The Circle: A Mathematical Exploration beyond the Line

by Alfred S. Posamentier, Robert Geretschlager

Overview

The circle has fascinated mathematicians since ancient times. This entertaining book describes in layperson's terms the many intriguing properties of this fundamental shape. If math has intimidated you, this may be the ideal book to help you appreciate the discipline through one of its most important elements.The authors begin with a brief review of the basic properties of the circle and related figures. They then show the many ways in which the circle manifests itself in the field of geometry—leading to some amazing relationships and truly important geometric theorems. In addition, they explore remarkable circle constructions and demonstrate how all constructions in geometry that usually require an unmarked straightedge and a compass can also be done with the compass alone. Among other things, the reader will learn that circles can generate some unusual curves - many even quite artistic. Finally, the role of circles in art and architecture and a discussion of the circle's place on the sphere bring "full circle" this presentation of a key element of geometry.

Product Details

ISBN-13:	9781633881679
Publisher:	Rowman & Littlefield Publishers, Inc.
Publication date:	08/23/2016
Pages:	300
Sales rank:	490,381
Product dimensions:	9.00(w) x 6.20(h) x 1.30(d)

About the Author

Alfred S. Posamentier has published over sixty books in the area of mathematics and mathematics education, including, most recently, Numbers: Their Tales, Types, and Treasures (with Bernd Thaller). After having been on the faculty of the City College of the City University of New York for forty years, where he was a professor of mathematics education and dean of the School of Education, he subsequently held the same positions at Mercy College, New York, for five years. He is currently Chief Liaison for International Affairs at Long Island University.

Robert Geretschläger is a mathematics teacher at Bundesrealgymnasium Keplerstrasse in Graz, Austria. He is also an adjunct member of the mathematics faculty at the Karl-Franzens University in Graz, the coach of the Austrian International Mathematical Olympiad team, and the president of the "Kangaroo" student mathematics contest in Austria. He is the author of Geometric Origami, among other works.

Table of Contents

Acknowledgments 9

Introduction 11

1 Circle Relationships in Basic Plane Geometry-and an Extension 15

2 The Circle's Special Role in Geometry 33

3 Famous Theorems about Circles 81

4 Circle-Packing Problems 113

5 Investigation of Equicircles 139

6 Circle Constructions: The Problem of Apollonius 155

7 Inversion: Circle Symmetry 175

8 Mascheroni Constructions: Using Only Compasses 197

9 Circles in Art and Architecture 217

10 Rolling Circles: Hypocycloids and Epicycloids Christian Spreitzer 233

11 Spherical Geometry-Circles on the Sphere 261

Afterword: A Cultural Introduction to the Circle Erwin Rauscher 281

Appendixes 297

Notes 323

Bibliography 329

Index 331

Editorial Reviews

"A fascinating book taking the reader full circle through the circle's baffling properties. As a layperson I was fascinated by the authors' subtle way of leading me through complex mathematical formulas—all connected with the circle but never circular. A cleverly written book offering new perspectives for the nonexperts.”—Arthur Mettinger, professor, University of Vienna, Department of English and American Studies; and academic president, FH Campus Wien University of Applied Sciences“This book should be read by everyone who thinks math and geometry are difficult or boring. The authors, leading mathematicians themselves, have taken the topic of circles and shown us how fascinating—magic, indeed—and central to our experience of the world circles are. Posamentier and Geretschläger have explored the world of circles and their many applications in everyday life as well as in geometry, in a way that the most nervous layperson can understand and enjoy. I for one will never look at architecture or art in the same way after reading this book. I especially commend it to every teacher or confused student of mathematics.” —Baroness Pauline Perry, former president of Lucy Cavendish College, Cambridge University “After reading this wonderful book, which helped me to recall my early memories of the geometry of circles, I was absolutely enchanted by the wonderful relationships involving circles—as well as their constructions and their appearances in our culture and on the sphere—that we should have been exposed to in our school days but never were! This is the book that all who have even the slightest interest in geometry will love, to have their knowledge of circles enriched. Great book!” —Charlotte Frank, PhD, senior advisor, McGraw-Hill Education; former director of curriculum and instruction, New York City Board of Education; and former regent, University of the State of New York “This is a remarkable addition to the geometry literature, giving a very incisive new insight into the role of the circle and how it relates to the line and other geometrical forms. Whatever your background, this is sure to help you reinforce your comprehension of all things geometric.” —Peter Taylor (AO), emeritus professor, University of Canberra “Alfred S. Posamentier and Robert Geretschläger are a congenial mathematical author team; they are impressive not only for their superior content competence but also for their extraordinary talent to present mathematics to the general audience as a fascinating, enriching adventure. This book not only enhances the reader's knowledge but also is remarkable for its modest tone and splendid style.” —Rudolf Taschner, professor of mathematics, Vienna University of Technology, and 2004 Austrian Scientist of the Year “This volume is a treasure trove on everything you want to know about circles. Starting from simple results in Euclidean geometry, it reaches far both in space and time, revisiting many famous historical problems and considering more modern topics such as inversive geometry and combinatorial geometry.. .. A fantastic collaboration between two noted mathematicians.”—Andy Liu, Professor Emeritus, Department of Mathematical and Statistical Sciences, University of Alberta, and vice president, International Mathematics Tournament of the Towns

From the Publisher