Divine Proportions: Rational Trigonometry to Universal Geometry

324 Pages·2005·3.04 MB·English

by  N J Wildberger

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Divine Proportions
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Divine Proportions
Rational Trigonometry to
Universal geometry
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CY N J Wildberger
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Assoc/Prof N J Wildberger
School of Mathematics
UNSW
Sydney 2052
Australia
email: [email protected]
webpages: http://web.maths.unsw.edu.au/~norman/
Copyright 2005 Wild Egg Pty Ltd
Typeset in Latex
Cover design by Alex Snellgrove
Printed on 100% acid-free paper by BPA Print Group
Printed in Australia
National Library of Australia Cataloguing-in-Publication Data
Wildberger, Norman John,
Divine proportions : rational trigonometry to universal
geometry
Bibliography.
Includes Index.
ISBN 0 9757492 0 X.
1. Trigonometry. 2. Geometry. I. Title.
516.2
Mathematics Subject Classification: 51XX, 14XX
Published in Australia by Wild Egg Pty Ltd
http://wildegg.com
To my parents, John and Trudy Wildberger

Preface
Divine Proportions: Rational Trigonometry to Universal Geometry introduces a
remarkable new approach to trigonometry and Euclidean geometry, with dramatic
implications for mathematics teaching, industrial applications and the direction of
mathematical research in geometry.
The key insight is that geometry is a quadratic subject. So rational trigonometry
replaces the quasi-linear notions of distance and angle with the related, but more
elementary, quadratic concepts of quadrance and spread, thus allowing the development
of Euclidean geometry over any field. This text covers the key definitions and results of
this new theory in a systematic way, along with many applications including Platonic
solids, projectile motion, Snell’s law, the problems of Snellius-Pothenot and Hansen,
and three-dimensional volumes and surface areas.
The message of this book is controversial, but it will be hard to deny the power of its
content. Teachers and students will benefit from a simpler and cleaner theory, which
takes less than half of the usual time to learn. Engineers, surveyors and scientists now
have a fascinating new set of tools that increase accuracy and reduce computation time
for geometric problems. Mathematicians finally have a logically coherent framework for
metrical geometry, which opens up exciting areas for investigation in algebraic
geometry, number theory, combinatorics and special functions.
Even though the content is in some sense elementary, this text is intended for a
mathematically mature audience. The novelty will provide some challenge even for
professional mathematicians.
With Divine Proportions it should become clearer why so many students are turned off
mathematics by trigonometry. The current curriculum attempts to arm them with
machine guns to hunt rabbits. Most students sense intuitively that this is not what
they need, but until now there has been no reasonable alternative to focus the minds of
educators. This book finally addresses the failure of modern geometry to win the
minds of young people, and provides the mathematical foundation for a dynamic and
elegant new approach to teaching trigonometry and geometry.
N J Wildberger
Sydney, 2005
vii
viii PREFACE
Acknowledgements
It is a pleasure to warmly thank Walter Bloom, Michael Duffy, Kimmo Eriksson,
Shuxiang Goh, Jack Hall, Chris Halse-Rogers, Vinh Le and John Wildberger for careful
readings of portions of the manuscript and for suggesting valuable improvements.
Helen Armstrong, Peter Brown, Daniel Chan, Peter Donovan, Hendrik Grundling,
Bruce Harvey, Rick Hill, Michael Hirschhorn, Krista Ruffolo, George Szekeres and
Martin Wildberger contributed useful comments and discussions. Dolores Knox, Thai
Son Huang, Ming Huo, Alex Ma and Ross Moore helped with layout, diagrams and
computer programming. Alex Snellgrove created the cover design. Many thanks to
them all for making this a more sensible, readable and attractive book.
This work owes a debt to the efforts of many expositors and researchers of geometry.
Some of the texts which have been particularly helpful to me have been Introduction to
geometry [Coxeter], Geometry Revisited [Coxeter-Greitzer], The Elements [Euclid], the
Concise Encyclopedia of Mathematics [Weisstein], Surveying Problems and Solutions
[Shepherd], Metric Affine Geometry [Snapper-Troyer], Trigonometric Delights [Maor],
and last but not least, my favourite 20th century mathematics text, 100 Great
Problems of Elementary Mathematics [Dorrie].
Without wonderful developments in computer software, this project could not have
been completed in the given time frame. I would like to thank Donald Knuth for his
invention of TeX, the many developers of LaTeX, MacKichan Software for Scientific
Workplace, the makers of CorelDraw and Geometric SketchPad, and the people at
Microsoft for a dependable operating system.
Thanks go to the Australian Research Council for financial support. The School of
Mathematics at UNSW provided a supportive and friendly work environment, and I
thank all my colleagues who offered encouragement despite the heretical nature of the
project. Finally I would like to thank Kim and Ali, for moral support during the
writing of this book.