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Ch. 4: The Erlanger Programm

Ch. 4: The Erlanger Programm . References: Euclidean and Non-Euclidean Geometries: Development and History 4 th ed By Greenberg Modern Geometries: Non-Euclidean, Projective and Discrete 2 nd ed by Henle Roads to Geometry 2 nd ed by Wallace and West

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Ch. 4: The Erlanger Programm

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  1. Ch. 4: The Erlanger Programm References: Euclidean and Non-Euclidean Geometries: Development and History 4th edBy Greenberg Modern Geometries: Non-Euclidean, Projective and Discrete 2nd ed by Henle Roads to Geometry 2nd ed by Wallace and West http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Klein.html

  2. Felix Christian Klein(1849-1925) • Born in Dusseldorf, Prussia • Studied Mathematics and Physics at the University of Bonn • 1872 Appointed to a chair at the University of Erlanger

  3. Erlanger Programm (1872) • Inaugural lecture on ambitious research proposal. • A new unifying principle for geometries. • Properties of a space that remain invariant under a group of transformations.

  4. Congruence • Congruent figures have identical geometric properties. • Measurement comes first in Euclidean geometry. • Congruence comes first in the Erlanger Programm.

  5. Congruence and Transformation • Superposition: Two figures are congruent when one can be moved so as to coincide with the other. • Three Properties of Congruence • Reflexivity A  A for any figure A. • Symmetry If A  B, then B  A. • Transitivity If A  B and B  C, then A  C.

  6. Transformation Group (p.38) • Let S be a nonempty set. A transformation group is a collection G of transformations T:SS such that • G contains the identity • The transformations in G are invertible and their inverses are in G, and • G is closed under composition

  7. Some Definitions (p.38) • A geometry is a pair (S,G) consisting of a nonempty set S and a transformation group G. The set S is the underlying space of the geometry. The set G is the transformation group of the geometry. • A figure is any subset A of the underlying set S of a geometry (S,G). Two figures A and B are congruent if there is a transformation T in G such that T(A)=B, where T(A) is defined by the formula T(A)={Tz: z is a point from A}.

  8. Euclidean Geometry • Underlying set is, C, the complex plane. • Transformation group is the set E of transformations of the form Tz=eiz+b Where  is a real constant and b is a complex constant. This type of transformation is called a rigid motion.

  9. Rigid Motion Tz=eiz+b • Composition of rotation and translation • Verify that this is a group – identity, inverses and closure.

  10. More Examples Next Time Invariants

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