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Surprising Geometric Constructions

Surprising Geometric Constructions

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Geometric constructions are fundamental to Euclidean geometry and appear in secondary-school textbooks. Most students of mathematics will also know that three constructions are impossible: trisecting angle, squaring a circle and doubling a cube. There are many interesting and surprising geometrical constructions that are probably unknown to students and teachers. This document presents these constructions in great detail using only secondary-school mathematics.

Part I presents constructions with the familiar straightedge and compass, such as approximations to pi. Of particular interest are the Mohr-Mascheroni theorem that a compass alone is sufficient and that a straightedge is not needed, as well as Steiner's theorem that a straightedge is sufficient if there exists a single circle in the plane.

In recent years, the art of origami---paper folding---has been given a mathematical formalization as described in Part II. It may come as a surprise that constructions with origami are more powerful than constructions with straightedge and compass.

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Keywords

  • Mathematics / Geometry

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