21. Erin Barker's Math Forum - A Bravenet.com Forum http//pratt.edu/~arch543p/help/noneuclidean_geometry.html . Re Re non-euclidean_geometry by tom wheat · Nov 13, 02 - 849 PM. Prof. John V. http://pub1.bravenet.com/forum/44595714/fetch/95520/

Web Hosting Guestbooks Message Forum Blogs ... Advertise Erin Barker's Math Forum If you have math questions ranging from algebra to differential calculus I can be of assistance. Return to Website Reply Forum Subject: Re: non-euclidean_geometry Name: tom wheat Date Posted: Nov 13, 02 - 7:28 PM Message: http://zebu.uoregon.edu/~js/ast123/images/omega.gif Replying to: http://members.tripod.com/~noneuclidean/applications.html http://pratt.edu/~arch543p/help/non-euclidean_geometry.html Note: the following has been abstracted from the Grolier Encyclopedia. Non-Euclidean Geometry * Comparisons of the Geometries * Historical Development * The Geometry of the Universe Comparisons of the Geometries The sum of the measures of the angles of a triangle is 180 deg in Euclidean Geometry, less than 180 deg in hyperbolic geometry, and more than 180 deg in elliptic geometry. The area of a triangle in hyperbolic geometry is proportional to the deficiency of its angle sum from 180 deg, while the area of a triangle in elliptic geometry is proportional to the excess of its angle sum over 180 deg. In Euclidean geometry all triangles have an angle sum of 180deg irrespective of area. Thus, similar triangles with different areas can exist in Euclidean geometry. This kind of occurrence is not possible in hyperbolic or elliptic geometry. In two-dimensional geometries, lines that are perpendicular to the same given line are parallel in Euclidean geometry, are neither parallel nor intersecting in hyperbolic geometry, and intersect at the pole of the given line in elliptic geometry. The appearance of the lines as straight or curved depends on the postulates for the space.

22. Upto11.net - Wikipedia Article For Article about noneuclidean_geometry is Unavailable. We could not find this article in our database. If you feel that you ve reached this page in error http://www.upto11.net/generic_wiki.php?q=non-euclidean_geometry

23. Non-Euclidean Geometry 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 http://physics.rug.ac.be/Fysica/Geschiedenis/HistTopics/Non-Euclidean_geometry.h

Non-Euclidean geometry Previous topic Next topic 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

24. AAS_Congruence AAS_Congruence D AAS Congruence AA_Similarity D Eta euclidean_geometry euclidean_geometry D Euclidean Geometry .. Noneuclidean_geometry Non-euclidean_geometry D Non-Euclidean Geometry http://www.uni-sw.gwdg.de/~hessman/rdf/math/math.txt

25. Fractal Of The Day (FotD) By Jim Muth END PARAMETER FILE= START 19.6 PARAMETERFORMULA FILE euclidean_geometry {; 3 long http://home.att.net/~Fractals_1/FotD_98-12-15.html

Fractal of the Day by Jim Muth Euclidean Geometry FOTD December 15, 1998 Fractal visionaries: A perfectly average late autumn day has just ended. The temperature of 44F 6.5C was average and the expected low tonight of 25F -4C is also just about average. In honor of the average day, I found an average fractal well actually I found it yesterday and here it is in all its one-day-later glory. I found this confused but interesting scene in a perturbed plane of the (-Z)^1.09+C Mandeloid. On the left, fractured spirals whirl like a dervish, while on the right, a perfect isoceles triangle rises from a flat plateau. I have named the picture "Euclidean Geometry" after the discipline that we so loved in High School. (At least I loved it.) And yes, I did give the colors a little boost in another program Unfortunately, the parameter file of today's image runs slow, requiring 1/2 hour on even the fastest Pentium , but salvation is at hand, for I have posted the finished GIF file to Usenet at: alt.binaries.pictures.fractals and to the Web at: http://home.att.net/~Paul.N.Lee/FotD/FotD.html

