The Non-Euclidean, Hyperbolic Plane: Its Structure and Consistency

The Non Euclidean Hyperbolic Plane Its Structure and Consistency The discovery of hyperbolic geometry and the subsequent proof that this geometry is just as logical as Euclid s had a profound in fluence on man s understanding of mathematics and the relation of ma

Non Euclidean geometry In mathematics, non Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry.As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. Non Euclidean Geometry from Wolfram MathWorld 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 Non Euclidean geometry mathematics Britannica Non Euclidean geometry, literally any geometry that is not the same as Euclidean geometry.Although the term is frequently used to refer only to hyperbolic geometry, common usage includes those few geometries hyperbolic and spherical that differ from but are very close to Euclidean Non Euclidean Geometry Mathematical Association of No living geometer writes clearly and beautifully about difficult topics than world famous Professor H S M Coxeter When non Euclidean geometry was first developed, it seemed little than a curiosity with no relevance to the real world. Euclidean and Non Euclidean Geometries This is the definitive presentation of the history, development and philosophical significance of non Euclidean geometry as well as of the rigorous foundations for it and for elementary Euclidean geometry, essentially according to Hilbert. Euclidean space In geometry, Euclidean space encompasses the two dimensional Euclidean plane, the three dimensional space of Euclidean geometry, and similar spaces of higher dimension.It is named after the Ancient Greek mathematician Euclid of Alexandria The term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry.Euclidean spaces also generalize to Euclidean Define Euclidean at Dictionary Euclidean definition, of or relating to Euclid, or adopting his postulates See . Euclidean Geometry from Wolfram MathWorld 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. Jnos Bolyai Hungarian mathematician Britannica Jnos Bolyai Jnos Bolyai, Hungarian mathematician and one of the founders of non Euclidean geometry a geometry that differs from Euclidean geometry in its definition of parallel lines The discovery of a consistent alternative geometry that might correspond to Geometria non euclidea Una geometria non euclidea una geometria costruita negando o non accettando alcuni postulati euclidei.Viene detta anche metageometria

  • Title: The Non-Euclidean, Hyperbolic Plane: Its Structure and Consistency
  • Author: Gordon Matthews
  • ISBN: 9780387905525
  • Page: 468
  • Format: Hardcover
  • The discovery of hyperbolic geometry, and the subsequent proof that this geometry is just as logical as Euclid s, had a profound in fluence on man s understanding of mathematics and the relation of mathematical geometry to the physical world It is now possible, due in large part to axioms devised by George Birkhoff, to give an accurate, elementary development of hyperboliThe discovery of hyperbolic geometry, and the subsequent proof that this geometry is just as logical as Euclid s, had a profound in fluence on man s understanding of mathematics and the relation of mathematical geometry to the physical world It is now possible, due in large part to axioms devised by George Birkhoff, to give an accurate, elementary development of hyperbolic plane geometry Also, using the Poincare model and inversive geometry, the equiconsistency of hyperbolic plane geometry and euclidean plane geometry can be proved without the use of any advanced mathematics These two facts provided both the motivation and the two central themes of the present work Basic hyperbolic plane geometry, and the proof of its equal footing with euclidean plane geometry, is presented here in terms acces sible to anyone with a good background in high school mathematics The development, however, is especially directed to college students who may become secondary teachers For that reason, the treatment is de signed to emphasize those aspects of hyperbolic plane geometry which contribute to the skills, knowledge, and insights needed to teach eucli dean geometry with some mastery.

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