61. The Math Forum - Math Library - Non-Euclidean Geom. Online materials covering Euclidean and nonEuclidean geometry as well as some other topics. It is intended to support a mathematics course at the level of college algebra, with 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 - 75 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>> Books by Lillian Lieber - Lillian Lieber, Hugh Gray Lieber, Paul Dry Books

62. Physics With Transforms By Rodolfo Padilla A new method of correlating physics formulas to derive one formula from a related formula using Euclidean geometry to represent the inter-relationship of physics formulas. http://physicstransforms.tripod.com/

Build your own FREE website at Tripod.com Share: Facebook Twitter Digg reddit document.write(lycos_ad['leaderboard']); document.write(lycos_ad['leaderboard2']); Physics with Transforms by Rodolfo Padilla Home Table of Contents Explanation the Transform ... go to Book by Rodolfo Padilla in Spanish / vea libro de Rodolfo Padilla en Espanol s="na";c="na";j="na";f=""+escape(document.referrer)

63. Euclidean And Non-Euclidean Geometry Euclidean Geometry (the high school geometry we all know and love) is the study of geometry based on definitions, undefined terms (point, line and plane) and the assumptions http://www.regentsprep.org/regents/math/geometry/GG1/Euclidean.htm

Euclidean and Non-Euclidean Geometry Topic Index Geometry Index Regents Exam Prep Center Euclidean Geometry (the high school geometry we all know and love) is the study of geometry based on definitions, undefined terms (point, line and plane) and the assumptions of the mathematician Euclid (330 B.C.) Euclid's text Elements was the first systematic discussion of geometry. While many of Euclid's findings had been previously stated by earlier Greek mathematicians, Euclid is credited with developing the first comprehensive deductive system. Euclid's approach to geometry consisted of proving all theorems from a finite number of postulates (axioms). Euclidean Geometry is the study of flat space . We can easily illustrate these geometrical concepts by drawing on a flat piece of paper or chalkboard. In flat space, we know such concepts as: the shortest distance between two points is one unique straight line. the sum of the angles in any triangle equals 180 degrees.

64. Question Corner -- Non-Euclidean Geometry NonEuclidean Geometry Asked by Brent Potteiger on April 5, 1997 I have recently been studying Euclid (the father of geometry), and was amazed to find out about the http://www.math.toronto.edu/mathnet/questionCorner/noneucgeom.html

Navigation Panel: (These buttons explained below Question Corner and Discussion Area Non-Euclidean Geometry Asked by Brent Potteiger on April 5, 1997 I have recently been studying Euclid (the "father" of geometry), and was amazed to find out about the existence of a non-Euclidean geometry. Being as curious as I am, I would like to know about non-Euclidean geometry. Thanks!!! All of Euclidean geometry can be deduced from just a few properties (called "axioms") of points and lines. With one exception (which I will describe below), these properties are all very basic and self-evident things like "for every pair of distinct points, there is exactly one line containing both of them". This approach doesn't require you to get into a philosophical definition of what a "point" or a "line" actually is. You could attach those labels to any concepts you like, and as long as those concepts satisfy the axioms, then all of the theorems of geometry are guaranteed to be true (because the theorems are deducible purely from the axioms without requiring any further knowledge of what "point" or "line" means). Although most of the axioms are extremely basic and self-evident, one is less so. It says (roughly) that if you draw two lines each at ninety degrees to a third line, then those two lines are parallel and never intersect. This statement, called

65. The Ontology And Cosmology Of Non-Euclidean Geometry A philosophical essay. http://www.friesian.com/curved-1.htm

The Ontology and Cosmology of Non-Euclidean Geometry Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence. David Hume, An Enquiry Concerning Human Understanding , Section IV, Part I, p. 20 [L.A. Shelby-Bigge, editor, Oxford University Press, 1902, 1972, p. 25] [ note Until recently, Albert Einstein's complaints in his later years about the intelligibility of Quantum Mechanics often led philosophers and physicists to dismiss him as, essentially, an old fool in his dotage. Happily, this kind of thing is now coming to an end as philosophers and mathematicians of the caliber of Karl Popper and Roger Penrose conspicuously point out the continuing conceptual difficulties of quantum theory [cf. Penrose's searching discussion in The Emperor's New Mind reductio ad absurdum argument against A fine statement about all this can be found in Joseph Agassi's foreword to the recent Einstein Versus Bohr , by the dissident physicist Mendel Sachs (Open Court, 1991): It is amazing that such things need to be said, and it is particularly revealing that the responses Agassi got to his questions reminded him of the intolerance of religious dogmatism.

