1. NonEuclid - Hyperbolic Geometry Article + Software Applet Features software that simulates hyperbolic straightedge and compass constructions. Provides basic information about noneuclidean geometry. of Hyperbolic Geometry. for use in High School and Undergraduate Education. Hyperbolic Geometry is a geometry of http://math.rice.edu/~joel/NonEuclid

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. NonEuclid has moved. The new location is: http://cs.unm.edu/~joel/NonEuclid/

2. The Geometer's Sketchpad® - Euclidean And Non-Euclidean Geometry Euclidean and Noneuclidean geometry with The Geometer's Sketchpad® 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

3. Euclidean Geometry 442 euclidean geometry. Topics include foundations of euclidean geometry, finite geometries, congruence, similarities, http://www.calpoly.edu/~math/math442.html

Euclidean Geometry Math 442 Euclidean Geometry Topics include foundations of Euclidean geometry, finite geometries, congruence, similarities, polygonal regions, circles and spheres. Other topics addressed are constructions, mensuration, and the parallel postulate. This course is appropriate for prospective and in-service mathematics teachers. Prerequisite: Completion of Math 248 (Methods of Proof); high school geometry, and additional experience with methods of mathematical proof. Expected Outcomes Students should develop an appreciation of: the concept of abstracting important features of physical objects and incorporating these abstractions into a geometrical system. axiomatic systems. geometric constructions. the meaning of types of proof. the historical background of geometry. Math Undergraduate Program

4. Euclidean Geometry euclidean geometry. GT shape. BT mathematical sciences http://www.nrc.ca/irc/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]

5. Non-Euclidean Geometry A historical account with links to biographies of some of the people involved.Category Science Math Geometry Non-Euclidean......Noneuclidean geometry. Nor is Bolyai's work diminished because Lobachevskypublished 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

6. Euclidean Geometry Note the following has been abstracted from the Grolier Encyclopedia. euclidean geometry is the study of points, lines, planes, and other geometric figures, using a modified version of the assumptions of Euclid (c.300 BC). http://acnet.pratt.edu/~arch543p/help/euclidean_geometry.html

Note: the following has been abstracted from the Grolier Encyclopedia. Euclidean Geometry Euclidean geometry is the study of points, lines, planes, and other geometric figures, using a modified version of the assumptions of Euclid (c.300 BC). The most controversial assumption has been the parallel postulate: there is one and only one line that contains a given point and is parallel to a given line. The development of Euclidean geometry extends at least from 10,000 BC to the 20th century. In the 4th century BC, Plato founded an Academy in Athens, emphasized geometry, and used the five regular Polyhedrons in his explanation of the scientific phenomena of the universe. Aristotle, a student of Plato at the Academy, identified the rules for logical reasoning. The 13 books of Euclid's Elements are based on the mathematics that was considered at Plato's Academy. The geometry in the Elements was a logical system based on ten assumptions. Five of the assumptions were called common notions (Axioms, or self-evident truths), and the other five were postulates (required conditions). The resulting logical system was taken as a model for deductive reasoning and had a profound effect on all branches of knowledge. Although it has been necessary to refine the postulates as concepts of existence, continuity, order, and other aspects of Geometry have changed, the resulting geometry is still called Euclidean geometry. Modifications of Euclid's parallel postulate provide the basis for

7. Non-Euclidean Geometry References A bibliographic reference list of books and articles on non-Euclidean geometries.Category Science Math Geometry Non-Euclidean......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

8. Euclidean Geometry This site provides as motivational introduction to geometry in a form (Euclidean) which is more accessible than nonEuclidean geometries. http://www.math.ubc.ca/~robles/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

9. Euclidean Geometry 14 Apr 1999 euclidean geometry, by Chris. 16 Apr 1999 euclidean geometry, by mark heise http://mathforum.com/epigone/geometry-college/primpwummi

a topic from geometry-college Euclidean geometry post a message on this topic post a message on a new topic 14 Apr 1999 Euclidean geometry , by Chris 16 Apr 1999 Euclidean geometry , by mark heise 15 Jun 2000 Euclidean Geometry , by ryansew830 15 Jun 2000 Re: Euclidean Geometry , by Lambrou Michael 15 Jun 2000 Re: Euclidean Geometry , by Ben Saucer 15 Jun 2000 Re: Euclidean Geometry , by Ben Saucer 14 Apr 1999 RE: Euclidean geometry , by Paul Yiu The Math Forum

10. Non-Euclidean Geometry An introduction to Non-euclidean geometrywritten by Jacob Graves as an MSc project in 1997. Requirements Description An introduction to Non-euclidean geometry written by Jacob Graves as an MSc project in 1997.Category Science Math Geometry Non-Euclidean......Noneuclidean geometry. 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

