
Geometry
Credits 
6 credit points 
Instructors 
Bogaart, T. van den (Universiteit Utrecht), Edixhoven, S.J. (Universiteit Leiden) 
Email 
t.vandenbogaart@uu.nl, edix@math.leidenuniv.nl 
Aim 
The aim of this course is to familiarize future teachers with the following approaches to geometry: (1) constructions and axioms; (2) coordinates and vectors; (3) perspective and projective planes; and (4) transformations and noneuclidean planes. This knowledge will help the student to see the connections between school geometry and more modern branches of mathematics. 
Description 
We will discuss these four themes: (1) After a brief review of some classical constructions we will examine the flaws of Euclids axiom system. Next, we will discuss Hilberts solution to this problem. We will see how the perception of the relation between geometry and numbers has changed since Euclids times. (2) We will briefly review coordinates, distances, vector spaces, inner products and matrices. (3) We will motivate projective planes by considering perspective in drawing. Projective planes are then introduced axiomatically. Next we treat some basic topics such as homogeneous coordinates, projections, broken linear transformations, double ratio and some standard theorems (Pappus and Desargues). (4) After studying transformations (isometries) of the Euclidean plane and the sphere we will introduce the peculiar geometry of the hyperbolic plane. (5) Depending on the interests of the participants these themes can be extended or other, related topics will be added, such as finite projective planes and elliptic curves with modern applications. 
Organization 
3 hours mixture of lecture and exercise class 
Examination 
A midterm written exam, and a take home assignment plus oral exam at the end. 
Literature 
John Stillwell, The four pillars of geometry, Springer (2005), ISBN 0387255303 
Prerequisites 
Basic linear algebra; familiarity with elementary group theory is not assumed, but can be helpful. 
