1. Non-Euclidean Geometry An introduction to Noneuclidean geometry written by Jacob Graves as an MSc project in 1997. http://cvu.strath.ac.uk/courseware/info/noneucgeom.html

Non-Euclidean Geometry Description: An introduction to Non-Euclidean Geometry written by Jacob Graves as an MSc project in 1997. Requirements: You must be using a browser that supports Java and have Java enabled. Begin Non-Euclidean Geometry

2. Non-Euclidean (hyperbolic) Geometry Applet Noneuclidean geometry. This applet allows click-and-drag drawing in the Poincare model of the (hyperbolic) non-Euclidean plane, and also motion. The circular arcs drawn by mouse drags are the geodesics ( in Euclidean space; however, in this non-euclidean geometry the Euclidean picture of it makes terms of the non-euclidean geometry, despite appearances, these motions preserve http://www.math.umn.edu/~garrett/a02/H2.html

Non-Euclidean Geometry This applet allows click-and-drag drawing in the Poincare model of the (hyperbolic) non-Euclidean plane, and also motion . The circular arcs drawn by mouse drags are the geodesics (straight lines) in this model of geometry. In "move" mode, click-and-drag slides the whole picture in the direction of the mouse drag. This is analogous to ordinary "sliding" of objects in Euclidean space; however, in this non-Euclidean geometry the Euclidean picture of it makes things appear to become smaller as they move toward the edge. But, in fact, in terms of the non-Euclidean geometry, despite appearances, these motions preserve distances and angles. The preservation of angles should be detectable if one keeps in mind that the angles are angles between the arcs of circles at their point of intersection. Since the bounding circle is "infinitely far away", the motion of the picture does not exactly parallel the mouse drag motion, but instead moves about the same non-Euclidean distance as the Euclidean distance moved by the mouse. So the picture will appear to lag behind the mouse. The University of Minnesota explicitly requires that I state that "The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by the University of Minnesota."

3. The Ontology And Cosmology Of Non-Euclidean Geometry The Ontology and Cosmology of Noneuclidean geometry. Though Reserved. The Ontology and Cosmology of Non-euclidean geometry, Note. 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 a 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.

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

KSEG Free Interactive Geometry Software Update: April 29 Italian Translation Updated Giancarlo Bassi has sent an update to his translation. Download the new qm file and the help file if you use Italian. Update: April 20 Welsh Translation Kevin Donnelly sent me a translation of KSEG into Welsh. The .qm file is here. Update: April 18 KSEG 0.4 out! The new features are view panning, zooming, exporting to an image (including high-quality antialiased output), as well as one-step construction of segment, ray, or arc endpoints and circle or arc centers (this is useful for Constructions). I have also changed printing (I think for the better :)it is now more accurate and prints the view, rather than scaling everything. High-quality 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. KSEG can be used in the classroom, for personal exploration of geometry, or for making high-quality figures for LaTeX. It is very fast, stable, and the UI has been designed for efficiency and consistency. I can usually make a construction in KSEG in less than half the time it takes me to do it with similar programs. Despite the name, it is Qt based and does not require KDE to run.

5. Non-Euclidean Geometry Classic text available from the MAA. http://www.maa.org/pubs/books/nec.html

Non-Euclidean Geometry H.S.M. Coexeter Series: Spectrum A classic is back in print! No living geometer writes more clearly and beautifully about difficult topics than world famous professor H. S. M. Coxeter. When non-Euclidean geometry was first developed, it seemed little more than a curiosity with no relevance to the real world. Then to everyone's amazement, it turned out to be essential to Einstein's general theory of relativity! Coxeter's book has remained out of print for too long. Hats off to the MAA for making this classic available once more. -Martin Gardner Coxeter's geometry books are a treasure that should not be lost. I am delighted to see "Non-Euclidean Geometry" back in print. -Doris Schattschneider H. S. M. Coxeter's classic book on non-Euclidean geometry was first published in 1942, and enjoyed eight reprintings before it went out of print in 1968. The MAA is delighted to be the publisher of the sixth edition of this wonderful book, updated with a new section 15.9 on the author's useful concept of inversive distance. Throughout most of this book, non-Euclidean geometries in spaces of two or three dimensions are treated as specializations of real projective geometry in terms of a simple set of axioms concerning points, lines, planes, incidence, order and continuity, with no mention of the measurement of distances or angles. This synthetic development is followed by the introduction of homogeneous coordinates, beginning with Von Staudt's idea of regarding points as entities that can be added or multiplied. Transformations that preserve incidence are called colineations. They lead in a natural way to elliptic isometries or "congruent transformations". Following a recommendation by Bertrand Russell, continuity is described in terms of

