21. 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/

22. Introduction To The Works Of Euclid Covers the life of Euclid and a discussion of euclidean geometry. http://www.obkb.com/dcljr/euclid.html

An Introduction to the Works of Euclid with an Emphasis on the Elements (first posted to the web in 1995) jump to: outline of paper text of paper suggestions for further study bibliography ... anchor here Outline of paper Introduction I. The Life of Euclid II. The Works of Euclid Other than the Elements ... Bibliography About this paper This is a paper I wrote in college for a History of Science course (although I've taken the liberty of modifying it slightly from time to time since I put it online). I know it's not publishable or anything, but it's still one of my favorite papers because it was so difficult to do. (I wrote it on a computer with about 12K of free RAM and only a cassette tape drive for storage!) In fact, the whole History of Science course was quite an experience. Students wishing to use this paper for their own reports on Euclid should know how to avoid plagiarism and how to cite online sources . In addition, I urge students to seek out the original printed sources yes, that means going to the library and not rely merely on what I say in this paper. (I'm always surprised by the number of junior high and high school students who e-mail me saying they can't find any information about Euclid!) Note that is used to denote square roots and all Greek letters used as symbols ( alpha beta , ...) are spelled out. Superscripts are implemented by using the appropriate HTML tags and may not display properly in some browsers. In this case, hopefully the meaning will be clear from the context.

23. Euclidean Geometry -- From MathWorld euclidean geometry. Twodimensional euclidean geometry is called plane geometry, and three-dimensional euclidean geometry is called solid geometry. http://mathworld.wolfram.com/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 General Geometry Euclidean Geometry A geometry in which Euclid's fifth postulate holds, sometimes also called parabolic geometry . Two-dimensional Euclidean geometry is called plane geometry , and three-dimensional Euclidean geometry is called solid geometry Hilbert proved the consistency of Euclidean geometry. Elements Elliptic Geometry Geometric Construction Geometry ... search Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, 1952. Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions with Numerous Examples, 2nd rev. enl. ed. Coxeter, H. S. M. and Greitzer, S. L.

24. Klein, Felix (1849-1925) -- From Eric Weisstein's World Of Scientific Biography Pl¼cker's assistant at Bonn who studied Analytic Geometry, describing geometry as the study of properties of figures which remain invariant under a Group of Transformations. He systemized Noneuclidean geometry. http://www.treasure-troves.com/bios/KleinFelix.html

Branch of Science Mathematicians Nationality German Klein, Felix (1849-1925) German mathematician who began his career as assistant at Bonn. Klein studied analytic geometry describing geometry as the study of properties of figures which remain invariant under a group of transformations He systemized non-Euclidean geometry and wrote a book on the icosahedron in 1884. He also worked on the development of group theory and collaborated with Lie in Erlanger Programm. He also is known in topology for the one-sided Klein bottle In addition to all his other work, he found time to write a classic history of mathematics. Additional biographies: MacTutor (St. Andrews) Bonn References Fricke, R. and Klein, F. Leipzig: B. G. Teubner, 1897-1912. Klein, F. Arithmetic, Algebra, Analysis. New York: Dover, N.D. Klein, F. Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis. New York: Dover. Klein, F. Famous Problems of Elementary Geometry. New York: Chelsea, 1956. Klein, F. Gesammelte Mathematische Abhandlungen. Berlin: Springer-Verlag, 1973.

25. 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. 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."

26. Euclidean Geometry The traditional presentation of euclidean geometry is as an axiomatic system, setting Introduction to euclidean geometry. Today euclidean geometry is usually constructed rather http://www.sciencedaily.com/encyclopedia/euclidean_geometry

Match: sort by: relevance date Free Services Subscribe by email RSS newsfeeds PDA-friendly format loc="/images/" A A A Find Jobs In: Healthcare Engineering Accounting College Contract / Freelance Customer Service Diversity Engineering Executive Healthcare Hospitality Human Resources Information Tech International Manufacturing Nonprofit Retail All Jobs by Job Type All Jobs by Industry Relocating? Visit: Moving Resources Moving Companies Mortgage Information Mortgage Calculator Real Estate Lookup Front Page Today's Digest Week in Review Email Updates ... Outdoor Living Encyclopedia Main Page See live article Euclidean geometry In mathematics Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. Mathematicians sometimes use the term to encompass higher dimensional geometries with similar properties. Euclidean geometry usually refers to geometry in the plane which is also called plane geometry . It is plane geometry which is the topic of this article. Euclidean geometry in three dimensions is traditionally called solid geometry . For information on higher dimensions see Euclidean space . Plane geometry is the kind of geometry usually taught in high school . Euclidean geometry is named after the Greek mathematician Euclid . Euclid's text Elements is an early systematic treatment of this kind of geometry Table of contents showTocToggle("show","hide")

