1. 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 http://www-history.mcs.st- andrews.ac.uk/Mathematicians/Klein.html 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 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 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. 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. 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 a)Reflexivity A  A for any figure A. b)Symmetry If A  B, then B  A. c)Transitivity If A  B and B  C, then A  C. Superposition: Two figures are congruent when one can be moved so as to coincide with the other. Three Properties of Congruence a)Reflexivity A  A for any figure A. b)Symmetry If A  B, then B  A. c)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 a)G contains the identity b)The transformations in G are invertible and their inverses are in G, and c)G is closed under composition Let S be a nonempty set. A transformation group is a collection G of transformations T:S  S such that a)G contains the identity b)The transformations in G are invertible and their inverses are in G, and c)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}. 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 T z=e i  z+b Where  is a real constant and b is a complex constant. This type of transformation is called a rigid motion. Underlying set is, C, the complex plane. Transformation group is the set E of transformations of the form T z=e i  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 T z=e i  z+b Composition of rotation and translation Verify that this is a group – identity, inverses and closure. T z=e i  z+b Composition of rotation and translation Verify that this is a group – identity, inverses and closure.

10. More Examples Next Time Invariants Next Time Invariants