66. Non-Euclidean Geometry Definition Of Non-Euclidean Geometry In The Free Online E nonEuclidean geometry, branch of geometry geometry Gr.,=earth measuring, branch of mathematics concerned with the properties of and relationships between points, lines http://encyclopedia2.thefreedictionary.com/non-Euclidean geometry

67. Euclidean And Non-Euclidean Geometry With The Geometer S Sketchpad Conference talk by Scott Steketee with downloadable sketches. http://www.dynamicgeometry.com/talks/Euc_Wien98/

Home Getting Started General Resources Instructor Resources ... General Resources Recent Talks Home Getting Started Considering Sketchpad Learning Sketchpad ... Site Map Recent Talks Sketchpad 4.0 Talks 2008. Scott Steketee. " Do the Wave: Trigonometry Comes Alive with Sketchpad. " Presented at the NCTM 2008 Annual Meeting, this talk explores several dynamic activities that can improve student understanding of sinusoids and other trigonometric functions. Download: Handout (.pdf file) Download: Activity Worksheets (.pdf file) Download: Activity Sketches (.zip file) Download: Presentation Sketch (.gsp file) 2007. Nicholas Jackiw. " Discrete Mathematics and Dynamic Geometry. " Presented at the NCTM 2007 Annual Meeting, this talk explores connections between Sketchpad's land of continuous geometric transformation and subjects drawn from discrete mathematics, where structures are finite (non-continuous). Topics include elementary number theory (multiples, factors, prime numbers, GCDs, Fibonacci numbers, and flower patterns), algorithms, recurrence relations, and Boolean or binary logic. Download: Presentation Sketches (.gsp file)

68. Forumgeom Free-access electronic journal about elementary Euclidean geometry. http://forumgeom.fau.edu/

Editorial Board About the Journal Instructions to Authors Submission of Papers Refereeing Statistics Links Forum Geometricorum indexed and reviewed by Mathematical Reviews Volume 1 (2001) Volume 2 (2002) Volume 3 (2003) ... Download and Viewing Instructions Subscription: If you want to receive email notifications of new publications, please send a blank email with subject line: Subscribe to FG. Editors' Corner Last modified by Paul Yiu, January 7, 2010.

69. NonEuclid: 1: Non-Euclidean Geometry 1.1 Euclidean Geometry The geometry with which we are most familiar is called Euclidean geometry. Euclidean geometry was named after Euclid, a Greek mathematician who lived in http://cs.unm.edu/~joel/NonEuclid/noneuclidean.html

3: What is Non-Euclidean Geometry 1.1 Euclidean Geometry: The geometry with which we are most familiar is called Euclidean geometry. Euclidean geometry was named after Euclid, a Greek mathematician who lived in 300 BC. His book, called "The Elements", is a collection of axioms, theorems and proofs about squares, circles acute angles, isosceles triangles, and other such things. Most of the theorems which are taught in high schools today can be found in Euclid's 2000 year old book. Euclidean geometry is of great practical value. It has been used by the ancient Greeks through modern society to design buildings, predict the location of moving objects and survey land. 1.2 Non-Euclidean Geometry: non-Euclidean geometry is any geometry that is different from Euclidean geometry. Each Non-Euclidean geometry is a consistent system of definitions, assumptions, and proofs that describe such objects as points, lines and planes. The two most common non-Euclidean geometries are spherical geometry and hyperbolic geometry. The essential difference between Euclidean geometry and these two non-Euclidean geometries is the nature of parallel lines: In Euclidean geometry, given a point and a line, there is exactly one line through the point that is in the same plane as the given line and never intersects it. In spherical geometry there are no such lines. In hyperbolic geometry there are at least two distinct lines that pass through the point and are parallel to (in the same plane as and do not intersect) the given line.

