refined Euclidean_geometry Information, explanation, recent texts, monographs, and related patents.
Information & explanations, latest texts & monographs on Euclidean_geometry (including recent related patents.)


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") 1 Axiomatic approach 2 Modern Introduction to Euclidean Geometry 2.1 The construction 3 Classical theorems 4 See Also 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 or axioms of the Euclidean system are:
  • Any two points can be joined by a straight line.
  • Any straight line segment can be extended indefinitely in a straight line.
  • Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
  • All right angles are congruent.
  • If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
    • The fifth postulate is equivalent to parallel postulate, which can be phrased as follows Through a point not on a given straight line, one and only one line can be drawn that never meets the given line. The parallel postulate seems less obvious than the others and many geometers tried in vain to prove it from them. In the 19th century it was shown that this could not be done, by constructing hyperbolic geometry where the parallel postulate is false, while the other axioms hold. (If one simply drops the parallel postulate from the list of axioms then you get more general geometry called absolute geometry). Another thing that was observed was that Euclid's five axioms are actually somewhat incomplete. For instance, one of his theorems is that any line segment is part of a triangle; he constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as third vertex. His axioms, however, do not guarantee that the circles actually intersect. Many revised systems of axioms were constructed, the most standard ones are Hilbert's axioms and Birkhoff's axioms. Modern Introduction to Euclidean Geometry Today Euclidean geometry is usually constructed rather than axiomatized, by means of analytic geometry. If one introduces geometry this way one can then prove the Euclidean (or any other) axioms as theorems in this particular model. This way does not have the beauty of axiomatic one but it is extremely short. The construction First let us define the set of points as set of pairs of real numbers (x,y). Then given two points P=(x,y) and Q= (z,t) one can define distances using the following formula: This is known as the Euclidean metric. All other notions as a straight line, angle, circle can be defined in terms of points as pairs of real numbers and the distances between them. For example straight line through P and Q can be defined as a set of points A such that the triangle APQ is degenerate, i.e. Classical theorems See Also Non-Euclidean geometry
    This article is adapted from from Wikipedia All Wikipedia article text is available under the terms of the GNU Free Documentation License

    Geometry, Relativity and the Fourth Dimension by Rudolf Rucker

    Janos Bolyai, Euclid, and the Nature of Space by Jeremy J. Gray

    Geometry and the Imagination (CHEL/87.H) by David Hilbert

    Euclidean and Non-Euclidean Geometries: Development and History by Marvin Jay Greenberg

    Non-Euclidean Geometry by H. S. M. Coxeter

    Euclidean and Non-Euclidean Geometry by Patrick J. Ryan

    Hyperbolic Geometry (Springer Undergraduate Mathematics Series) by James W. Anderson

    Riemannian Geometry (Graduate Texts in Mathematics, 171) by Peter Petersen

    Roads to Geometry (2nd Edition) by Edward C. Wallace

    Euclidean and Non-Euclidean Geometries: Development and History by Marvin Jay Greenberg

    Elementary Euclidean Geometry : An Introduction by C. G. Gibson

    Experiencing Geometry: In Euclidean, Spherical and Hyperbolic Spaces (2nd Edition) by David W. Henderson

    Geometry for College Students by Peter B. Geltner

    Taxicab Geometry : An Adventure in Non-Euclidean Geometry by Eugene F. Krause

    Hyperbolic Manifolds and Discrete Groups by Michael Kapovich


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