6. Non-Euclidean Geometry With LOGO This document is a Review of Noneuclidean geometry with LOGO by Helen Sims-Coomber and Ralp Martin prepared by Pam Bishiop of CTI Mathematics which appeared http://www.bham.ac.uk/ctimath/reviews/logo.html

Non-Euclidean Geometry with LOGO Helen Sims-Coomber and Ralph Martin, Department of Computing Mathematics, University of Wales, College of Cardiff. This article describes a version of LOGO currently under development at Cardiff that uses non-Euclidean geometry. The ultimate aim is that a final version could be given to mathematics students to help them visualise non-Euclidean geometry. The programming language LOGO with its Turtle Graphics facilities is well known in educational circles. The turtle is a small triangular pointer that appears on the display screen. Simple commands are used to move it (FORWARD or BACK) and rotate it (LEFT or RIGHT); it leaves a trail behind it as it moves around the screen. Using the turtle to draw in this way provides an easy introduction to computing for young children, but LOGO is equally suitable for older students. Many sophisticated areas of mathematics (including topology, relativity and differential geometry) can be explored through the use of turtle graphics (see [1]). The system under development at Cardiff is specifically designed for exploring non-Euclidean geometry. Euclidean geometry is the kind taught in schools. Most students will be familiar with the properties of Euclidean parallel lines; given a straight line, L, and a point, P, not on the line, we can construct exactly one line through P parallel to L. The distinguishing feature of non-Euclidean geometry is the behaviour of parallel lines. There are two main types of non-Euclidean geometry: hyperbolic geometry, in which more than one line parallel to L can be constructed, and elliptic geometry, in which parallel lines do not exist at all.

7. The Geometer's Sketchpad® - Euclidean And Non-Euclidean Geometry Conference talk by Scott Steketee with downloadable sketches. http://www.keypress.com/sketchpad/talks/Euc_Wien98/

Home Resources Technical Support JavaSketchpad ... Site Map Resources Bibliography 101 Project Ideas Sketchpad Links Sketchpad for Cassiopeia ... TI Graphing Calculators JavaSketchpad About JSP JSP Gallery JSP Download Center JSP Developer's Grammar ... JSP Links Instructor Resources Download Instructor's Evaluation Edition Workshop Guide Professional Development Recent Talks Technical Support FAQ Product Updates Tech Support Request Form General Information Product Info How to Order Curriculum Modules Other Key Sites Key Curriculum Press Key College Publishing KCP Technologies Keymath.com 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

8. Non-Euclidean Geometry - Mathematics And The Liberal Arts A resource for student research projects and for teachers interested in using the history of mathematics in their courses. http://math.truman.edu/~thammond/history/NonEuclideanGeometry.html

Non-Euclidean Geometry - Mathematics and the Liberal Arts See the page The Parallel Postulate . To expand search, see Geometry . Laterally related topics: Symmetry Analytic Geometry Trigonometry Pattern ... Tilings , and The Square The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews , published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet Make comment on this category Make comment on this project

9. Non-Euclidean Geometry Noneuclidean geometry. Nor is Bolyai s work diminished because Lobachevsky published a work on non-euclidean geometry in 1829. Neither http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Non-Euclidean_geometry.html

Non-Euclidean geometry Geometry and topology index History Topics Index 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

10. Non-Euclidean Geometry -- From Eric Weisstein's Encyclopedia Of Scientific Books From theTreasureTroves collection. http://www.treasure-troves.com/books/Non-EuclideanGeometry.html

Non-Euclidean Geometry see also Non-Euclidean Geometry Anderson, James W. Hyperbolic Geometry. New York: Springer-Verlag, 1999. 230 p. $?. Bonola, Roberto. Non-Euclidean Geometry, and The Theory of Parallels by Nikolas Lobachevski, with a Supplement Containing The Science of Absolute Space by John Bolyai. New York: Dover, 1955. 268 p., 50 p., and 71 p. Borsuk, Karol. Foundations of Geometry: Euclidean and Bolyai-Lobachevskian Geometry. Projective Geometry. Amsterdam, Netherlands: North-Holland, 1960. 444 p. Carslaw, H.S. The Elements of Non-Euclidean Plane Geometry and Trigonometry. London: Longmans, 1916. Coxeter, Harold Scott Macdonald. Non-Euclidean Geometry, 6th ed. Washington, DC: Math. Assoc. Amer., 1988. 320 p. $30.95. Greenberg, Marvin J. Euclidean and Non-Euclidean Geometries: Development and History, 3rd ed. San Francisco, CA: W.H. Freeman, 1994. $?. Iversen, Birger. Hyperbolic Geometry. Cambridge, England: Cambridge University Press, 1992. 298 p. $?. Manning, Henry Parker. Introductory Non-Euclidean Geometry.

