Constructions. History Geometric constructions: what can be built with just a straight-edge and a compass Ancient Greeks asked many questions about constructions:

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Presentation transcript:

Constructions

History Geometric constructions: what can be built with just a straight-edge and a compass Ancient Greeks asked many questions about constructions: 1.Can we trisect an arbitrary angle? 2.Is it possible to double the cube? 3.Can we square the circle?

Rules for Constructions 0. Start with 2 distinct points in the plane 1.Can draw a line through any 2 already constructed points. 2.Can draw a circle with center an already constructed point and through another already constructed point. 3.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