70. KSEG A program for exploring Euclidean geometry. http://www.mit.edu/~ibaran/kseg.html

KSEG Free Interactive Geometry Software Update: May 1, 2007 Mongolian Batnasan sent me a translation into Mongolian. Update: Februrary 3, 2006 KSEG 0.403 and Traditional Chinese I finally got around to a bit of maintenance on KSEG: it now uses qmake for building and should build on Qt 3.x without the compatibility headers. Yao Chang-Li sent me a Traditional Chinese translation and help file . It is included in the .403 file. Update: May 15, 2005 KSEG in Chinese Xu Xianghua kindly contributed Chinese translations of the KSEG UI and the help. Download the UI translation here and the help here Update: July 3, 2004 KSEG 0.401 for Windows! By popular demand, despite my dislike for microsoft, I've ported KSEG to windows using the old Qt noncommercial version 2.3 (I hope I'm not violating anything). The source is uglier and there are some bugs not in the Linux version. Download the whole thing here (it should run out of the box). Sample Output Images (not screenshotsthose are below) of a well-known theorem and a strange locus, both exported with KSEG: Description: KSEG is a Free (GPL) interactive geometry program for exploring Euclidean geometry. It runs on Unix-based platforms (according to users, it also compiles and runs on Mac OS X and should run on anything that Qt supports). You create a construction, such as a triangle with a circumcenter, and then, as you drag verteces of the triangle, you can see the circumcenter moving in real time. Of course, you can do a lot more than thatsee the feature list below.

71. The Derivation Of Plane Euclidean Geometry From The Real Numbers THE DERIVATION OF . PLANE EUCLIDEAN GEOMETRY . FROM THE REAL NUMBERS FOREWORD TO THE WEB VERSION The manuscript on this site was originally composed http://euclidean-geometry.org/

MPBodyInit('index_files') THE DERIVATION OF PLANE EUCLIDEAN GEOMETRY FROM THE REAL NUMBERS FOREWORD TO THE WEB VERSION The manuscript on this site was originally composed on a MacBook using Microsoft Word and MathType5, and then converted to web pages using MathType’s Export to MathPage feature. The “equations” are GIF files accessible by any browser. If your browser supports MathType's MathZoom feature, by clicking an equation the equation increases in size (zooms), which feature is useful for easier viewing of subscripts, superscripts, summation limits, hats, etc. Occasionally you will come across what appears to be a single equation which zooms in two pieces. This is an artifact of the formatting of the original Word document. More than one equation can be zoomed at a time. Click any zoomed equation a second time and it will revert to normal size. Shift-click any zoomed equation and all zoomed equations revert to normal size. If your browser doesn't support MathZoom (Safari does not!), download and use Firefox. As the “equations” are treated as units by your browser, spacing different from that intended during composing can appear on your screen. To minimize such oddities, the author recommends that you adjust your window to the following width, i.e. so that both arrowheads display on the window’s margins:

72. Euclidean Geometry Financial Definition Of Euclidean Geometry. Euclidean Geometr Whereas Euclidean geometry consists of smooth, straight lines, fractal geometry consists of rough or fragmented shapes that can be split into parts, each of which is a reduced http://financial-dictionary.thefreedictionary.com/Euclidean Geometry

73. Untitled Elementary exposition of Euclidean geometry, Newton vs. Einstein, and the possibility to travel into the future, and see into the past, in relativity. By Alex MacLellan. http://themaclellans.com/timetravel.html

Space-Time Physics and the Future of Time Travel By Alex MacLellan Human beings have always tried to explain the world around them. They have worked to organize it, to measure it, and to understand it. These efforts at understanding such basic concepts as space and time and light have led to many discoveries. In the last one hundred years many changes have been made to how we understand these concepts. You may not be able to move as quickly and easily through time as the hero of H.G. Well's Time Machine, but scientists have been able to understand how certain variables can affect time - to slow it down and speed it up - leading them to believe that it is possible that time travel may become possible. Before looking at time travel, an understanding of the terms and history are required. Space, time, and light have been studied and the definitions have evolved as research has advanced. Looking at these concepts as they have been explained through history will help us understand what may be possible in the future. Portrait of Euclid Over history many people have come up with many different views about what time is and what space is, and how to understand them. Space was defined be Euclid, a Greek mathematician. He was the creator of geometry in 295 B.C.

74. Non-Euclidean Geometry -- From Wolfram MathWorld In three dimensions, there are three classes of constant curvature geometries. All are based on the first four of Euclid's postulates, but each uses its own version of the http://mathworld.wolfram.com/Non-EuclideanGeometry.html

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Interactive Demonstrations Non-Euclidean Geometry In three dimensions, there are three classes of constant curvature geometries . All are based on the first four of Euclid's postulates , but each uses its own version of the parallel postulate . The "flat" geometry of everyday intuition is called Euclidean geometry (or parabolic geometry ), and the non-Euclidean geometries are called hyperbolic geometry (or Lobachevsky-Bolyai-Gauss geometry ) and elliptic geometry (or Riemannian geometry). Spherical geometry is a non-Euclidean two-dimensional geometry. It was not until 1868 that Beltrami proved that non-Euclidean geometries were as logically consistent as Euclidean geometry SEE ALSO: Absolute Geometry Elliptic Geometry Euclid's Postulates Euclidean Geometry ... Spherical Geometry REFERENCES: Bolyai, J. "Scientiam spatii absolute veritam exhibens: a veritate aut falsitate Axiomatis XI Euclidei (a priori haud unquam decidenda) indepentem: adjecta ad casum falsitatis, quadratura circuli geometrica." Reprinted as "The Science of Absolute Space" in Bonola, R. Non-Euclidean Geometry, and The Theory of Parallels by Nikolas Lobachevski, with a Supplement Containing The Science of Absolute Space by John Bolyai.