11. Non-Euclidean Geometry References References for Noneuclidean geometry. R Bonola, Non-euclidean geometry A Critical and Historical Study of its Development (New York, 1955). http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/References/Non-Euclidean_geo

References for Non-Euclidean geometry R Bonola, Non-Euclidean Geometry : A Critical and Historical Study of its Development (New York, 1955). T R Chandrasekhar, Non-Euclidean geometry from early times to Beltrami, Indian J. Hist. Sci. N Daniels,Thomas Reid's discovery of a non-Euclidean geometry, Philos. Sci. F J Duarte, On the non-Euclidean geometries : Historical and bibliographical notes (Spanish), Revista Acad. Colombiana Ci. Exact. Fis. Nat. H Freudenthal, Nichteuklidische Geometrie im Altertum?, Archive for History of Exact Sciences J J Gray, Euclidean and non-Euclidean geometry, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 877-886. J J Gray, Ideas of Space : Euclidean, non-Euclidean and Relativistic (Oxford, 1989). J J Gray, Non-Euclidean geometry-a re-interpretation, Historia Mathematica J J Gray, The discovery of non-Euclidean geometry, in Studies in the history of mathematics (Washington, DC, 1987), 37-60. T Hawkins, Non-Euclidean geometry and Weierstrassian mathematics : the background to Killing's work on Lie algebras

12. Euclidean Geometry From MathWorld euclidean geometry from MathWorld A geometry in which Euclid's fifth postulate holds, sometimes also called parabolic geometry. Twodimensional euclidean geometry is called plane geometry, and http://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/EuclideanGeometry.

13. Physics With Transforms JoseTaniaRodolfo@hotmail.com A new method of correlating physics formulas to derive one formula from a related formula using euclidean geometry to represent the interrelationship of physics formulas. http://physicstransforms.tripod.com/

document.isTrellix = 1; var cm_role = "live" var cm_host = "tripod.lycos.com" var cm_taxid = "/memberembedded" Check out the NEW Hotbot Tell me when this page is updated Physics with Transforms JoseTaniaRodolfo@hotmail.com Table of Contents Explanation the Transform Physics with Transforms side 1 ... About Me e-mail address: JoseTaniaRodolfo@hotmail.com Click to receive e-mail when this page is updated Powered by NetMind s="na";c="na";j="na";f=""+escape(document.referrer)

14. Euclidean & Non Euclidean Geometry EUCLIDEAN NONeuclidean geometry. Most of us are familiar with the term geometry. The only real difference between one high-schooler and another's opinion on geometry is the connotation of the word. try to clarify exactly what euclidean geometry is. euclidean geometry consists of all the known http://www.dsdk12.net/project/euclid/GEOEUC~1.HTM

Most of us are familiar with the term geometry. The only real difference between one high-schooler and another's opinion on geometry is the connotation of the word. It may be interesting to know for someone whose knowledge on geometery comes only fro m the course they took in high school that what they learned to be a straight line is curved in other types of geometry. These types of geometry are called non-Euclidean because they do not follow the rules that were concretely established in Euclid's El ements. Before I address the topic of Euclidean geometry, I'd like to try to clarify exactly what Euclidean geometry is. Euclidean geometry consists of all the known rules, definitions, propositions, and thereoms before and up to the time of the Greek scholar Euclid. Euclid compiled all of this information into a thirteen-volume set of books entitled Euclid's Elements. The books start with those studies of Pythagoras. Of course, Pythagoras' most known contribution to geometry is the Pythagoram Thereom - a2 + b2 = c2. Other mathematicians whose studies are found in the Elements are Apollonius, who contributed to conics, and Archimedes, who gave his knowledge of mechanics and the areas of circles. I think the best way to describe Euclidean geometry is to say that it is all based on the daily human perception of the world, and with the relationship between objects. For example, if I am standing on a floor in school, it is a given that the floor is level. If I am looking at a desktop parallel to the floor then one can only draw the assumption that the desktop is also level. I drew all of these assumptions based on the location and relativity of the objects near me. If I take a ruler and lay it on its side on a desktop, and they align, the desktop is straight. Straight is known also as a 180 degree angle.

