Felix Klein A Legacy of Innovation in Mathematics and Education - FAUbox
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
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:
ax 5 + bx 4 + cx 3 + dx 2 + 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 PostulateKarl 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 PoincaréEratosthenes (276-194 BC)
A surface can be represented by a flat polygon,
identifying the boundary points appropriately.
Torus
Flat TorusMöbius 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 Felix Klein
Mathematics
from a Higher Elementary
Standpoint Mathematics
Volume II: Geometry
from a Higher
Standpoint
Volume III: Precision Mathematics
and Approximation MathematicsThe 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 StandpointTo know more…
Renate Tobies
A comprehensive and
Felix Klein
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
RRRYou can also read
