41. 51M05: General Euclidean Geometry Introduction. We use this category to hold files concerning nonplanareuclidean geometry topics. The files on this page are more http://www.math.niu.edu/~rusin/known-math/index/51M05.html

Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields POINTERS: Texts Software Web links Selected topics here 51M05: General Euclidean geometry Introduction We use this category to hold files concerning non-planar Euclidean geometry topics. The files on this page are more like samples of the techniques one may use for 3D problems (or n-dimensional: much of what is here is really independent of the number of dimensions.) History Applications and related fields The actions of the point groups among the crystallographic groups are the basis for the construction of the Platonic solids and the regular divisions of the sphere in R^3. For more information, consult the polyhedra and spheres pages. Subfields Parent field: 51M - Real and Complex Geometry Textbooks, reference works, and tutorials Software and tables For computational geometry see 68U05: Computer Graphics Pointer to Mesa , a 3-D graphics library (similar to OpenGL). Other web sites with this focus Selected topics at this site A FAQ: Shortest distance between two lines in 3-dimensional space.

42. Non-Euclidean (hyperbolic) Geometry Applet Noneuclidean geometry. But, in fact, in terms of the non-euclidean geometry,despite appearances, these motions preserve distances and angles. 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. garrett@math.umn.edu

43. MATH 6118: Non-euclidean Geometry MATH 6118 090 Non-euclidean geometry SPRING 2002. Chapter 2 EuclideanGeometry from January 9, 16, and 23, available as one PDF file. http://www.math.uncc.edu/~droyster/courses/spring02/

MATH 6118 - 090 Non-Euclidean Geometry SPRING 2002 Dr. David C. Royster droyster@email.uncc.edu General Class Information Class Syllabus . Click here for a PDF version for printing. List of topics to be covered each day.Click here for a PDF version for printing. Class Worksheets and Lecture Notes Chapter 1: History from January 9, 2002, available as a PDF file. Chapter 2: Euclidean Geometry from January 9, 16, and 23, available as one PDF file. This is the amended file that contains the graphics. Chapter 3: Euclidean Constructions from January 30, available as one PDF file. Lab Day: Introduction to Geometer's Sketchpad from February 6, available as a PDF file. Chapter 4: Introduction to Hyperbolic Geometry from February 13, available as one PDF file. Chapter 5: Poincare Models of Hyperbolic Geometry from February 27, available as one PDF file. Chapter 6: Hyperbolic Analytic Geometry from March 27, available as one PDF file. Assignments Assignment 2 Homework 2 Solutions: PDF file Assignment 3 Homework 3 Solutions: PDF file Geometer's Sketchpad Activities Assignment 4 Solutions using JavaSketchPad Assignment 6 (What happened to Assignment 5?)

44. Non-Euclidean Geometry Seminar Seminar notes by Greg Schreiber.Category Science Math Geometry NonEuclidean...... We began with an exposition of euclidean geometry, first from Euclid's perspective(as given in his Elements) and then from a modern perspective due to Hilbert http://www.math.columbia.edu/~pinkham/teaching/seminars/NonEuclidean.html

Seminar on the History of Hyperbolic Geometry Greg Schreiber In this course we traced the development of hyperbolic (non-Euclidean) geometry from ancient Greece up to the turn of the century. This was accomplished by focusing chronologically on those mathematicians who made the most significant contributions to the subject. We began with an exposition of Euclidean geometry, first from Euclid's perspective (as given in his Elements) and then from a modern perspective due to Hilbert (in his Foundations of Geometry). Almost all criticisms of Euclid up to the 19th century were centered on his fifth postulate, the so-called Parallel Postulate.The first half of the course dealt with various attempts by ancient, medieval, and (relatively) modern mathematicians to prove this postulate from Euclid's others. Some of the most noteworthy efforts were by the Roman mathematician Proclus, the Islamic mathematicians Omar Khayyam and Nasir al-Din al-Tusi, the Jesuit priest Girolamo Sacchieri, the Englishman John Wallis, and the Frenchmen Lambert and Legendre. Each one gave a flawed proof of the parallel postulate, containing some hidden assumption equivalent to that postulate. In this way properties of hyperbolic geometry were discovered, even though no one believed such a geometry to be possible. References: Four general references were used throughout this course: Bonola's Non-Euclidean Geometry, Jeremy Gray's Ideas of Space, Greenberg's Euclidean and Non-Euclidean Geometries, and McCleary's Geometry from a Differential Viewpoint. In addition, original works of these mathematicians were used whenever possible, as well as biographies of them. These books included Euclid's Elements, Hilbert's Foundations of Geometry, Proclus's A Commentary on the First Book of Euclid's Elements, Saccheri's Euclid Vindicated, Bolyai's Science of Absolute Space, Lobachevskii's Geometrical Researches in the Theory of Parallels, and Riemann's "On the Hypotheses Which Lie at the Foundations of Geometry," among others.

