Topics in Geometry
University of Joensuu 12.2. - 2.3.2007
Professor Eric Lehman (University of Caen, France)
Updated 11.10.2020 MEP
Year 2000 course with the same name
Volume: 22 hours lectures/exercises + exam or seminar presentation
(2 ov / 4 pt)
The course schedule
| Dates (updated 24.02.07) |
Times |
Places |
| Mondays 12.2. and 19.2. |
10-12 |
M8 |
| Tuesdays 13.2. 20.2. and 27.2. |
8-10, 8-10, 10-12 |
M8, M8, M352 |
| Wednesdays 14.2. 21.2. and 28.2(exam) |
10-12, 10-12, 10.00-12.00 |
M5, M352, M5 |
| Thursdays 15.2. and 22.2. |
12-14 |
M352, M8 |
| Fridays 16.2., 23.2. (and 2.3. feedback, exam ) |
12-14 |
M8 |
Examination is on Wednesday 28th of February 10.00 - 12.00 at
M5.
The start is 10.00 sharp.
Description of content (tentative) 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.
Some dynamic geometry computer exercises may be included.
Lecture Notes (version 9.3.2007)
Chapter 1. Parallel projections and central projections in plane
geometry:
a necessary and sufficient condition for three lines to be parallel,
cross ratio and applications: theorems of Menelaus, of Ceva and of the
complete quadrilateral
1. Graduation from line to another
line
2. Successive lines
parallel to the sides of a triangle
3. Lines
and points generated by intersections starting with a quadrangle
(dynamic sketch exercises)
Chapter 2. Orthocentric quadrangles:
centers of a triangle, orthocentric quadrangular, tetrahedron and orthocentric
tetrahedron
1. Triangle special points
(unfinished version: for illustration)
2. 9 Point Circle and
orthocentric quadrangle
3. Feuerbach's
theorem (in- and ex-circles)
Chapter 3. Algebraic description of a euclidean space of dimension
3
use of complex numbers to describe a euclidean plane, use of Clifford
algebra to describe a 3-dimensional euclidean space
Chapter 4. An example of a real affine plane
Chapter 5. All kinds of angles:
oriented angles of lines and cocyclicity, oriented angles of rays,
unoriented angles, measures of angles, how to define an angle?
Simson
line
Oriented
and unoriented angles of lines and rays
Chapter 6. History of geometry and classification of geometries:
examples of theorems of different natures: affine, euclidean, projective,
group acting on a set, each geometry has it's own group
?Chapter 6. Pencils of lines and pencils of circles:
relative positions of lines and circles in the plane, orthogonal lines
and circles, pencils, spheres and planes in space
Chapter 6. Geometric transformations: isometries, inversion in the plane,
inversion in space
Chapter 7. The Elements of Euclid: the XIII books, book I, the fifth postulate
Chapter 8. History of geometry and classification of geometries:
examples of theorems of different natures: affine, euclidean, projective,
group acting on a set, each geometry has it's own group
Chapter 9. Penrose tilings or Bézier curves or differential geometry
or what else will be asked
Prerequisite: common sense; basic vocabulary: point, line, vector,
circle; basic algebra. More elaborate concepts will be used from time to
time, but are not compulsary to understand the main parts of the course
and will be explained and defined at request of the audience.
Note: passing this course gives 1 ov (2 pt) on the English language
studies (in "common studies" of the Faculty of Science, only in the "old"
study system, see opinto-opas 2004-2005).
Contact person in Joensuu: Martti Pesonen (etunimi.sukunimi@uef.fi)
Department of Physics and Mathematics
http://cs.uef.fi/matematiikka/kurssit/TopicsInGeometry/