15. NonEuclid - Hyperbolic Geometry Article & Applet Plane. Basic Concepts What is Noneuclidean geometry - euclidean geometry, Spherical Geometry, Hyperbolic Geometry, and others. http://www.cs.unm.edu/~joel/NonEuclid/

NonEuclid is Java Software for Interactively Creating Ruler and Compass Constructions in both the for use in High School and Undergraduate Education. Hyperbolic Geometry is a geometry of Einstein's General Theory of Relativity and Curved Hyperspace. Authors: Joel Castellanos - Graduate Student, Dept. of Computer Science , University of New Mexico Joe Dan Austin - Associate Professor, Dept. of Education, Rice University Ervan Darnell - Graduate Student, Dept. of Computer Science, Rice University Italian Translation by Andrea Centomo, Scuola Media "F. Maffei", Vicenza Funding for NonEuclid has been provided by: CRPC, Rice University Institute for Advanced Study / Park City Mathematics Institute Run NonEuclid Applet (click button below): If you do not see the button above, it means that your browser is not Java 1.3.0 enabled. This may be because: 1) you are running a browser that does not support Java 1.3.0, 2) there is a firewall around your Internet access, or 3) you have Java deactivated in the preferences of your browser. Both and Microsoft Internet Explorer 6.0

16. NonEuclid: Non-Euclidean Geometery NonEuclid What is Noneuclidean geometry. 1.1 euclidean geometry he Geometry with which we are most familiar is called euclidean geometry. http://www.cs.unm.edu/~joel/NonEuclid/noneuclidean.html

What is Non-Euclidean Geometry 1.1 Euclidean Geometry: he 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 was of great practical value to the ancient Greeks as they used it (and we still use it today) to design buildings and survey land. 1.2 Spherical Geometry: non-Euclidean Geometry is any geometry that is different from Euclidean Geometry. One of the most useful non-Euclidean geometries is Spherical Geometry which describes the surface of a sphere. Spherical Geometry is used by pilots and ship captains as they navigate around the world. Working in Spherical Geometry has some non intuitive results. For example, did you know that the shortest flying distance from Florida to the Philippine Islands is a path across Alaska? The Philippines are South of Florida - why is flying North to Alaska a short-cut? The answer is that Florida, Alaska, and the Philippines are collinear locations in Spherical Geometry (they lie on a "Great Circle"). Another odd property of Spherical Geometry is that

17. Thabit Gives information on background and contributions to noneuclidean 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 Born: 826 in Harran, Mesopotamia (now Turkey) Died: 18 Feb 901 in Baghdad, (now in Iraq) Click the picture above to see a larger version Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Thabit ibn Qurra was a native of Harran and a member of the Sabian sect. The Sabian religious sect were star worshippers from Harran often confused with the Mandaeans (as they are in [1]). Of course being worshipers of the stars meant that there was strong motivation for the study of astronomy and the sect produced many quality astronomers and mathematicians. The sect, with strong Greek connections, had in earlier times adopted Greek culture, and it was common for members to speak Greek although after the conquest of the Sabians by Islam, they became Arabic speakers. There was another language spoken in southeastern Turkey, namely Syriac, which was based on the East Aramaic dialect of Edessa. This language was Thabit ibn Qurra's native language, but he was fluent in both Greek and Arabic. Some accounts say that Thabit was a money changer as a young man. This is quite possible but some historians do not agree. Certainly he inherited a large family fortune and must have come from a family of high standing in the community.

18. Forumgeom Freeaccess electronic journal about elementary euclidean geometry. http://forumgeom.fau.edu/

Editorial Board About the Journal Instructions to Authors Submission of Papers Frequently Asked Questions 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 30, 2004.

19. Non-Euclidean Geometry Noneuclidean geometry. Geometry and topology index. History Topics Index. In about 300 BC Euclid wrote The Elements, a book which was to become one of the most famous books ever written. hypothesis of the acute angle and derived many theorems of non-euclidean geometry without realising what he was doing http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Non-Euclidean_geometry.ht

Non-Euclidean geometry Geometry and topology index History Topics Index 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

20. Non-Euclidean Geometry -- From MathWorld Noneuclidean geometry. It was not until 1868 that Beltrami proved that non-Euclidean geometries were as logically consistent as euclidean geometry. http://mathworld.wolfram.com/Non-EuclideanGeometry.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Geometry Non-Euclidean Geometry 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 Absolute Geometry Elliptic Geometry Euclid's Postulates ... search . "Welcome to the Non-Euclidean Geometry Homepage." http://members.tripod.com/~noneuclidean/

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