 Home  - Basic_Math - Euclidean Geometry
e99.com Bookstore
 Images Newsgroups
 21-40 of 83    Back | 1  | 2  | 3  | 4  | 5  | Next 20

 Euclidean Geometry:     more books (100)

lists with details

1. 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/

Extractions: 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.

2. Upto11.net - Wikipedia Article For
http://www.upto11.net/generic_wiki.php?q=non-euclidean_geometry

3. 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

Extractions: 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

4. 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

5. 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

6. 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

7. 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

Extractions: 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

8. 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

9. Spartanburg SC | GoUpstate.com | Spartanburg Herald-Journal
http://www.goupstate.com/apps/pbcs.dll/section?category=NEWS&template=wiki&text=

10. 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

11. SciNet Science & Technology Search, News, Articles
http://www.scinet.cc/dir/Science/Mathematics/Geometry/more10.html

12. 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/

Extractions: Google Adwords allows businesses to advertise on any number of sites, including Expo Central, regardless of the size of their budget. Ad campaigns can be set up to target based on keywords, geographical location of the visitor, and/or individual web sites. They handle the most common sizes of ads and allow text, image or video. Get the power of Google on your desktop PCs! Google Pack provides a single installer for many essential PC applications from Google and other trusted partners. These applications improve your browsing with increased speed and security, provide better searching capability for the web and for your own PC and enable you to collect and organize your personal data, including photos. Home Science Math Geometry > Non-Euclidean Directory of Web Resources @ Expo Central : Non-Euclidean

13. 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/

14. 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

15. Aaron Gershfield - Non-Euclidean_geometry
http://aarongershfield.com/wiki/wiki.php?wiki=Non-Euclidean_geometry

16. 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.

17. 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

18. Non-euclidean Geometry - Definition Of Non-euclidean Geometry - Labor Law Talk D
http://dictionary.laborlawtalk.com/Non-euclidean_geometry

Extractions: 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.

19. 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

20. Introduction To Non-Euclidean Geometry - EscherMath