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Geometric Constructions in History: Ancient Greeks and Rules for Constructible Figures

Learn about the ancient Greeks' geometric constructions and the rules for constructing figures using only a straight-edge and compass. Discover what can be built with these tools and explore questions like trisecting angles and doubling the cube. Get insights into constructible figures and examples such as equilateral triangles, squares, and more.

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Geometric Constructions in History: Ancient Greeks and Rules for Constructible Figures

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  1. Constructions

  2. 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?

  3. 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

  4. 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.

  5. Examples • Equilateral triangle • Square • Bisect angle • Pentagon • 15-gon • mn-gon when m and n are relatively prime

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