45. Euclidean & Non Euclidean Geometry EUCLIDEAN NONeuclidean geometry. Before I address the topic of Euclideangeometry, I'd like to try to clarify exactly what euclidean geometry is. 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.

46. NRICH Secondary Topics Measures Euclidean Geometry GeometryCoordinate + Geometry-Euclidean + Graph Theory + Groups + Investigation+ Logic - Measures + Area - euclidean geometry + Pythagoras Theorem http://www.nrich.maths.org.uk/topic_tree/Measures/Euclidean_Geometry/

47. NRICH Secondary Topics Euclidean Geometry Top Level + 3D + Algebra + Analysis + Calculus + Combinatorics + Complex Numbers euclidean geometry + 3D + Properties of Shapes + Geometry + Geometry http://www.nrich.maths.org.uk/topic_tree/Euclidean_Geometry/

48. Applications Of Non-Euclidean Geometry The Applications Of Noneuclidean geometry. Table of Contents. 1.Where EuclideanGeometry Is Wrong. 5.Celestial Mechanics. Where euclidean geometry Is Wrong. http://members.tripod.com/~noneuclidean/applications.html

The Applications Of Non-Euclidean Geometry Table of Contents Where Euclidean Geometry Is Wrong The Theory of General Relativity Spherical Geometry Celestial Mechanics Where Euclidean Geometry Is Wrong Since Euclid first published his book Elements in 300 B.C. it has remained remarkably correct and accurate to real world situations faced on Earth. The one problem that some find with it is that it is not accurate enough to represent the three dimensional universe that we live in. It has been argued that Euclidean Geometry, while good for architecture and to survey land, when it is moved into the third dimension, the postulates do not hold up as well as those of hyperbolical and spherical geometry. Both of those geometries hold up to a two dimensional world, as well as the third dimension. Back To Top Cosmology - Cosmology is the study of the origin, constitution, structure, and evolution of the universe. Back To Top The Theory of General Relativity Einstein's Theory Of General Relativity is based on a theory that space is curved. The cause is explained by the theory itself. Einstein's General Theory of Relativity can be understood as saying that: Matter and energy distort space The distortions of space affect the motions of matter and energy.

49. What Is Non-Euclidean Geometry? What is Noneuclidean geometry? Non l. Other than this difference, non-Euclideangeometry holds to the rest of Euclid's postulates http://members.tripod.com/~noneuclidean/whatisit_main.html

What is Non-Euclidean Geometry? Non Euclidean geometry is geometry that refutes Euclid's fifth postulate: Given a point p and a line l, there is exactly one line through p that is parralel to l. Other than this difference, non-Euclidean geometry holds to the rest of Euclid's postulates: Two points determine a line A straight line can be extended with no limitation Given a point and a distance a circle can be drawn with the point as center and the distance as radius All right angles are equal To see how this movement got started, please see the History section. Now, there are two ways that you can refute this. One way is to say: Given a line l and point p, there are no lines parralel to l through p. This leads to Spherical/Elliptical Geometry The other is to say: Given a line l and a point p, there are infinite lines through p parralel to l. This leads to Hyperbolic Geometry This page has been visited Bibliography Here are some pages I found most useful in creating this webpage: The Math Forum Homepage The Geometry Center Homepage NonEuclid Homepage The Hyperbolic Geometry Exhibit ... The math.sci newsgroup accessible via Dejanews The geom.pre-college newsgroup

50. Non-Euclidean Geometry Noneuclidean geometry. Despite element. This course intends to be a businesslikeintroduction to non-euclidean geometry for nonexperts. http://www.mccme.ru/mathinmoscow/courses/noneucld.htm

