History • Geometric constructions: what can be built with just a straight-edge and a compass • Ancient Greeks asked many questions about constructions: • Can we trisect an arbitrary angle? • Is it possible to double the cube? • Can we square the circle?
Rules for Constructions 0. Start with 2 distinct points in the plane • Can draw a line through any 2 already constructed points. • Can draw a circle with center an already constructed point and through another already constructed point. • Can construct points which are at intersection of 2 distinct constructed lines, 2 distinct constructed circles, or a constructed line and a constructed circle
Definitions • A figure is constructible if we can construct it by applying rules 0 and a finite number of steps 1-3. • The sequence of steps is called a construction. • The 2 points in step 0 are called the base points.
Examples • Equilateral triangle • Square • Bisect angle • Pentagon • 15-gon • mn-gon when m and n are relatively prime
Tilings • Regular tiling of the plane consists of regular polygons