Facts on File Geometry Handbook

289 Pages·2008·1.37 MB·English

by  Gorini, Catherine A.

THE FACTS ON FILE
GEOMETRY
HANDBOOK
CATHERINE A. GORINI, Ph.D.
Maharishi University of Management, Fairfield, Iowa
To Roy Lane for introducing me to the fascination and joys of geometry
To my parents for their love, support, and encouragement
To Maharishi Mahesh Yogi for the gift of pure knowledge
The Facts On File Geometry Handbook
Copyright ©2003 by Catherine A. Gorini, Ph.D.
All rights reserved. No part of this book may be reproduced or utilized in any
form or by anymeans, electronic or mechanical, including photocopying,
recording, or by any information storage or retrieval systems, without
permission in writing from the publisher. For information contact:
Facts On File, Inc.
An imprint of Infobase Publishing
132 West 31st Street
New York NY 10001
ISBN-10: 0-8160-4875-4
ISBN-13: 978-0-8160-4875-5
Library of Congress Cataloging-in-Publication Data
Gorini, Catherine A.
Facts on file geometry handbook / Catherine A. Gorini
p. cm.
Includes bibliographical references and index.
ISBN 0-8160-4875-4
1. Geometry.I. Title: Geometry handbook. II. Title.
QA445.2 .G67 2003
516—dc21 2002012343
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Cover design by Cathy Rincon
Illustrations by David Hodges
Printed in the United States of America
VB Hermitage 10 9 8 7 6 5 4 3
This book is printed on acid-free paper.
CONTENTS
Acknowledgments iv
Introduction v
SECTION ONE Glossary 1
SECTION TWO Biographies 181
SECTION THREE Chronology 227
SECTION FOUR Charts & Tables 247
APPENDIX Recommended Reading 273
INDEX 277
ACKNOWLEDGMENTS
When asked by the emperor Ptolemy if there were not a shorter road to geometry
than through the Elements,Euclid replied that there was “no royal road to
geometry.” This is indeed a good fortune for all, because it has meant that the one
common road to geometry is crowded with students, teachers, amateurs,
dilettantes, experts, and nobility, all working together, sharing ideas, making
contributions large and small. I have enjoyed all my journeys on this road and
would like to acknowledge the fellow travelers who have made this road feel royal
to me, particularly Ed Floyd, David Henderson, Walter Meyer, Doris
Schattschneider, Daniel Schmachtenberger, Marjorie Senechal, and Bob Stong.
This book could not have been written without the help of many individuals, only
afew of whom I can mention here. Great appreciation goes to all the ladies of
Maharishi Spiritual University of America at Heavenly Mountain, North Carolina,
particularly Meg Custer and Marcia Murphy, for providing a quiet and sublimely
blissful atmosphere in which to work. I am indebted to Michelle Allen, Jim Bates,
Lynn Ellis, Amit Hooda, Janet Kernis, Abbie Smith, Joe Tarver, and Swati Vivek
for their help with library research and graphics. Elizabeth Frost-Knappman and
Jim Karpen played an essential role in transforming a seed of an idea into a
reality, and for this I am deeply grateful. Frank K. Darmstadt, Senior Editor at
Facts On File, provided crucial guidance and all manner of expertise with
unfailing patience.
iv
INTRODUCTION
Geometry is the study of shape, form, and space, and is of interest for its own
sake as well as for its many applications in the sciences and the arts. Geometry
is an integral part of mathematics, drawing upon all other areas in its
development, and in turn contributing to the development of other parts of
mathematics.
The roots of geometry lie in many different cultures, including the ancient
Vedic, Egyptian, Babylonian, Chinese, and Greek civilizations. The
development of geometry as a deductive science began with Thales and reached
maturity with the Elementsof Euclid around 300 B.C.E.