Non-Euclidean Geometry Despite the fact that non-Euclidean geometry has found its use in numerous applications (the most striking example being 3-dimensional topology), it has retained a kind of exotic and romantic element. This course intends to be a businesslike introduction to non-Euclidean geometry for nonexperts. Prerequisites from linear algebra, point set topology, group theory, metric spaces. Axioms for plane geometry. The inversive models. The hyperboloid and the Klein model. The geometry of the sphere. Some computations in the hyperbolic plane and on the sphere. Hyperbolic isometries. Convex polygons. Isoperimetric inequality in non-Euclidean geometry. Hyperbolic surfaces. Books: I. S. Iversen, Hyperbolic geometry , Cambridge Univ. Press 1993 H. S. Coxeter, Non-Euclidean Geometry , Toronto Univ. Press, 1957 Math in Moscow Home Page List of courses

51. Euclidean Geometry euclidean geometry. From Sir Thomas L. Heath's translation of Euclid'sElements Postulates. Let the following be postulated 1. To http://www.mtholyoke.edu/courses/jmorrow/euclid.html

HOME COURSE INFORMATION Euclidean Geometry From Sir Thomas L. Heath's translation of Euclid's Elements Postulates Let the following be postulated: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to one another. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Some definitions from the same source: 1. A point is that which has no part. 2. A line is breadthless length. 3. The extremities of a line are points 4. A straight line is a line which lies evenly with the the points on itself. 5. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. 6. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right Common Notions 1. Things which are equal to the same thing are also equal to one another.

52. Key College Publishing: Posamentier/Advanced Euclidean Geometry Advanced euclidean geometry Excursions for Students and Teachers WithIllustrations in The Geometer’s Sketchpad ®. Alfred S. Posamentier http://www.keycollege.com/catalog/titles/advanced_euclidean_geom.html

Home Customer Service Ordering Information Contact Us ... Site Map Product Information Mathematics Products Statistics Products Software Products Partners in Publishing ... Author Web Sites Resource Centers The Geometer's Sketchpad Fathom Dynamic Statistics The Heart of Mathematics DevMAP ... Mathematics Across the Curriculum Project Resource Store for Future Teachers Supplemental Materials Core Resource and Secondary Texts Software Resources General Information Professional Development Customer Service Contact Us Company Profile ... Employment Opportunities Other Key Sites Key Curriculum Press Keymath.com KCP Technologies Advanced Euclidean Geometry: Excursions for Students and Teachers With Illustrations in Alfred S. Posamentier, City College, The City University of New York Advanced Euclidean Geometry fills this void by providing a thorough review of the essentials of the high school geometry course and then expanding those concepts to advanced Euclidean geometry, to give teachers more confidence in guiding student explorations and questions. The text contains hundreds of illustrations created in Sketchpad This title is available through your college bookstore and is also available packaged with a specially priced Student Bundle Package of version 4. Contact your Key College Publishing sales representative.

53. Key College Publishing: Advanced Euclidean Geometry Print This Page, Advanced euclidean geometry Excursions for Studentsand Teachers With Illustrations in The Geometer’s Sketchpad ®. http://www.keycollege.com/catalog/printer_friendly/advanced_euclidean_geom.html

Print This Page Advanced Euclidean Geometry: Excursions for Students and Teachers With Illustrations in Alfred S. Posamentier, City College, The City University of New York Advanced Euclidean Geometry fills this void by providing a thorough review of the essentials of the high school geometry course and then expanding those concepts to advanced Euclidean geometry, to give teachers more confidence in guiding student explorations and questions. The text contains hundreds of illustrations created in Sketchpad This title is available through your college bookstore and is also available packaged with a specially priced Student Bundle Package of version 4. Contact your Key College Publishing sales representative. Instructor Resources will be available to qualified adopters and will also be available online. Publication will follow publication of the text. Advanced Euclidean Geometry and Dynamic Geometry are registered trademark of Key Curriculum Press. Sketchpad is a trademark of Key Curriculum Press.

