Transcript Ch. 4: The Erlanger Programm References:

Ch. 4: The Erlanger
Programm
References:
•Euclidean and Non-Euclidean Geometries:
Development and History 4th ed By 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.standrews.ac.uk/Mathematicians/Klein.html
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
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.
Congruence
• Congruent figures have identical
geometric properties.
• Measurement comes first in Euclidean
geometry.
• Congruence comes first in the Erlanger
Programm.
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.
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
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}.
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.
Rigid Motion
Tz=eiz+b
• Composition of rotation and translation
• Verify that this is a group – identity,
inverses and closure.
More Examples
Next Time
Invariants