11. Euclidean Geometry? By David Garcia by David Garcia Subject euclidean geometry? Author David Garcia dggarcia@prodigy.net Date 22 Apr 98 162118 0400 (EDT) HELP!!! http://mathforum.com/epigone/geometry-college/waywhethimp

Euclidean Geometry? by David Garcia reply to this message post a message on a new topic Back to geometry-college Subject: Euclidean Geometry? Author: dggarcia@prodigy.net Date: 22 Apr 98 16:21:18 -0400 (EDT) HELP!!! prove that if there exists a triangle such that the segment joining the midpoints of any two sides is equal in measure to one-half the third side then the geometry is Euclidean..... The Math Forum

12. Discussion Noneuclidean geometry. This tutorial consists of html constructed constructedapplet which demonstrates it visually. Development of euclidean geometry. http://cvu.strath.ac.uk/courseware/msc/jgraves/

Non-Euclidean Geometry This tutorial consists of html constructed pages which explain non-Euclidean geometry, and a JAVA constructed applet which demonstrates it visually Development of Euclidean Geometry Description of Euclidean Geometry basic geometry ), and the more complicated ones which relied on axiom 5 in their proof ( Euclidean geometry Problems with Euclidean Geometry Many mathematicians after Euclid (and even Euclid himself) where not comfortable with axiom five, it is quite a complicated statement and axioms are meant to be small, simple and straightforward. Axiom five is more like a theorem than an axiom, and as such it should have to be proved to be true and not assumed. The problem that Euclid and every mathematician after him found for 200 years was that it could not be proven from the 4 axioms before it. However, all the theorems that can be proved from it worked and many mathematicians were happy just to leave it. It is something that seems obviously true and yet was impossible to prove mathematically in a satisfactory way. Development of Hyperbolic Geometry Description of Hyperbolic Geometry Hyperbolic geometry is hard to describe. Its basic premise, that there can be multiple parallel lines through a point, is itself very hard to accept. In purely mathematical terms it is not so difficult. It consists of all Euclid's theorems that can be proved from the first four axioms (

13. The Ontology And Cosmology Of Non-Euclidean Geometry A philosophical essay.Category Science Math Geometry Non-Euclidean......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.

14. NRICH | Secondary Topics | Euclidean Geometry The Nrich Maths Project Cambridge, England. Mathematics resources for children, parents and teachers to enrich learning. Published on the 1st of each month. Problems, children's solutions, interactivities, games, articles, news http://nrich.maths.org/topic_tree/Euclidean_Geometry

NRICH Prime NRICH Club ... Get Printable Page December 02 Magazine Site Update News Events Problems Solutions ... Games Archive Problems Solutions Articles Inspirations ... Interactivities Web board Ask NRICH Asked NRICH NRICH Club Register Tough Nuts About Help! ... Where is NRICH? Associated Projects Maths Thesaurus MOTIVATE EuroMaths Millennium Maths ... Other Maths sites Display maths using fonts images Help Back Issues Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Bernard's Bag(P) - solutions(P) Penta Probs(P) - solutions(P) Let Me Try(P) - solutions(P) Kid's Mag(P) Play Games(P) Staff Room(P) 6 Problems - solutions 15+Challenges - solutions Articles Games LOGOland Editorial News Click on the folders to browse problem topics from the secondary site. You can then go directly to each of the problems. Top Level Algebra Analysis Calculus Combinatorics Complex Numbers Euclidean Geometry Properties of Shapes Geometry Geometry Euclidean Geometry-Coordinate Geometry-Euclidean Groups Investigation Investigations Logic Measures Mechanics Number Pre-calculus Probability Programs Properties of Shapes Statistics Trigonometry Unclassified algebra geometry-coordinate number

15. Non-Euclidean Geometry With LOGO Review of a new version of LOGO developed at Cardiff.Category Science Math Geometry Non-Euclidean......This document is a Review of Noneuclidean geometry with LOGO by Helen Sims-Coomberand 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.