54. Non-Euclidean Geometry Noneuclidean geometry. Introduction Unlike other alone. However, Euclideangeometry was defined as using all five of the axioms. The http://www.geocities.com/CapeCanaveral/7997/noneuclid.html

Non-Euclidean Geometry Introduction: Unlike other branches of math, geometry has been connected with two purposes since the ancient Greeks. Not only is it an intellectual discipline, but also, it has been considered an accurate description of our physical space. However in order to talk about the different types of geometries, we must not confuse the term geometry with how physical space really works. Geometry was devised for practical purposes such as constructions, and land surveying. Ancient Greeks, such as Pythagoras (around 500 BC) used geometry, but the various geometric rules that were being passed down and inherited were not well connected. So around 300 BC, Euclid was studying geometry in Alexandria and wrote a thirteen-volume book that compiled all the known and accepted rules of geometry called The Elements, and later referred to as Euclid’s Elements. Because math was a science where every theorem is based on accepted assumptions, Euclid first had to establish some axioms with which to use as the basis of other theorems. He used five axioms as the 5 assumptions, which he needed to prove all other geometric ideas. The use and assumption of these five axioms is what it means for something to be categorized as Euclidean geometry, which is obviously named after Euclid, who literally wrote the book on geometry. The first four of his axioms are fairly straightforward and easy to accept, and no mathematician has ever seriously doubted them. The first four of Euclid’s axioms are:

55. Geomview Manual - Non-Euclidean Geometry Noneuclidean geometry. Geomview supports hyperbolic and sphericalgeometry as well as euclidean geometry. The three buttons at the http://www.geomview.org/docs/html/geomview_70.html

Go to the first previous next last section, table of contents Non-Euclidean Geometry Geomview supports hyperbolic and spherical geometry as well as Euclidean geometry. The three buttons at the bottom of the Main panel are for setting the current geometry type. In each of the three geometries, three models are supported: Virtual Projective , and Conformal . You can change the current model with the Model browser on the Camera panel. Each Geomview camera has its own model setting. The default model is all three spaces is Virtual . This corresponds to the camera being in the same space as, and moving under the same set of transformations as, the geometry itself. In Euclidean space Virtual is the most useful model. The other models were implemented for hyperbolic and spherical spaces and just happen to work in Eucldiean space as well: Projective is the same as Virtual but by default displays the unit sphere, and Conformal displays everything inverted in the unit sphere. In hyperbolic space, the Projective model setting gives a view of the projective ball model of hyperbolic 3-space imbedded in Euclidean space. The camera is initially outside the unit ball. In this model, the camera moves by Euclidean motions and geometry moves by hyperbolic motions. Conformal model is similar but shows the conformal ball model instead.

56. Geomview Manual - Non-Euclidean Geometry Go to the previous, next section. Noneuclidean geometry. Geomview supportshyperbolic and spherical geometry as well as euclidean geometry. http://www.geom.uiuc.edu/software/geomview/docs/geomview_10.html

Go to the previous next section. Non-Euclidean Geometry Geomview supports hyperbolic and spherical geometry as well as Euclidean geometry. The three buttons at the bottom of the Main panel are for setting the current geometry type. In each of the three geometries, three models are supported: Virtual Projective , and Conformal . You can change the current model with the Model browser on the Camera panel. Each Geomview camera has its own model setting. The default model is all three spaces is Virtual . This corresponds to the camera being in the same space as, and moving under the same set of transformations as, the geometry itself. In Euclidean space Virtual is the most useful model. The other models were implemented for hyperbolic and spherical spaces and just happen to work in Eucldiean space as well: Projective is the same as Virtual but by default displays the unit sphere, and Conformal displays everything inverted in the unit sphere. In hyperbolic space, the Projective model setting gives a view of the projective ball model of hyperbolic 3-space imbedded in Euclidean space. The camera is initially outside the unit ball. In this model, the camera moves by Euclidean motions and geometry moves by hyperbolic motions. Conformal model is similar but shows the conformal ball model instead.

