Geometric ConstructionsSpringer Science & Business Media, 06.12.2012 - 206 Seiten Geometric constructions have been a popular part of mathematics throughout history. The ancient Greeks made the subject an art, which was enriched by the medieval Arabs but which required the algebra of the Renaissance for a thorough understanding. Through coordinate geometry, various geometric construction tools can be associated with various fields of real numbers. This book is about these associations. As specified by Plato, the game is played with a ruler and compass. The first chapter is informal and starts from scratch, introducing all the geometric constructions from high school that have been forgotten or were never seen. The second chapter formalizes Plato's game and examines problems from antiquity such as the impossibility of trisecting an arbitrary angle. After that, variations on Plato's theme are explored: using only a ruler, using only a compass, using toothpicks, using a ruler and dividers, using a marked rule, using a tomahawk, and ending with a chapter on geometric constructions by paperfolding. The author writes in a charming style and nicely intersperses history and philosophy within the mathematics. He hopes that readers will learn a little geometry and a little algebra while enjoying the effort. This is as much an algebra book as it is a geometry book. Since all the algebra and all the geometry that are needed is developed within the text, very little mathematical background is required to read this book. This text has been class tested for several semesters with a master's level class for secondary teachers. |
Inhalt
1 | |
The Ruler and Compass | 29 |
The Compass and the MohrMascheroni Theorem | 53 |
The Ruler | 69 |
The Ruler and Dividers | 83 |
The PonceletSteiner Theorem and Double Rulers | 97 |
The Ruler and Rusty Compass 107 | 106 |
Sticks | 109 |
The Marked Ruler | 123 |
Paperfolding | 145 |
The Back of the Book 161 | 160 |
Suggested Reading and References | 189 |
Index 199 | 198 |
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Häufige Begriffe und Wortgruppen
algebra angle bisector appear earlier bisect called chapter circle coefficients compass construction compass number compass point congruent consider contains coordinates cubic definition desired determined dividers point drawing equation Euclid example Exercise field Figure folding follows geometry give given point Hence inscribed integers iterated Lemma length marked ruler mathematics means midpoint Note numbers obtained operation paper line paper point parabola parallel parallelogram passes perpendicular point of intersection positive prime problem proof prove pythagorean quadratic extension rational reflection regular requires result roots ruler and compass ruler and dividers ruler construction ruler line ruler point rusty compass segment sequence side solution solve square starter set stick point sum of squares Suppose tangent Theorem theory tool toothpick triangle trisection vertices