Gary B. Meisner - The Golden Ratio: The Divine Beauty of Mathematics

THE DIVINE BEAUTY OF MATHEMATICS

Gary B. Meisner

Founder of Goldennumber.net and PhiMatrix TM

W hat makes a single number so captivating that it has persisted in our imaginations for more than two thousand years? So universal that it is found in the writings of an ancient Greek mathematician, the musings of a revolutionary cosmological scientist, the designs of a twentieth-century architect, and the intrigue of a best-selling thriller novel-turned-movie blockbuster? So pervasive that it appears simultaneously in the greatest architectural monument of the ancient world, the paintings of historys most eminent Renaissance artist, and the atomic arrangement of recently discovered quasicrystalline minerals? And so controversial that it engenders confusing and polarizing claims about its appearances and applications?

You may think, or have been told, that the evidence has already been presented, the answers have already been found, and that this case is closed. The golden ratio is not a new topicmuch has been written about it since ancient times. What could possibly be new? The answers may surprise you. Fortunately, technology and knowledge continue to advance at an ever-increasing pace, constantly providing new information that was previously unavailable. Just as new technologies in DNA evidence can reveal new truths that completely overturn a past verdict in a criminal case, new technology is giving us the information and tools to show that past verdicts on this topic were also lacking in their completeness and accuracy. Were about to overturn some past convictions, toonot convictions of felons held in a prison, but rather convictions of belief held in the mind. Beliefs can be their own form of prison, and we often dont know how imprisoned our minds are until we see the world from a variety of different perspectives.

Our new tools for collecting forensic evidence are the Internet, new software applications on much faster computing technology, and a growing global community of people sharing information. In 1997, the Internet was used by only 11 percent of the developed world and only 2 percent globally. in 2001, and followed in 2004 with my PhiMatrix software, which allows the analysis of digital images in just seconds. There is now a mind-boggling collection of images to study, many of which were not readily available in high-resolution until the last five to ten years. Many of the insights Ill share with you were contributed by users around the world who had no way of connecting with each other until very recently. So, indeed, some of the information written on this topic just a decade or two ago can now be shown to be incomplete in its facts and conclusions. And I fully suspect that technologies and information available ten or twenty years from now will bring new insights that werent readily available as I write these words today.

Whether youre a mathematician, designer, phi aficionado, or phi skeptic, I hope youll find something new, interesting, and informative in this book, and I hope it challenges you to see and apply this number in new ways. Furthermore, I hope to kindle a fire in you as we journey across time and space, exploring the very unusual and unique mathematical properties of this ubiquitous numberknown by various monikers through the agesthat has inspired so many of historys greatest minds.

This c. 100 CE fragment from papyri found at Oxyrhynchus, Egypt, shows a diagram from Book II, Proposition 5 of Euclids Elements. The first reference to the extreme and mean ratio appears in the definitions and Proposition 30 of Book VI.

WHAT IS PHI?

Lets begin this ongoing journey of discovery with a basic understanding of this intriguing number, get to know some of the people throughout history whose lives it touched, and explore where it appears and the ways in which it has been used over the millennia. Represented in shorthand by the decimal 1.618, phi is an irrational number followed by an infinite number of digits, and is accurate enough for almost any practical purpose we ask it to serve, takes much less time to write, and saves an infinite number of trees when printed. The familiar number 3.14, which relates a circles circumference to its diameter, is represented by the Greek letter (pi). Similarly, 1.618 is represented by another Greek letter, (phi), although it has taken on other aliases in different eras of history. In mathematics circles it is sometimes denoted by the Greek letter (tau). Today, it is most often called the golden ratio, but it has also been known in recent times as the golden number, golden proportion, golden mean, golden section, and golden cut. Further back in time, it was even described as divine.

This divine, golden number is unique in its mathematical properties and frequent appearances throughout geometry and nature. Most everyone learned about the number pi () in school, but relatively few curricula include phi, where well use the uppercase Greek symbol to designate 1.618, and the lowercase to designate its reciprocal, 1/1.618 or 0.618. This is perhaps in part because grasping all its manifestations can transport one beyond an academic setting into the realm of the spiritual. Indeed, unveils an unusually frequent constant of design that applies to so many aspects of life, art, and architecture, but lets begin with the purest and simplest of facts about , which are found in the field of geometry.

MAKING THE GOLDEN RATIO GOLDEN

The golden ratio wasnt golden until the 1800s. It is believed that German mathematician Martin Ohm (17921872) was the first person to use the term golden in reference to it when he published in 1835 the second edition of the book Die Reine Elementar-Mathematik (The Pure Elementary Mathematics), famed for containing the first known usage of goldener schnitt (golden section) in a footnote.

History records the ancient Greek mathematician Euclid as describing it firstand perhaps bestin Book VI of his mathematics treatise Elements:

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.

So, wheres the magic and wonder in that? Lets start with an example. If I asked you to divide a line, you could do so in many places. If you cut it in half, youll create this:

The whole line is 1. Lets call it A.
The first segment is /. Lets call it B.
The second segment is also /. Lets call it C.

Here, the ratio of A to B is 2 to 1, and the ratio of B to C is 1 to 1