In the Elements,Euclid sets out a list of statements called postulates and
common notions, which are fundamental, self-evident truths of geometry. These
statements express common experiences and intuitions about space: It is flat, it
extends infinitely in all directions, and it has everywhere the same structure.
From the postulates and common notions, Euclid derives 465 propositions that
form the body of Euclidean geometry.Euclid’spresentation is systematic,
beginning with simple statements that depend only on the postulates and
common notions and concluding with intricate and exquisite propositions. Each
proposition is supported by a logical deduction or proof, based only on the
postulates, the common notions, and other propositions that have already been
proved.
Because of its clarity and logical rigor, the axiomatic method of Euclidean
geometry soon became the model for how a discipline of knowledge should be
organized and developed. The theorems of Euclidean geometry came to be
regarded as absolute truth and assumed a special role as the crowning achievement
of human thought. Trying to emulate the success of Euclid, philosophers and
scientists alikesought to find the self-evident principles from which their
disciplines could be derived.
During the 19th century,however, the discovery of non-Euclidean geometries
brought an end to the special and unique role of Euclidean geometry.Nicolai
Lobachevsky, János Bolyai, and Bernhard Riemann showed that there are other
geometries, just as systematic and just as rigorous as Euclidean geometry,but with
surprisingly different conclusions.
These new geometries revolutionized the way mathematicians viewed
mathematics and the nature of mathematical truth. A mathematical theory must
vv
IInnttrroodduuccttiioonn
be based on a collection of postulates or axioms, which are not self-evident
truths, but instead are rules assumed to be true only in the context of a specific
theory. Thus, the axioms of Euclidean geometry are different from those of
elliptic geometry or hyperbolic geometry. A mathematical theorem, proven by
logical deduction, is not true, but rather only valid, and even then valid only in
the realm governed by the axioms from which it was derived. Different axioms
give rise to different, even contradictory, theorems.
The discovery of non-Euclidean geometries strengthened the study of geometry,
giving it new vigor and vitality and opening new avenues of investigation.
However, non-Euclidean geometries were only one aspect of the development
of geometry in the 19th century, which was truly a golden age for geometry,
with projective geometry, affine geometry, vector spaces, and topology all
emerging as important and independent branches of geometry.
Today,after more gradual growth during the first part of the 20th century,
geometry is once again flourishing in a new golden age.
There are many factors supporting this renewed interest in geometry. Computers
have become both an inspiration and tool for geometry, responsible for
computational geometry,computer graphics, and robotics. Many newer areas of
geometry,such as combinatorial geometry, discrete geometry, differential
geometry, algebraic topology, dynamical systems and fractals, graph theory, and
knot theory are the natural unfoldment of discoveries of earlier centuries. Other
areas of current interest havetheir origin in applications: crystallography,
frameworks, minimal surfaces, sphere packings, and the string theories of
modern physics.
All these areas and more are included in this handbook, which will serve
students of geometry,beginner or advanced, and all those who encounter
geometric ideas in their pursuit of the sciences, arts, technology, or other areas
of mathematics.
Euclidean geometry, trigonometry, projective geometry, analytic geometry,
non-Euclidean geometry,vectors, differential geometry, topology,
computational geometry, combinatorial geometry, knot theory, and graph
theory are just some of the branches of geometry to be found here. The
selection of topics has focused on what a student would first encounter in any
of these areas. Terms from other parts of mathematics that are needed to
understand geometric terms are included in the glossary, but the reader is
referred to The Facts On File Algebra Handbookand The Facts On File
Calculus Handbookfor a more comprehensive discussion of these topics.
Nowhere is it assumed here that the reader has studied calculus.
IInnttrroodduuccttiioonn
vi
Introduction
GLOSSARY
Concrete or abstract, every mathematical concept needs a precise name that can
be used to distinguish it from everything else. The great variety of technical terms
used in geometry reflects the long history and organic growth of geometry.
Many ordinary words, such as side, face, edge, and foot, have precise
geometrical meanings. Still other words, such as segment, graph, and pole, have
taken on more than one mathematical meaning. Fortunately, the meanings of
such words can be determined from the context in which they appear.