27. Discussion Noneuclidean geometry. This tutorial consists of html constructed constructed applet 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 (

28. Euclidean Geometry - Wikipedia, The Free Encyclopedia euclidean geometry. From Wikipedia, the free encyclopedia. In mathematics Modern Introduction to euclidean geometry. Today Euclidean http://en.wikipedia.org/wiki/Euclidean_geometry

Euclidean geometry From Wikipedia, the free encyclopedia. In mathematics Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. Mathematicians sometimes use the term to encompass higher dimensional geometries with similar properties. Euclidean geometry usually refers to geometry in the plane which is also called plane geometry . It is plane geometry which is the topic of this article. Euclidean geometry in three dimensions is traditionally called solid geometry . For information on higher dimensions see Euclidean space Plane geometry is the kind of geometry usually taught in high school . Euclidean geometry is named after the Greek mathematician Euclid . Euclid's text Elements is an early systematic treatment of this kind of geometry Table of contents 1 Axiomatic approach 2 Modern introduction to Euclidean geometry 2.1 The construction 3 Classical theorems ... edit Axiomatic approach The traditional presentation of Euclidean geometry is as an axiomatic system , setting out to prove all the "true statements" as theorems in geometry from a set of finite number of axioms The five postulates of the Elements are: Any two points can be joined by a straight line Any straight line segment can be extended indefinitely in a straight line.

29. Non-Euclidean Geometry A historical account with links to biographies of some of the people involved. http://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/Non-Euclidean_geometr

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

30. Non-Euclidean Geometry - Wikipedia, The Free Encyclopedia Noneuclidean geometry. (Redirected from Non-euclidean geometry). External link. MacTutor Archive article on non-euclidean geometry. See also. http://en.wikipedia.org/wiki/Non-euclidean_geometry

Non-Euclidean geometry From Wikipedia, the free encyclopedia. (Redirected from Non-euclidean geometry 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. Behavior of lines with a common perpendicular in each of the three types of geometry Table of contents 1 History 2 Reference 3 External link 4 See also ... edit History While Euclidean geometry (named for the

31. 51M05: General Euclidean Geometry Introduction. We use this category to hold files concerning nonplanar euclidean 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.

32. Non-Euclidean Geometry Resources in noneuclidean geometry. http://www.westford.mec.edu/schools/tips/noneucld.html

Non-Euclidean Geometry Taxicab Geometry This site is an introduction to non-Euclidean geometry with real world examples. http://www2.gvsu.edu/~vanbelkj/Project.html Non-Euclidean Geometry This site covers Euclid's Elements through the end of the 1800s with over 23 references. http://www-history.mcs.st-and.ac.uk/history/HistTopics/Non-Euclidean_geometry.html Euclidean/Non-Euclidean Geometry This is a brief introduction and comparison. http://www.tdsb.on.ca/nymthp/non-euclidean.html Euclidean and Non-Euclidean Geometry with The Geometer's Sketchpad The sketches are downloadable in Mac and PC formats. http://www.keypress.com/sketchpad/talks/Euc_Wien98/index.html Non-Euclidean Geometry The site includes Euclid's The Elements and Fifth Postulate as well as the history of Non-Euclidean geometry. http://csis.pace.edu/~ryeneck/mahony/LMPROJ8.HTM http://dsdk12.net/project/euclid/GEOEUC~1.HTM

33. 51M04: Elementary Euclidean Geometry (2-dimensional) links Selected topics here 51M04 Elementary euclidean geometry (2dimensional). Introduction. Ordinary plane geometry (such http://www.math.niu.edu/~rusin/known-math/index/51M04.html

Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields POINTERS: Texts Software Web links Selected topics here 51M04: Elementary Euclidean geometry (2-dimensional) Introduction Ordinary plane geometry (such as is studied in US secondary schools) holds an irresistible appeal, although many results derive what appear to be unimaginative conclusions from tortured premises. Nonetheless, from time to time something catches our eye and gets us to think about ordinary triangles and circles. History Applications and related fields Constructibility with compass and straightedge is dealt with elsewhere Tilings and packings in the plane are part of Convex Geometry Many topics regarding polygons (e.g. decompositions into triangles and so on) are treated as part of polyhedral geometry Subfields Parent field: 51M - Real and Complex Geometry Textbooks, reference works, and tutorials Software and tables A compendium of plane curves For computational geometry see 68U05: Computer Graphics Other web sites with this focus The Geometry Junkyard has a "pile" for planar geometry (and other related topics of interest!)

