Specific Objectives of course:

To familiarize mathematics students with the axiomatic approach to geometry from a logical, historical, and pedagogical point of view and introduce them with the basic concepts of Affine Geometry, Affine spaces and Platonic Ployhedra.

Course Outline:


Vector spaces and affine geometry:

Collinearity of three points, ratio AB/BC. Linear combinations and linear dependent set versus affine combinations and affine dependent sets. Classical theorems in affine geometry:  Thales,  Menelaus, Ceva,  Desargues. Affine  subspaces, affine maps. Dimension of a linear subspace and of an affine subspace.

Euclidean  geometry:

Scalar  product,  Cauchy-Schwartz inequality: norm of a vector, distance between two points, angles between two non-zero vectors. Pythagoras theorem, parallelogram law, cosine and sine rules. Elementary geometric loci.

Orthogonal  transformations:

Isometries  of   plane  (four  types),
Isometries of space (six types). Orthogonal bases.

Platonic   polyhedra:

Euler   theorem   on   finite   planar   graphs.
Classification of  regular  polyhedra  in  space.  Isometries  of  regular polygons and regular polyhedra.