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Felix Klein a Legacy of Innovation in Mathematics and Education

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Felix Klein a Legacy of Innovation in Mathematics and Education Felix Klein A Legacy of Innovation in Mathematics and Education Roberto Rodríguez del Río IES San Mateo Dept. de Análisis Matemático y Matemática Aplicada, UCM http://www.mat.ucm.es/~rrdelrio/ April, 14, 2021, Department Mathematik, Friedrich-Alexander Universität, Erlangen-Nüremberg Felix Klein , (1849-1925) Born in Düsseldorf, 22, 52, 432 The Language of the Symmetry Symmetry Mosaics in Alhambra, Granada, Spain A thing is symmetrical if there is something you can do to it so that after you have finished doing it it looks the same as before. Herman Weyl, Symmetry The mathematical language of Symmetry is Group Theory The Group of transformations (rotations) that leave fixed the square ABCD is G = {e, p, q, r} A equation that could not be solved: ax5 + bx4 + cx3 + dx2 + ex + f = 0 Évariste Galois (1811-1832) Niels Henrik Abel (1802-1829) Mario Livio, The Equation That Couldn’t Be Solved, 2005 Place de l’Étoile, Paris, 1857 Felix Klein and Sophus Lie (1842-1899) Felix Klein and Sophus Lie visited Paris in (summer) 1870 to learn about Group Theory with Camille Jordan (1838-1922). Two types of mathematicians, two types of mathematics Karl Weierstraß (1815-1897) The Erlangen Program The Euclid’s Elements Parallel Postulate Karl Friedrich Gauß (1777-1855) János Bolyai (1802-1860). Nikolái Lobachevski (1792-1856) Bernhard Riemann (1826-1866) The “so-called” Non-euclidian Geometries Parabolic (euclidean) Geometry Elliptical Geometry Hyperbolic Geometry F. Klein, Über die sogenannte Nicht-Euklidische Geometrie, Mathematische Annalen, On the so-called non-Euclidean geometry, 1871-1873 Euclidean models for Non-euclidean Geometries Klein disk for Klein disk for Hyperbolic Geometry Elliptic Geometry • Erlangen, October, 1872, appointment of F. Klein as a Full Professor, (23 y.o.) • Inaugural Lecture • The Erlangen Program The Erlangen Program A comparative review of recent researches in geometry Euclidean Geometry in Erlangen Program M = {Points of the plane} (Isometry Group) G = {rotations, translations, reflexions} The transformation of Group G preserves distances, area, perpendicularity, parallelism, etc. My 1872 Programme, appearing as a separate publication, had but a limited circulation at first. With this I could be satisfied more easily, as the views developed in the Programme could not be expected at first to receive much attention. F. Klein, A comparative review of recent researches in geometry https://arxiv.org/abs/0807.3161 (Complete English Translation) - Klein, Editor of Mathematische Annalen, 1872 - Klein left Erlangen in 1875 - He got a chair in Munich The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin. The Klein’s bottle and Henri Poincaré Eratosthenes (276-194 BC) A surface can be represented by a flat polygon, identifying the boundary points appropriately. Torus Flat Torus Möbius Strip Klein’s Bottle Kleinsche Fläche (Klein Surface) Kleinsche Flasche (Klein Bottle) Automorphic functions and Henri Poincaré (1854-1912) Klein disk with Poincaré disk with hyperbolic parallel lines hyperbolic parallel lines Göttingen: «The Mecca of mathematicians» Mathematical tradition in Göttingen, before Klein Karl Friedrich Gauß (1777-1855) Sophie Germain (1776-1831) Richard Dedekind (1831-1916) Bernard Riemann (1826-1866) Sofia Kovalevskaya (1850-1891) Grace Chisholm Young (1868-1944) Emmy Noether (1882-1935) “The man of the future” David Hilbert (1862-1943) Hermann Minkowski (1864-1909) The Legacy of Felix Klein in Teaching Mathematics The Klein Project, 2008 https://www.mathunion.org/icmi/activities/klein-project/activities/klein-project Felix Klein, Elementary Mathematics from a Higher Standpoint Felix Klein Elementary Mathematics from a Higher Standpoint Volume I: Arithmetic, Algebra, Analysis Felix Klein Elementary Mathematics Felix Klein from a Higher Elementary Standpoint Mathematics from a Higher Volume II: Geometry Standpoint Volume III: Precision Mathematics and Approximation Mathematics The child cannot possibly understand if numbers are explained axiomatically as abstract things devoid of meaning, with which one can operate according to formal rules. On the contrary, he associates numbers with concrete representations. They are nothing else than quantities of nuts, apples, and other good things, and in the beginning they can be and should be put before him only in such tangible form. […] Mathematics should be associated with everything that is seriously interesting to a person at the particular stage of his development. Felix Klein, Elementary Mathematics from a Higher Standpoint To know more… Renate Tobies Felix Klein A comprehensive and Visionen für Mathematik, Anwendungen und Unterricht well documented book about Felix Klein R. Tobies, Felix Klein. Visionen für Mathematik, Anwendungen und Unterricht, Springer, 2019. A collection of Klein’s main ideas on teaching mathematics R. Rodríguez del Río, Felix Klein. Una nueva visión de la geometría, RBA, 2017 (Felix Klein. A New Vision of the Geometry) Thank you! April, 2021 RRR.
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