34. References For Non-Euclidean Geometry A bibliographic reference list of books and articles on nonEuclidean geometries. http://www-groups.cs.st-and.ac.uk/~history/HistTopics/References/Non-Euclidean_g

35. What Is Non-Euclidean Geometry? What is noneuclidean geometry? Consequently, in a euclidean geometry every point has one and only one line parallel to any given line. http://njnj.essortment.com/noneuclideange_risc.htm

What is non-Euclidean geometry? An introduction to the history and mathematics of non-Euclidean geometries. Euclid's geometrical thesis, "The Elements" (c. 300 B.C.E), proposed five basic postulates of geometry. Of these postulates, all were considered self-evident except for the fifth postulate. The fifth postulate asserted that two lines are parallel (i.e. non-intersecting) if a third line can intersect both lines perpendicularly. Consequently, in a Euclidean geometry every point has one and only one line parallel to any given line. For centuries people questioned Euclid's fifth postulate. Even Euclid seemed suspicious of the fifth postulate because he avoided solving problems with it until his 29th example. Mathematicians stumbled with ways to prove the validity of the fifth postulate from the first four postulates, which we now call the postulates of absolute geometry. Those mathematicians who didn't fail were soon seen to have fallacious errors in their reasoning. These errors usually occurred because a mathematician had made self-fulfilling assumptions pertaining to parallel lines, rather than working with the other postulates. Essentially, they were forcing a result through the application of faulty logic. bodyOffer(29808) Though many mathematicians questioned Euclidean geometry, Euclidean thought prevailed through school mathematical programs. "The Elements" became the most widely purchased non-religious work in the world, and it still remains the most widely received of mathematical texts. Furthermore, mathematical inquiries into the nature of non-Euclidean geometries were often devalued as frivolous. The philosopher Immanuel Kant (1724-1804) called Euclid's geometry, "the inevitable necessity of thought." Such philosophical opinions impeded mathematical progress in the field of geometry. Karl Friedrich Gauss (1777-1855), who began studying non-Euclidean geometries at the age of 15, never published any of his non-Euclidean works because he knew the mathematical precedent was against him.

36. COMPUTING IN EUCLIDEAN GEOMETRY Lecture Notes Series on Computing Vol. 4 COMPUTING IN euclidean geometry (2nd Edition) edited by Ding-Zhu Du (Univ. Minnesota Inst. http://www.wspc.com/books/compsci/2463.html

Home Browse by Subject Bestsellers New Titles ... Browse all Subjects Search Keyword Author Concept ISBN Series New Titles Editor's Choice Bestsellers Book Series ... Lecture Notes Series on Computing - Vol. 4 COMPUTING IN EUCLIDEAN GEOMETRY edited by Ding-Zhu Du This book is a collection of surveys and exploratory articles about recent developments in the field of computational Euclidean geometry. Topics covered include the history of Euclidean geometry, Voronoi diagrams, randomized geometric algorithms, computational algebra, triangulations, machine proofs, topological designs, finite-element mesh, computer-aided geometric designs and Steiner trees. This second edition contains three new surveys covering geometric constraint solving, computational geometry and the exact computation paradigm. Contents: On the Development of Quantitative Geometry from Phythagoras to Grassmann (W-Y Hsiang) Computational Geometry: A Retrospective (B Chazelle) Randomized Geometric Algorithms (K L Clarkson) Voronoi Diagrams and Delaunay Triangulations (S Fortune) Geometric Constraint Solving in R and R Polar Forms and Triangular B-Spline Surfaces (H-P Seidel) Readership: Computer scientists and mathematicians.

37. Non-Euclidean Geometries, Models NonEuclidean Geometries As Good As Might Be. From our perspective, the situation was exactly the same as with euclidean geometry. http://www.cut-the-knot.org/triangle/pythpar/Model.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Non-Euclidean Geometries As Good As Might Be In 1823, Janos Bolyai wrote to his father: "Out of nothing I have created a new universe." By which he meant that starting from the first 4 of Euclid's postulates and a modified fifth, he developed an expansive theory that, although quite unusual, did not seem to lead to any logical contradiction. Gauss expressed his conviction in consistency of the theory he had in mind in a letter in 1824. However, hesitant of the public reaction to the idea that, by the side of Euclidean, there is another geometry, he never published anything on the subject. From our perspective, the situation was exactly the same as with Euclidean geometry. Euclid built an axiomatic theory by deriving a lot of theorems from his five postulates. No one had ever proved that continuing in Euclid's footstep would not lead to a contradiction. However, a 2000 year history elevated Elements on a pedestal of infallibility. Kant even stipulated that the universe had been built according to Euclid. Lobachevsky was quite aware of the problem and in later publications tried without success to redefine the notions of line and plane. A breakthrough came in 1868 with the publication of Saggio di interpretazoine della geometria non euclidea by the Italian mathematician Eugenio Beltrami (1835-1900). Beltrami discovered that Lobachevsky's geometry admits an interpretation in terms of Euclidean geometry. From here it follows that if Lobachevsky's geometry leads to a contradiction, Euclidean geometry is as well contradictory. In other words, consistency of Euclidean geometry implies consistency of the geometry of Lobachevsky. The construction is now known as Beltrami-Klein model of Lobachevsky geometry. Sometimes it's called the