Often, new concepts are named after individuals. Many of these, such as the
Euler line, Mandelbrot set, and Penrose tiles, are named for the person who
discovered or first studied them. On the other hand, a surprising number of
discoveries, including the Simson line and the Reuleaux triangle, are named for
individuals who had very little or even nothing to do with them.
Many geometric objects or concepts that were invented in modern times have
newly devised names based on Greek or Latin roots. Words such as fractal,
polyomino, and polytope have such an origin.
Still other geometric creations were given colorful names based on some
distinctive property or feature of their appearance; horseshoe map, monkey
saddle, batman, kissing number,tangle, and jitterbug are good examples of this.
There are close to 3,000 definitions given here, from abscissa to zonohedron.
Each term is defined as simply as possible and illustrations are used throughout
to clarify meanings. Cross-references are given in small capital letters.
BIOGRAPHIES
Geometry today is the result of the work of many individuals, each making
contributions great or small, some remembered and some virtually forgotten. The
towering greats of mathematics, Isaac Newton, Leonhard Euler, Carl Friedrich
Gauss, and David Hilbert, contributed to many areas of mathematics in addition
to geometry.There are other great figures, such as Archimedes, Nicolai
Lobachevsky, Felix Klein, and H. S. M. Coxeter, whose work was primarily in
geometry.Other mathematicians, such as Maria Agnesi, Jean Robert Argand,
Giovanni Ceva, and Helge von Koch, are remembered primarily for a single
concept bearing their name.
Manydiscoveries in geometry were the by-products of work in other areas.
Leonardo da Vinci and M. C. Escher were artists, Joseph Plateau was a
physicist, Pierre Bezier was an engineer, and Edward Lorenz is a meteorologist,
yet all made significant contributions to geometry. Still others, such as Henri
Introduction
vii
Introduction
Brocard and Robert Ammann, pursued geometry more as a hobby than as a
profession, but nevertheless made significant contributions to geometry that
now bear their names.
All these and more, for a total of more than 300 individuals, have brief
biographies here. Included are more than 60 current figures who are now alive
and still contributing to the development of geometry.
CHRONOLOGY
Geometry has had a long and rich development from ancient to modern times,
continually acquiring new knowledge without relinquishing past glories. In fact,
achievements of the past are inevitably reworked into new results. For example,
the straight lines of Euclid have become the geodesics of differential geometry,
and Euclid’s parallel lines have given rise to the ultraparallels of hyperbolic
geometry and parallel transport in differential geometry.
Old problems continue to inspire mathematicians. In recent decades,
mathematicians have found solutions to problems first posed by Kepler, Euler,
Tait, and Poincaré. Each new discovery reveals more territory to explore, and
geometry seems to progress at a faster and faster rate each decade.
More than 200 important milestones in the long history of geometry are listed
in this chronology. The list spans more than 4,000 years, with more than a
quarter of the achievements since 1950.
CHARTS AND TABLES
The results of geometry are recorded in formulas and theorems, numbers and
symbols. In this section, the reader will find more than 100 fundamental
theorems from all areas of geometry and detailed instructions for more than
50 compass-and-straightedge constructions. There are also tables of
geometric formulas, trigonometry tables, the values of important constants,
and the meanings of commonly used symbols.
RECOMMENDED READING
There is a wealth of material available to anyone who wants to explore
geometry further. The list given here includes many resources, among them
books at an introductory level, resource materials for the classroom teacher,
software and websites, and carefully selected advanced books for those who
want to probe geometry more deeply.
Each book has its own special angle and flavor, and the full range of geometry,
ancient to modern, theoretical to applied, is covered. There are books here to
appeal to every taste, need, and interest—everyone can find something suitable
for continuing the study of geometry.
Introduction
viii
SECTION ONE
GLOSSARY
1