57. Euclidean Geometry Resources euclidean geometry resources. Recommended References. see index fortotal category for your convenience Best Retirement Spots http://futuresedge.org/mathematics/Euclidean_Geometry_.html

Euclidean Geometry resources. Recommended References. [see index for total category] for your convenience: Best Retirement Spots Web Hosting ULTRAToolBox Resources on Diet and Nutrition Pain Relief Allergies Tech Refresh , and finally - a must check - Mediterranean diet Discovery. Euclidean Geometry applications, theory, research, exams, history, handbooks and much more Introduction: Introduction to Hyperbolic Geometry (Universitext) by Arlan Ramsay Introduction to Non-Euclidean Geometry by Harold E. Wolfe Compact Riemann Surfaces: An Introduction to Contemporary Mathematics (Universitext) by Jurgen Jost Introduction to Non-Euclidean Geometry by Harold Eichholtz Wolfe Introduction to Non-Euclidean Geometry by David. Gans Applications: Theory: Hardy Spaces on the Euclidean Space (Sprringer Monographs in Mathematics) by Akihito Uchiyama Probability Theory of Classical Euclidean Optimization Problems (Lecture Notes in Mathematics (Springer-Verlag), 1675) by Joseph Yukich Shadows of the Circle: Conic Sections, Optimal Figures and Non-Euclidean Geometry by Vagn Lundsgaard Hansen Spectral Asymptotics on Degenerating Hyperbolic 3-Manifolds (Memoirs of the American Mathematical Society, 643)

58. Non-Euclidean Geometry encyclopediaEncyclopedia noneuclidean geometry. non-euclidean geometry,branch of geometry in which the fifth postulate of Euclidean http://www.infoplease.com/ce6/sci/A0835830.html

Trace your family history with Ancestry.com All Infoplease All Almanacs General Entertainment Sports Biographies Dictionary Encyclopedia Infoplease Home Almanacs Atlas Dictionary ... Fact Monster Kids' reference Info:Daily Fun facts Homework Center Newsletter You've got info! Help Site Map Visit related sites from: Family Education Network Encyclopedia non-Euclidean geometry non-Euclidean geometry, branch of geometry in which the fifth postulate of Euclidean geometry, which allows one and only one line parallel to a given line through a given external point, is replaced by one of two alternative postulates. Allowing two parallels through any external point, the first alternative to Euclid 's fifth postulate, leads to the hyperbolic geometry developed by the Russian N. I. Lobachevsky in 1826 and independently by the Hungarian Janos Bolyai in 1832. The second alternative, which allows no parallels through any external point, leads to the elliptic geometry developed by the German Bernhard Riemann in 1854. The results of these two types of non-Euclidean geometry are identical with those of Euclidean geometry in every respect except those propositions involving parallel lines, either explicitly or implicitly (as in the theorem for the sum of the angles of a triangle). Sections in this article: Introduction Hyperbolic Geometry Elliptic Geometry Non-Euclidean Geometry and Curved Space Bibliography Nonesuch Press nonintercourse Search Infoplease Info search tips Search Biographies Bio search tips About Us Contact Us Link to Infoplease ... Privacy

59. Non-Euclidean Geometry: Elliptic Geometry encyclopediaEncyclopedia—noneuclidean geometry Elliptic Geometry.In elliptic geometry there are no parallels to a given line http://www.infoplease.com/ce6/sci/A0860016.html

Trace your family history with Ancestry.com All Infoplease All Almanacs General Entertainment Sports Biographies Dictionary Encyclopedia Infoplease Home Almanacs Atlas Dictionary ... Fact Monster Kids' reference Info:Daily Fun facts Homework Center Newsletter You've got info! Help Site Map Visit related sites from: Family Education Network Encyclopedia non-Euclidean geometry Elliptic Geometry In elliptic geometry there are no parallels to a given line L through an external point P, Sections in this article: Introduction Hyperbolic Geometry Elliptic Geometry Non-Euclidean Geometry and Curved Space Bibliography Hyperbolic Geometry non-Euclidean geometry Non-Euclidean Geometry and Curved Space Search Infoplease Info search tips Search Biographies Bio search tips About Us Contact Us Link to Infoplease ... Privacy

60. Euclidean Geometry, Fractals And Adhesion euclidean geometry, Fractals and Adhesion. Sperling540 In EuclideanGeometry, a line is expressed as distance to the first power http://web.umr.edu/~wlf/Math/fractal.html

Euclidean Geometry, Fractals and Adhesion [Sperling-540] In Euclidean Geometry, a line is expressed as distance to the first power, and area is expressed as distance to the second power. Some irregular structures such as coastlines and interdiffusion boundaries (also called an interphase ) are better expressed as fractional powers between one and two. R. P. Wool and J. M. Long, Polym. Prepr., 31(2), 558 (1990) In two dimensions, D subscript f is 7/4. Last Update- January 5, 1995- wld

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