26. Kifb Browser WordNet senses for the word euclidean_geometry. nouns. elementary geometry, parabolic geometry, Euclidean geometry geometry based on Euclid s axioms eg, http://ckip.iis.sinica.edu.tw/cgi-bin/kifb/kifb_cgi?english=euclidean_geometry&l

27. Math Mutation 35 One Messed-Up Triangle You Have Probably Heard Geometry on Wikipedia /a li a href= http//en.wikipedia. org/wiki/Noneuclidean_geometry Non-Euclidean Geometry on Wikipedia /a http://www.aracnet.com/~eseligma/mm/mm35.txt

Euclidean Geometry on Wikipedia Non-Euclidean Geometry on Wikipedia Math Mutation 35: One Messed-Up Triangle You have probably heard the term "Non-Euclidean Geometry" mentioned in science fiction novels or modern physics discussions about oddly curved spaces. The name seems to connote some kind of weird form of geometry that doesn't match with our usual assumptions about the world. And indeed, non-euclidean geometry does make it possible to construct bizarre figures that seem to defy our basic assumptions. But what exactly does "non-euclidean geometry" mean? To start with, let's review what Euclidean geometry is. Basically, this is the type of geometry taught in most high schools, based on the works of the greek mathematician Euclid over two thousand years ago. Central to this system of geometry are five postulates, or basic assumptions about how the world works, which are used as the basis for proving more complex theorems. Four of the postulates are very simple: any two points determine a line, any line segment can be extended indefinitely, any line segment can be the radius of a circle, and all right angles are congruent. The fifth postulate, or "parallel postulate", is a little more complicated to state: Given a line and a point not on the line, exactly one line can be drawn through that point that is parallel to the first line. Quite a mouthful compared to the other postulates, don't you think? Because it is so much more complex to state, many mathematicians over the years tried to find ways to get rid of it entirely, and prove it based on the other postulates. If you draw a few pictures on a piece of paper, you'll soon realize that the fifth postulate has to be true. Well, drawing pictures on paper might not be the best way to figure this out, since you'll soon see that it's hard to even draw something the supposedly violates this postulate. For example, suppose that instead of the parallel postulate, *no* line can be drawn through an external point that is parallel to a given line. That would mean that if you draw two lines forming a right angle, and then a third line that also forms a right angle with the second, then the third and first lines must intersect somewhere, forming a triangle with two right angles! Otherwise, the third line would be parallel to the first, violating our modified postulate. Due to absurdities like this, for many years it was assumed that the fifth postulate must be true, and the only open question was whether it should be a postulate or a theorem. But in the nineteenth century, mathematicians were coming to a growing realization. While mathematics is often very useful for describing the real world, ultimately, it is a system for deducing the consequences of your basic assumptions, or postulates. So why not try modifying the fifth postulate, deducing the consequences, and seeing where that takes you? Mathematicians Janos Bolyoi and Nicolai Lobachevsky independently pursued this idea in the 1820s and 1830s, developing whole geometries based on modifications of the parallel postulate. At first, it looked like they were just playing some silly symbolic game, and it was several decades before their work was widely accepted. But gradually their colleagues realized that these new geometries were very usefully modelling properties of different types of surfaces. For example, let's look again at our 'absurd' example of a triangle with two right angles. How could such a thing be possible? On a flat plane, it really is absurd. But look at a globe of the Earth. Draw lines, which are actually great circle segments on the surface of a sphere, from the equator to the North Pole at the and 90 degree meridians. Each of these lines forms a 90 degree angle with the equator, yet they intersect at the pole and we really do have a triangle with two right angles! And the angle at the pole is also 90 degrees, so there are actually three right angles in this triangle. In other words, our modified geometry may not make sense when viewed on a flat plane, but is an accurate description of the properties of shapes on the surface of a sphere. And you can now amaze your friends by drawing triangles with three right angles, though they may get mad at you for defacing their globes. The development of non-euclidean geometries became vital when Albert Einstein began working on his general theory of relativity in the early 20th century. In Einstein's models, the three-dimensional space we live in is curved in the fourth dimension in regions where matter is present so these odd forms of geometry are what actually describe the real world, rather than Euclid's so-called "obvious" models! Of course, whether in a small region of the surface of a sphere, or in a small region of relativistic space, Euclid's conclusions are an excellent approximation of reality for most practical purposes. And it's still true that Euclid made an immeasurable contribution by showing how interesting and useful conclusions could be deduced from a simple set of basic postulates. But the real world is a lot more complicated than Euclid thought. And this has been your math mutation for today. References: Euclidean Geometry on Wikipedia Non-Euclidean Geometry on Wikipedia