75. Thabit Summary Gives information on background and contributions to non-euclidean geometry, spherical trigonometry, number theory and the field of statics. Was an important translator of Greek materials, including Euclid s Elements, during the Middle Ages. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Thabit.html

Al-Sabi Thabit ibn Qurra al-Harrani Click the picture above to see a larger version Thabit ibn Qurra was an important Islamic mathematician who worked on number theory, astronomy and statics. Full MacTutor biography [Version for printing] List of References (29 books/articles) Mathematicians born in the same country Honours awarded to Thabit (Click below for those honoured in this way) Lunar features Crater Thebit Other Web sites Encyclopaedia Britannica Muslim scientists Previous (Chronologically) Next Main Index Previous (Alphabetically) Next Biographies index JOC/EFR � November 1999 The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Thabit.html

76. Klein, Felix (1849-1925) -- From Eric Weisstein's World Of Scientific Biography Pl cker s assistant at Bonn who studied Analytic Geometry, describing geometry as the study of properties of figures which remain invariant under a Group of Transformations. He systemized Non-Euclidean Geometry. http://scienceworld.wolfram.com/biography/KleinFelix.html

Branch of Science Mathematicians Nationality German Klein, Felix (1849-1925) German mathematician who began his career as assistant at Bonn. Klein studied analytic geometry describing geometry as the study of properties of figures which remain invariant under a group of transformations He systemized non-Euclidean geometry and wrote a book on the icosahedron in 1884. He also worked on the development of group theory and collaborated with Lie in Erlanger Programm. He also is known in topology for the one-sided Klein bottle In addition to all his other work, he found time to write a classic history of mathematics. Additional biographies: MacTutor (St. Andrews) Bonn Fricke, R. and Klein, F. Leipzig: B. G. Teubner, 1897-1912. Klein, F. Arithmetic, Algebra, Analysis. New York: Dover, n.d. Klein, F. Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis. New York: Dover. Klein, F. Famous Problems of Elementary Geometry. New York: Chelsea, 1956. Klein, F. Gesammelte Mathematische Abhandlungen. Berlin: Springer-Verlag, 1973. Klein, F.

77. History Of Non Euclidean Geometry History of NonEuclidean Geometry http//www-groups.dcs.st-and.ac.uk/~history/ HistTopics/Non- http://www.slideshare.net/dr.f/history-of-non-euclidean-geometry

78. Euclidean Geometry Euclidean Geometry This site provides as motivational introduction to geometry in a form (Euclidean) which is more accessible than nonEuclidean geometries. http://www.geom.uiuc.edu/~crobles/hyperbolic/eucl/

Up: The Hyperbolic Geometry Exhibit Euclidean Geometry This site provides as motivational introduction to geometry in a form (Euclidean) which is more accessible than non-Euclidean geometries. Here we will establish definitions and concepts that we can apply, via analogy, to our discussion of hyperbolic geometry. This overview includes: A brief history of the parallel postulate . A familiarity with the parallel postulate is especially important as it is those geometries formed under the negation of Hilbert's parallel postulate that we define as hyperbolic geometries. Isometries of the plane. Reflection Translation Glide Reflection Rotation Isometries as products of reflections Table of Contents Up: The Hyperbolic Geometry Exhibit Created: Jul 15 1996 - Last modified: Jul 15 1996

79. Euclidean Geometry Definition Of Euclidean Geometry In The Free Online Encyclope Euclidean geometry. Study of points, lines, angles, surfaces, and solids based on Euclid 's axioms. Its importance lies less in its results than in the systematic method Euclid http://encyclopedia2.thefreedictionary.com/Euclidean geometry

80. Help With Links In CSS Popup Text Boxes - Mac Forums 7 posts 3 authors - Last post May 22, 2009I would like to create a link in the text when you roll over the rectangle with the pointer and be able to click on Euclidean geometry in http://forums.macrumors.com/showthread.php?t=697409

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