38. Non-Euclidean Geometry Noneuclidean geometry Taxicab Geometry This site is an introduction to non-euclidean geometry with real world examples. http//www2 http://westford.mec.edu/schools/tips/noneucld.html

Non-Euclidean Geometry Taxicab Geometry This site is an introduction to non-Euclidean geometry with real world examples. http://www2.gvsu.edu/~vanbelkj/Project.html Non-Euclidean Geometry This site covers Euclid's Elements through the end of the 1800s with over 23 references. http://www-history.mcs.st-and.ac.uk/history/HistTopics/Non-Euclidean_geometry.html Euclidean/Non-Euclidean Geometry This is a brief introduction and comparison. http://www.tdsb.on.ca/nymthp/non-euclidean.html Euclidean and Non-Euclidean Geometry with The Geometer's Sketchpad The sketches are downloadable in Mac and PC formats. http://www.keypress.com/sketchpad/talks/Euc_Wien98/index.html Non-Euclidean Geometry The site includes Euclid's The Elements and Fifth Postulate as well as the history of Non-Euclidean geometry. http://csis.pace.edu/~ryeneck/mahony/LMPROJ8.HTM http://dsdk12.net/project/euclid/GEOEUC~1.HTM

39. Question Corner -- Euclidean Geometry In Higher Dimensions euclidean geometry in Higher Dimensions. I would like to know where I can find out a little more than high school maths on euclidean geometry. http://www.math.toronto.edu/mathnet/questionCorner/eucgeom.html

Navigation Panel: (These buttons explained below Question Corner and Discussion Area Euclidean Geometry in Higher Dimensions Asked by Victor Humberstone on February 10, 1997 I would like to know where I can find out a little more than high school maths on Euclidean Geometry. In particular, I would like to understand n -dimensional symmetrical `solids' (esp 4, 5 dimensions.) My son has recently been asking about a drawing of a `hypercube' (a 4-D cube) in an old book by George Gamov in which such an object was drawn and wants to understand how to extend the concept. I can't help! Can you help me to help him? Euclidean Geometry in higher dimensions is best understood in terms of coordinates and vectors. In fact, it is these which even give meaning to geometric concepts in higher dimensions. So, let me start with a quick overview of those (which you, as a physics graduate, will know anyway and may want to skip, but others reading the page may not): In 3 dimensions, we all have an intuitive understanding of what length and angle mean, and it is not at all clear how to extend these concepts to higher dimensions.

40. Question Corner -- Non-Euclidean Geometry Noneuclidean geometry. Asked by Brent Potteiger on April 5, 1997 Being as curious as I am, I would like to know about non-euclidean geometry. Thanks!!! http://www.math.toronto.edu/mathnet/questionCorner/noneucgeom.html

Navigation Panel: (These buttons explained below Question Corner and Discussion Area Non-Euclidean Geometry Asked by Brent Potteiger on April 5, 1997 I have recently been studying Euclid (the "father" of geometry), and was amazed to find out about the existence of a non-Euclidean geometry. Being as curious as I am, I would like to know about non-Euclidean geometry. Thanks!!! All of Euclidean geometry can be deduced from just a few properties (called "axioms") of points and lines. With one exception (which I will describe below), these properties are all very basic and self-evident things like "for every pair of distinct points, there is exactly one line containing both of them". This approach doesn't require you to get into a philosophical definition of what a "point" or a "line" actually is. You could attach those labels to any concepts you like, and as long as those concepts satisfy the axioms, then all of the theorems of geometry are guaranteed to be true (because the theorems are deducible purely from the axioms without requiring any further knowledge of what "point" or "line" means). Although most of the axioms are extremely basic and self-evident, one is less so. It says (roughly) that if you draw two lines each at ninety degrees to a third line, then those two lines are parallel and never intersect. This statement, called

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