28. AstronomyCompendium » December 15 » Code Between 1820 and 1823 he prepared a treatise on a complete system of http//en.wikipedia.org/wiki/Noneuclidean_geometry non-Euclidean geometry. http://astronomycompendium.wikispaces.com/page/code/December 15

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30. Sigma WordNet Mapping Browser member topic 106005518 euclidean_geometry, elementary_geometry, parabolic_geometry member topic 106006777 - non-euclidean_geometry http://sigma.ontologyportal.org:4010/sigma/WordNet.jsp?synset=106000644

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Top Science Mathematics > Geometry Geometry Geometry is a branch of mathematics that is concerned with the properties of configurations of geometric objects - points, (straight) lines, and circles being the most basic of these. < Prev Page Next Page > Pages: Projective Geometry Rudolf Steiner's approach. http://www.anth.org.uk/NCT/ More Info.... Riemannian Geometry A set of postscript lecture notes for a graduate level course on Riemannian geometry. http://www.maths.lth.se/matematiklu/personal/sigma/Riemann.ps More Info.... Geometry and the Imagination in Minneapolis Geometry exercises for a two-week summer workshop led by John Conway, Peter Doyle, Jane Gilman and Bill Thurston at the Geometry Center in Minneapolis, June 1991. http://www.geom.uiuc.edu/docs/doyle/mpls/handouts/handouts.html More Info.... Geometry Activities By Ephraim Fithian. http://homepage.mac.com/efithian/geometry.html More Info.... Geometry Formulas and Facts Excerpts from the 30th Edition of the CRC Standard Mathematical Tables and Formulas (1995), namely, the geometry section minus differential geometry. http://www.geom.uiuc.edu/docs/reference/CRC-formulas/

32. Non-Euclidean Web Directory. part of the MacTutor History of Mathematics archive. http//wwwhistory.mcs. st-andrews.ac.uk/history/HistTopics/References/Non-euclidean_geometry.html http://www.expocentral.com/directory/Science/Math/Geometry/Non,045Euclidean/

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33. Science : Math : Geometry : Non-Euclidean : - NoMoz.org http//wwwhistory.mcs.st-andrews.ac.uk/history/HistTopics/References/Non- euclidean_geometry.html Lock this listing - So it can t be http://www.nomoz.org/Top/Science/Math/Geometry/Non-Euclidean/

Arts Business Computers Games ... ABOUT Options: Directory Entire Web Top Science Math Geometry ... Non-Euclidean Non-Euclidean 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/ NonEuclid - Hyperbolic Geometry Article + Software Applet - NonEuclid is a software simulation offering straightedge and compass constructions in hyperbolic geometry. http://cs.unm.edu/~joel/NonEuclid/ References for Non-Euclidean Geometry - A bibliographic reference list of books and articles on non-Euclidean geometries, part of the MacTutor History of Mathematics archive. http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/References/Non-Euclidean_geometry.html 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 Spherical Trigonometry, Arc Distance Formula

34. Katalog Wartosciowych Stron : Science : Math : Geometry : Non-Euclidean http//wwwhistory.mcs.st-andrews.ac.uk/history/HistTopics/References/Non- euclidean_geometry.html. Seminar on the History of Hyperbolic Geometry http://www.katalogdmooz.radexbis.pl/index.php?c=Science/Math/Geometry/Non-Euclid

35. Aaron Gershfield - Non-Euclidean_geometry Retrieved from http//en.widikepia.ogwriki.php?wiki=Noneuclidean_geometry . Categories Geometry Non-Eudcilean geometry Hbleyropic geometry http://aarongershfield.com/wiki/wiki.php?wiki=Non-Euclidean_geometry

