory Of General Relativity is based on a theory that space is curved. The cause is explained by the theory itself. Einstein's General Theory of Relativity can be understood as saying that: Matter and energy distort space The distortions of space affect the motions of matter and energy.

31. Geometry The uses of homogeneous barycentric coordinates in plane euclidean geometry, Int. J. Math. Educ. Sci. Technol., 31 (2000) 569 578. http://www.math.fau.edu/yiu/geometry.html

Paul Yiu Department of Mathematics Florida Atlantic University An Introduction to the Geometry of the Triangle, 2001 [ ps file pdf Notes on Euclidean Geometry, 1998 Elementary Mathematical Works of Leonhard Euler, 1999 ps file pdf with a chronologial listing of Euler's works [ ps file pdf Recreational Mathematics, 2003 Algebraic Topology, 2006 ... A Tour of Triangle Geometry February 2004 A Short Tour of Triangle Geometry Around the Nine-point Circle (GSP), April 2007 Regular Heptagon by Angle Trisection and Other Constructions [ ps pdf April 2007 Heron triangles which cannot be decomposed into two integer right triangles [ ps pdf ], February 2008 Geometry Papers: Construction of integer right triangles with consecutive legs in the tradition of Jiuzhang Suanshu, (Chinese) EduMath . no.3, (1996) 26 32. Heronian triangles with associated inradii in arithmetic progression Crux Math Isosceles triangles equal in perimeter and area Missouri J. Math. Sci

32. Interactive Non-Euclidean Geometry 1 Interactive Noneuclidean geometry is an interactive program for creating ruler and compass constructions in euclidean, spherical and hyperbolic geometry. http://arsenailna.cie.uma.es/~ccriado/Interactive Non.htm

Interactive Non-Euclidean Geometry 1.0 - Free download Interactive Non-Euclidean Geometry is an interactive program for creating ruler and compass constructions in euclidean, spherical and hyperbolic geometry. This program has been created by Carlos Criado-Cambon and Juan-Carlos Criado-Alamo. Universidad de Malaga (UMA). Spain For any comment write to: c_criado@uma.es. Introduction This program has been created for drawing in euclidean, and spherical geometries, and also in the four most popular models of hyperbolic geometry The four models of hyperbolic geometry You can draw and view simultaneously in these four models: Klein model. Poincaré model. Half-plane model. Hemisphere model. The ‘point’ tool If you want to plot points you can check this option. Move the mouse over any one of the models and the help shows you the position. Click over the site where you want to put the point with the left button. The points are the basic option of this program because all the other options need points to work. Selecting points Click over a point to select it.

33. Euclidean Geometry euclidean geometry is a mathematical wellknown system attributed to the Greek mathematician Euclid of Alexandria. Euclid s text Elements was the first http://www.sciencedaily.com/articles/e/euclidean_geometry.htm

Science Reference Share Blog Print Email Bookmark Euclidean geometry Euclidean geometry is a mathematical well-known system attributed to the Greek mathematician Euclid of Alexandria. See also: Mathematics Computer Modeling Mathematical Modeling Euclid's text Elements was the first systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content. The method consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions (theorems) from those axioms. Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could be fitted together into a comprehensive deductive and logical system.. For more information about the topic Euclidean geometry , read the full article at Wikipedia.org , or see the following related articles: Geometry read more Hyperbolic geometry read more ... read more Note: This page refers to an article that is licensed under the GNU Free Documentation License . It uses material from the article Euclidean geometry at Wikipedia.org. See the

34. Geometry Tutorial In order to enroll in the second year of the Great Books Tutorial, all students must take my euclidean geometry course. The course requires 510 hours a http://www.gbt.org/geo.html

Euclidean Geometry Tutorial This tutorial covers the Geometry in Euclid's Elements Euclid 's famous text was "the" book for the study of Geometry until the 19th century. It has been studied by a host of intellectual greats. His systematic approach to Geometry is not a only a tremendous study in how to think and reason, but it became the paradigm that later philosophers would attempt to follow in setting up their own systems of thought. There is really no other mathematical text that rivals its impact on intellectual history. This tutorial is highly recommended not only for its tremendous historical value, but also as a fine addition to the Geometry-starved Saxon program. The only fault I see with the Saxon programs is its meager treatment of Geometric proofs. Along with Saxon, most modern math texts are downplaying Geometric proofs because they are teaching to the SAT and it does not require proofs. But after going through the magnificent proofs of Euclid , you will see why his work is truly a mathematical classic.

35. Euclidean Geometry - Definition From The Merriam-Webster Online Dictionary Definition of euclidean geometry from the MerriamWebster Online Dictionary with audio pronunciations, thesaurus, Word of the Day, and word games. http://www.m-w.com/dictionary/euclidean geometry

Home Visit Our Sites Unabridged Dictionary Learner's Dictionary ... Contact Us Also Visit: Unabridged Encyclopedia Britannica Visual ESL: Learner's for Kids: Spell It! Word Central Dictionary Thesaurus Spanish/English Medical Search "euclidean geometry" in: Thesaurus Spanish/English Medical Dictionary Open Dictionary Browse words next to: euclidean geometry Browse the Dictionary: A B C D ... Z euclidean geometry One entry found. euclidean geometry Main Entry: euclidean geometry Function: noun Usage: often capitalized E Date: circa 1865  geometry based on Euclid's axioms  the geometry of a euclidean space Learn more about "euclidean geometry" and related topics at Britannica.com Pronunciation Symbols Products Premium Services ... Advertising Info

36. Non-Euclidean Geometry The two most important types of noneuclidean geometry are hyperbolic geometry and elliptical geometry. The different models of non-euclidean geometry can http://www.daviddarling.info/encyclopedia/N/non-Euclidean_geometry.html

