Kurssin alustava aikataulu (aikataulumuutoksista voidaan sopia yhdessä osallistujien kanssa):
Ensimmäinen luento tiistaina 14.3. klo
10-12 salissa K3, jonka jälkeen luennot jatkuvat viikoilla
11 - 17 tiistaisin ja perjantaisin klo 10-12 normaalisti
salissa
K3 paitsi seuraavina päivinä:
ti
28.3. salissa K1
ma 10.4.
klo 10 -12 mikroluokassa M18
(linkit demoihin alla sisällysluettelossa)
pe 14.4.
salissa M8
ti
18.4. salissa K4.
Kurssikuvaus:
The aims of the course are to present beautiful mathematics and to show how to derive abstract concepts from simple drawings. Each chapter will begin at a very elementary level and end with insights of more conceptual mathematics. The chapters will be (nearly completely) independent from each other. Exercises will be given on tuesdays and corrected on fridays.
The following plan may be changed at the request of the students.
Planned lectures:
Chapter 1. Parallel and central
projections in plane geometry
Theorem of Thales; theorem of Menelaus; cross ratio;
harmonic division; theorem of Ceva.
Chapter 2. Orthocentric quadrangulars
Centers of a triangle; Euler's straight line; 3 orthocentric
quadrangles associated with a triangle; Feuerbach's
theorem; tetraedrons and orthocentrical tetraedrons.
Chapter 3. How to define angles
and measures of angles in a plane?
Points from which a segment is viewed under a given angle;
oriented
and unoriented angles of lines or of rays; cocyclicity; Simson's
line; measures of angles; need of analysis.
Chapter 4. Inversion, a mean to
transform circles into straight lines and spheres into planes
Relative positions of two circles; bundles of circles;
orthogonal circles; inversion in the plane; z->1/conj(z);
inversion
of a point; inversions
of circles and lines; Steiner's porism; inversion in space; stereographic
projection; Riemann's sphere.
Tuosta on laajempi versiokin
Chapter 5. Computer Aided Geometric
Design
Skeletons; Bezier curves; rational Bezier curves; barycentric
computations; spline functions.
Chapter 6. Is geometry a physical
or mathematical theory?
Lines and
points generated by intersections starting with a quadrangle; curvature
of lines, of surfaces; Gauss' intrinsic curvature; Riemann's space; algebraic
geometry; differential geometry; groups and geometries.
Chapter 7. Geometry of space-time
Groups of Galileo, Lorentz and Poincaré; Minkowski
space and special theory of relativity; Langevin's traveller; Newton and
Einstein; general theory of relativity; groups and theoretical physics;
strings.