Francis D. Hauser - The Golden Ratio: The Facts and the Myths

The Golden Ratio

= 1.618

The Facts and the Myths

Francis D. Hauser, PhD

The Golden Ratio: The Facts and the Myths by Francis D. Hauser

Copyright 2015 Francis D. Hauser. All rights reserved.

ISBN-13: 978-1517518776
ISBN-10: 1517518776

Library of Congress Number: 2015915899
CreateSpace Independent Publishing Platform, North Charleston, SC

Synopsis

Euclid, a Greek mathematician, wrote The Elements twenty-three hundred years ago. This book is regarded as historys oldest and still referenced textbook. It is primarily about geometry and contains dozens of figures. Five of these are constructed using a line that Euclid says is cut in extreme and mean ratio. Today this is called the golden ratio and often referred to with the symbol .

Many myths have grown up around this ratio. For example, Kepler, of solar system fame, thought that was a tool of God. I wrote this book to learn about these myths.

From The Elements, I learned how Euclid used . I read about the myths. They arise from very famous people such as Pythagoras, Plato, Pacioli, Kepler, and Fibonacci. I found there is a common thread among the myths. is an irrational number (a number whose digits go on forever and never form a repeating pattern). can be used to draw pleasing figures using only an ungraded straightedge and compass. But because it is irrational, its numerical value cannot be written down using integers and fractions, which were the only numbers used in Euclids time.

Mathematicians before Euclid knew that irrational numbers existed. They could describe them with words and figures. But to many people, a number unable to be written down was absurd.

Because of Euclids book, the existence of became widely known. For centuries, when there was a close connection between religion, science, and math, even great scientists and engineers believed that was godlike.

This book discusses the myths from an engineering viewpoint. Were the pyramids built using ? Is the golden spiral really a logarithmic spiral? Are the Fibonacci numbers patterned after ?

The last chapter of the book shows how Euclid handled irrational numbers; how Euclid did algebra using geometry; and a simple visual proof of why there are only five Platonic solids.

About the Author

Francis D. Hauser earned his PhD in electrical engineering from the University of Denver in 1972.

He is a dynamic systems analyst in the motions and control of launch and reentry vehicles, multibody spacecraft, fixed and rotor-wing aircraft, large high-speed oceangoing watercraft, landcraft, and wind-driven turbines.

He is a university and college lecturer in graduate, undergraduate, and continuing-education courses on general optimization theory, Newtonian mechanics (statics and dynamics), linear algebra, Fourier analysis, and conventional, modern, and space-vehicle control.

The figures in this book were drawn using methods in Dr. Hausers other book: Excel with VBA for Engineers and Mathematicians, CreateSpace Independent Publishing, 2015. The application, Excel with VBA, resides in Microsoft Office.

Acknowledgment

Several years ago, my friend Melco Absin handed me a note that led to this book.
Do you know anything about this number, = 1.6180339887 ?

Abraham Lincoln Quoted Euclid

Things that are equal to the same thing are also equal to each other.
This is the First Common Notion in Book I of The Elements.

About the Cover

Table of Contents

Introduction

Euclid wrote The Elements twenty-three hundred years ago. It is regarded as historys oldest math textbook that is still valid and still referenced. It includes virtually all of the math that was known at the time, but is principally about geometry.

The earliest actual copy is a Greek edition by Theon, published sixteen hundred years ago. Historians think that parts of the manuscript were written by others, including Pythagoras, Eudoxus, and Thaetetus. Euclid collected these into a single manuscript along with his own work.

Euclid constructs more than 465 figures in The Elements, at least one for each of its 465 theorems (propositions). Five of these figures are constructed using a geometric quantity that mythology has come to call the golden ratio. Euclid doesnt use that name. Instead, he calls it the extreme and mean ratio, and defines it as follows: A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.

Lets define the whole line as L, the greater segment as a, and the lesser as b. Now, if we rewrite this proposition using math notation: L is said to have been cut in extreme and mean ratio when, as L is to a, so is a to b.

Referring to , L = a + b. When , L has been cut in extreme and mean ratio.

Figure 1: A Line Cut in the Extreme and Mean Ratio

Therefore, either or