36. Should Britain Apologize? : From Language To Literature : TranslatorsCafe.com Fo http//en.wikipedia.org/wiki/Noneuclidean_geometry Does this mean we should stop building houses? I don t think so. http://www.translatorscafe.com/cafe/MegaBBS/thread-view.asp?threadid=9478&Messag

37. Strony Nie Tylko O Egipcie : Science : Math : Geometry : Non-Euclidean http//wwwhistory.mcs.st-andrews.ac.uk/history/HistTopics/References/Non- euclidean_geometry.html. Seminar on the History of Hyperbolic Geometry - Seminar http://www.katalog.dahar.pl/index.php?c=Science/Math/Geometry/Non-Euclidean

38. Non-euclidean Geometry - Definition Of Non-euclidean Geometry - Labor Law Talk D This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article Noneuclidean_geometry . http://dictionary.laborlawtalk.com/Non-euclidean_geometry

keywords Noun non-Euclidean geometry - geometry based on axioms different from Euclid''s By Wordnet Dictionary The term non-Euclidean geometry (also spelled: non-Euclidian geometry ) describes both hyperbolic and elliptic geometry , which are contrasted with Euclidean geometry . The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. In Euclidean geometry, if we start with a point A and a line l , then we can only draw one line through A that is parallel to l . In hyperbolic geometry, by contrast, there are infinitely many lines through A parallel to l , and in elliptic geometry, parallel lines do not exist. (See the entries on hyperbolic geometry and elliptic geometry for more information.) Another way to describe the differences between these geometries is as follows: consider two lines in a plane that are both perpendicular to a third line. In Euclidean and hyperbolic geometry, the two lines are then parallel. In Euclidean geometry, however, the lines remain at a constant distance , while in hyperbolic geometry they "curve away" from each other, increasing their distance as one moves farther from the point of intersection with the common perpendicular. In elliptic geometry, the lines "curve toward" each other, and eventually intersect; therefore no parallel lines exist in elliptic geometry.

39. Science : Math : Geometry : Non-Euclidean > Elitarna Lista Elitarnych Stron D, References for NonEuclidean Geometry. U, http//www-history.mcs.st-andrews. ac.uk/history/HistTopics/References/Non-euclidean_geometry.html http://spamy.pomorskie.pl/index.php?c=Science/Math/Geometry/Non-Euclidean

40. Introduction To Non-Euclidean Geometry - EscherMath (as illustrated by Escher’s prints Circle Limit IIV). Retrieved from http//math.slu.edu/escher/index.php/Introduction_to_Non-euclidean_geometry http://math.slu.edu/escher/index.php/Introduction_to_Non-Euclidean_Geometry

Introduction to Non-Euclidean Geometry From EscherMath Jump to: navigation search So far we have looked at what is commonly called Euclidean geometry. There are occasions where this type of geometry doesn’t get one very far. Suppose we look at this sphere and want to measure the distance between the centers of two 5-pointed stars. You can’t just use a ruler, because you can’t put the ruler flat on the sphere to measure the length. If measuring length is already tricky, how would you find area? edit Famous Early Geometers Pythagoras (ca. 540 BC) Showed that in a right triangle the sum of the squares of the sides equals the square of the hypotenuse. Plato (ca 380 BC) Laid the basis for formal geometry. His name is associated with the Platonic solids. Above the entrance to his school of Philosophy (the Academy) was engraved : “Let no one ignorant of geometry enter my doors” Aristotle (ca 340 BC) The tutor of Alexander the Great, also trained many of the great geometers of the time. Euclid (ca 300 BC) The first to write down the postulates for what is now known as Euclidean geometry. He was associated with the famous School of Alexandria. Archimedes (ca 225 BC) Pliny called him “the God of Mathematics”. He was also associated with the School of Alexandria. His name is now associated with the Archimedean solids. He was killed during the Siege of Syracuse. He was so immersed in his math that he supposedly did not notice the city being taken over by the Romans.

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