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Felix Klein argued in his Erlangen program that one can consider various " geometries " by specifying an appropriate transformation group that leaves certain geometric properties invariant.
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Felix and Klein
* 1849 – Felix Klein, German mathematician ( d. 1925 )
See also the letter from Robert Gauss to Felix Klein on 3 September 1912.
In 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Chairman of Mathematics at the University of Göttingen, at that time the best research-center for mathematics in the world.
Also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called " self-inverse " fractals.
Historically, the concept of fundamental group first emerged in the theory of Riemann surfaces, in the work of Bernhard Riemann, Henri Poincaré and Felix Klein, where it describes the monodromy properties of complex functions, as well as providing a complete topological classification of closed surfaces.
The Klein bottle was first described in 1882 by the German mathematician Felix Klein.
* Klein Bottle animation from 2010 including a car ride through the bottle and the original description by Felix Klein: produced at the Free University Berlin.
Some of Lie's early ideas were developed in close collaboration with Felix Klein.
Mathematician Felix Klein wrote about Abel:
* June 22 – Felix Klein, German mathematician ( b. 1849 )
an: Felix Klein
ca: Felix Klein
cs: Felix Christian Klein
da: Felix Klein
de: Felix Klein
et: Felix Klein
es: Felix Klein
fr: Felix Klein
fy: Felix Klein
it: Felix Klein
ht: Felix Klein
la: Felix Klein
Felix and argued
Marcus Minucius Felix ( late 2nd-3rd c .), an Early Christian writer, argued for the existence of God based on the analogy of an ordered house in his The Orders of Minucius Felix.
The State of Tennessee argued that legislative districts were essentially political questions, not judicial ones, as had been held by a plurality opinion of the Court in which Justice Felix Frankfurter declared that, " Courts ought not to enter this political thicket.
In their book on Ali, Felix Dennis and Don Atyeo argued that " the blow generated enough power to lift Liston's left foot, upon which most of his weight was placed, off the canvas.
Spivak argued that the plot was part of a " conspiracy of Jewish financiers working with fascist groups ", referring specifically to Felix Warburg, the McCormack – Dickstein Committee, and certain members of the American Jewish Committee in collusion with J. P. Morgan.
Felix and Erlangen
Felix Klein's Erlangen program, considering the future of research in mathematics, is so called because Klein was then at the University of Erlangen-Nuremberg.
Several major strands of more abstract mathematics ( including invariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programme leading to the study of the classical groups ) built on projective geometry.
Projective geometry, like affine and Euclidean geometry, can also be developed from the Erlangen program of Felix Klein ; projective geometry is characterized by invariants under transformations of the projective group.
Only after Felix Klein's Erlangen program was affine geometry recognized for being a generalization of Euclidean geometry.
We identify as affine theorems any geometric result that is invariant under the affine group ( in Felix Klein's Erlangen programme this is its underlying group of symmetry transformations for affine geometry ).
In his Erlangen program, Felix Klein played down the tension between synthetic and analytic methods:
Simple Lie groups include many classical Lie groups, which provide a group-theoretic underpinning for spherical geometry, projective geometry and related geometries in the sense of Felix Klein's Erlangen programme.
Through some steps of application of the circle inversion map, a student of transformation geometry soon appreciates the significance of Felix Klein ’ s Erlangen program, an outgrowth of certain models of hyperbolic geometry
Furthermore, Felix Klein was so overcome by this facility of mappings to identify geometrical phenomena that he delivered a manifesto, the Erlangen program, in 1872.
He sought to apply the techniques of Pfaffian systems to the geometries of Felix Klein's Erlangen program.
At Erlangen he received a doctorate, supervised by Felix Klein, on non-Euclidean geometry.
The affine subspaces are model surfaces — they are the simplest surfaces in R < sup > 3 </ sup >, and are homogeneous under the Euclidean group of the plane, hence they are Klein geometries in the sense of Felix Klein's Erlangen programme.
* Felix Klein produces the Erlangen program on geometries.
Finally Felix Klein's " Erlangen program " identified the underlying theme of all of these geometries, defining each of them as the study of properties invariant under a given group of symmetries.
Felix Klein's Erlangen program attempted to identify such invariants under a group of transformations.
These model surfaces are Klein geometries in the sense of Felix Klein's Erlangen programme.
He went to the universities at Leipzig and Erlangen to work with Felix Klein and Ferdinand von Lindemann.
This motivation came, in part, from Felix Klein's Erlangen programme where one was interested in a notion of symmetry on a space, where the symmetries of the space were transformations forming a Lie group.
Such an approach to geometry was cultivated by Felix Klein in his Erlangen program.
In Felix Klein's Erlangen program, each possible group of symmetries defines a geometry in which objects that are related by a member of the symmetry group are considered to be equivalent.
The Erlangen program of Felix Klein appealed early on to Segre, and he became a promulgator.
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