16. Non-Euclidean Geometry Classic text available from the MAA.Category Science Math Geometry Non-Euclidean......Noneuclidean geometry. When non-euclidean geometry was first developed, itseemed little more than a curiosity with no relevance to the real world. 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

17. NRICH | Secondary Topics | Measures | Euclidean Geometry The Nrich Maths Project Cambridge, England. Mathematics resources for children, parents and teachers to enrich learning. Published on the 1st of each month. Problems, children's solutions, interactivities, games, articles, news http://nrich.maths.org/topic_tree/Measures/Euclidean_Geometry

@import url(../../../styles/default.css); @import url(../../../styles/minimal.css); @import url(../../../styles/speech.css); Accessibility About Register Help ... Get Printable Page NRICH Community Ask NRICH Asked NRICH Tough Nuts Associated Projects Maths Thesaurus MOTIVATE EuroMaths Millennium Maths Project ... Links Display maths using fonts images Help Back Issues Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Bernard's Bag(P) - solutions(P) Penta Probs(P) - solutions(P) Let Me Try(P) - solutions(P) Kid's Mag(P) Play Games(P) Staff Room(P) 6 Problems - solutions 15+Challenges - solutions Articles Games LOGOland Editorial News Click on the folders to browse problem topics from the secondary site. You can then go directly to each of the problems. Top Level Algebra Analysis Calculus Combinatorics Complex Numbers Geometry Geometry-Cartesian Geometry-Coordinate Geometry-Euclidean Graph Theory Groups Investigation Logic Measures Area Euclidean Geometry Pythagoras Theorem Perimeter Properties of Shapes Rates of Change Sequences Time Vectors Volume perimeter Mechanics Number Pre-calculus Probability Programs Properties of Shapes Pythagoras Sequences Space and Shape Statistics Symmetry Trigonometry Unclassified algebra number Email Nrich Return to top of page Back

18. Read This: Non-Euclidean Geometry Read This! The MAA Online book review column review of Non-euclidean geometry,by HSM Coxeter. Read This! Non-euclidean geometry by HSM Coxeter. http://www.maa.org/reviews/coxeterneg.html

Read This! The MAA Online book review column Non-Euclidean Geometry by H. S. M. Coxeter Reviewed by Robert Stolz Coxeter's Non-Euclidean Geometry begins with a wonderful historical overview of the development of non-Euclidean geometry in the first chapter. Only a few proofs are given or sketched in this chapter. They flow with the prose and play an integral part in the understanding of the beginnings of hyperbolic, spherical, elliptic and differential geometry, among others. The mathematician, as well as the non-mathematician, is able to gain insights into these various types of geometry by the end of this chapter. For the remainder of the book, the historical aspects are never far from the mathematics. Many chapters begin with an introduction which puts the contents of the chapter into historical perspective. Coxeter uses comparisons, especially to Euclidean geometry, to aid in the understanding of other geometries. An example is found in chapter 8 where descriptive geometry (described as "high school geometry with congruence and parallelism left out") is compared to projective geometry. Coxeter takes us through the developments of real projective geometry, elliptic geometry in one, two and three dimensions, descriptive geometry and hyperbolic geometry in one and two dimensions. Though he introduces the elliptic metric "by means of absolute polarity" and could have introduced the hyperbolic metric in a similar fashion, he chose to "reverse the process" in order to "follow the historical development more closely". This not only gives the reader a different perspective, but once again reminds us of the historical perspective.

19. Non-Euclidean Geometry Noneuclidean geometry. see also Non-euclidean geometry. Anderson, James W. HyperbolicGeometry. Non-euclidean geometry, 6th ed. Washington, DC Math. Assoc. http://www.ericweisstein.com/encyclopedias/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.

20. Key Curriculum Press | Advanced Euclidean Geometry Supplementals Advanced euclidean geometry. Advanced euclidean geometryExcursions for Secondary Teachers and Students. Alfred S http://www.keypress.com/catalog/products/supplementals/Prod_AdvancedEuclidean.ht

Home Customer Service Ordering Information Contact Us ... Site Map Resource Centers The Geometer's Sketchpad Fathom Dynamic Statistics Discovering Algebra Discovering Geometry ... IMP Professional Development General Information One-Day Workshops Three-Day Workshops Summer Institutes ... Custom Workshops General Information Ordering Information Customer Service Contact Us Meet Our Authors ... Press Room Other Key Sites Key College Publishing KCP Technologies Keymath.com Supplementals Advanced Euclidean Geometry Advanced Euclidean Geometry: Excursions for Secondary Teachers and Students Alfred S. Posamentier, City College, The City University of New York Advanced Euclidean Geometry provides a thorough review of the essentials of the high school geometry course and then expands those concepts to advanced Euclidean geometry, to give teachers more confidence in guiding student explorations and questions. The text contains hundreds of illustrations created with software and includes a Windows/Macintosh CD-ROM containing over 100 interactive sketches. User must have access to Advanced Euclidean Geometry 250 pages (Includes CD-ROM) Legal notices is a registered trademark of Key Curriculum Press.

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