Course Summary

1. Basic Euclidean Constructions (with a ruler and a compass).

* Bisecting a line segment.

* Constructing perpendicular or parallel lines.

* Subdividing segments into equal parts.

* Finding the center of a circle.

* Replicating angles.

* Other.

-> Can you bisect an angle ?

-> Can you trisect an angle ?

2. The Golden Section and the Fibonacci numbers.

* Constructing the golden cut of a line segment.

* The Golden rectangle and the golden triangles (constructions).

* Embedded golden rectangles and spirals.

* Fibonacci numbers.

3. Symmetries (in 2 dimensions).

* What are they? (rearrangements of the points in the plane preserving distances).

* Three basic kinds: rotations, reflections (with respect to lines) and translations (every symmetry is a composition of the basic symmetries).

* Symmetries of a 2D object = symmetries that permute the points within the object.

* The symmetries of a bounded objects made of rotations and reflections. Translations could be symmetries only of unbounded objects.

* Similarities: rearrangement of the points in the plane preserving angles (but not necessarily distances).

4. Tilings of the plane (and Escher).

* Tiling of a plane = a plane covered by nicely fitting 2 dimensional objects (usually filled polygons).

* Regular tiling: tiling with one type of a regular polygon; this can be done only with triangles, squares and hexagons.

* How to get Escher-like art.

Part 2.