SPACE AND TIME GRAVITATIONAL PHYSICS MATHEMATICS A ... CONTACT entire Web this site non-Euclidean geometry Any geometry in which Euclid's fifth postulate, the so-called parallel postulate , doesn't hold. (One way to say the parallel postulate is: Given a straight line and a point A not on that line, there is only one exactly straight line through A that never intersects the original line.) The two most important types of non-Euclidean geometry are hyperbolic geometry and elliptical geometry . The different models of non-Euclidean geometry can have positive or negative curvature . The sign of curvature of a surface is indicated by drawing a straight line on the surface and then drawing another straight line perpendicular to it: both these lines are geodesics . If the two lines curve in the same direction, the surface has a positive curvature; if they curve in opposite directions, the surface has negative curvature. Elliptical (and spherical) geometry has positive curvature whereas hyperbolic geometry has negative curvature. The discovery of non-Euclidean geometry had immense consequences. For more than 2,000 years, people had thought that

37. What Is Non-Euclidean Geometry? An introduction to the history and mathematics of nonEuclidean geometries. http://njnj.essortment.com/noneuclideange_risc.htm

Enter your search terms Submit search form Web essortment.com What is non-Euclidean geometry? An introduction to the history and mathematics of non-Euclidean geometries. Euclid's geometrical thesis, "The Elements" (c. 300 B.C.E), proposed five basic postulates of geometry. Of these postulates, all were considered self-evident except for the fifth postulate. The fifth postulate asserted that two lines are parallel (i.e. non-intersecting) if a third line can intersect both lines perpendicularly. Consequently, in a Euclidean geometry every point has one and only one line parallel to any given line. For centuries people questioned Euclid's fifth postulate. Even Euclid seemed suspicious of the fifth postulate because he avoided solving problems with it until his 29th example. Mathematicians stumbled with ways to prove the validity of the fifth postulate from the first four postulates, which we now call the postulates of absolute geometry. Those mathematicians who didn't fail were soon seen to have fallacious errors in their reasoning. These errors usually occurred because a mathematician had made self-fulfilling assumptions pertaining to parallel lines, rather than working with the other postulates. Essentially, they were forcing a result through the application of faulty logic. Though many mathematicians questioned Euclidean geometry, Euclidean thought prevailed through school mathematical programs. "The Elements" became the most widely purchased non-religious work in the world, and it still remains the most widely received of mathematical texts. Furthermore, mathematical inquiries into the nature of non-Euclidean geometries were often devalued as frivolous. The philosopher Immanuel Kant (1724-1804) called Euclid's geometry, "the inevitable necessity of thought." Such philosophical opinions impeded mathematical progress in the field of geometry. Karl Friedrich Gauss (1777-1855), who began studying non-Euclidean geometries at the age of 15, never published any of his non-Euclidean works because he knew the mathematical precedent was against him.

38. PlanetMath: Geometry These axioms today serve as the foundation of plane euclidean geometry. For instance, in the case of euclidean geometry, the property of orthogonality http://planetmath.org/encyclopedia/EuclideanGeometry.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... RSS Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About geometry (Topic) RayleighDistribution "geometry" is owned by CWoo full author list owner history view preamble View style: HTML with images page images TeX source See Also: finite projective plane projective plane point-free geometry comparison of common geometries Other names: Egyptian geometry Also defines: Greek geometry, Euclidean geometry Keywords: calculus Attachments: non-Euclidean geometry (Definition) by Wkbj79 Euclidean geometry of plane (Topic) by matte differential geometry (Topic) by rspuzio Euclidean geometry of space (Topic) by pahio analytic geometry (Topic) by pahio Log in to rate this entry. view current ratings Cross-references: separation hyperbola quadrature language ... feasible There are 136 references to this entry.

39. Geometry :: Non-Euclidean Geometry --  Britannica Student Encyclopaedia geometry, Noneuclidean geometry In the 19th century, many mathematicians began questioning one of Euclid s main premises that, simply stated, http://student.britannica.com/comptons/article-201469/geometry

document.writeln(AAMB1); Already a member? LOGIN Home Free Trial Guided Tour Britannica Online This Article's Table of Contents Expand all Collapse all Introduction EUCLIDEAN GEOMETRY ... Solids Non-Euclidean Geometry Construction Drafter's tools. The impossible problems. Print this Table of Contents Unsure of the meaning of a word? Double-click it to look it up in Merriam-Webster's Student Dictionary. document.writeln(AAMB3); geometry Student's Encyclopedia Page 9 of 12 Print Page Print Article E-mail Article Cite Article EUCLIDEAN GEOMETRY Non-Euclidean Geometry Expand your Research: Journals And Magazines The Web's Best Sites geometry... (75 of 34238 words) To read the full article, activate your FREE Trial Previous Page Solids Page 9 of 12 Non-Euclidean Geometry Next Page Construction document.writeln(AAMB2); To cite this page: MLA style: geometry http://student.britannica.com/comptons/article-201469 APA style: geometry ). In Retrieved http://student.britannica.com/comptons/article-201469

40. Euclidean Geometry euclidean geometry. GT shape BT mathematical sciences FT geometrie elementaire PT geometric solids geometric surfaces lines(geometry) point(geometry) http://irc.nrc-cnrc.gc.ca/thesaurus/euclidean_geometry.html

euclidean geometry GT shape BT mathematical sciences FT geometrie elementaire PT geometric solids geometric surfaces lines(geometry) point(geometry) ... trigonometry NT affine geometry descriptive geometry plane geometry solid geometry ... [Help]

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