Phi (pronounced fee) is perhaps the most astonishing and fascinating number in the world. This number (whose value is 1.6180339887…) was first defined by Euclid over two thousand years ago. It is the ratio which results from dividing a line segment in such a way that the two resulting segments are in the same proportion to each other as the entire original segment is to the longer of the two resulting segments. In other words, start with one line segment AB. Place a point C somewhere along AB such that AC (the longer segment) is to CB (the shorter segment) as AB (the whole original segment) is to AC (the longer segment). If the length of AC is 1, then the length of AB is 1.6180339887…, or phi.
(There. Are you confused yet? Get the book, and it will clear everything up for you.)
Euclid first called this relationship “extreme and mean ratio”. Since then, it has been called by a variety of names, including the “Divine Proportion”, the “Golden Section”, and the “Golden Ratio”, thus the title of this book. The number was given the name phi in the early twentieth century, because the Greek letter phi is the first letter in the name of the Greek sculptor Phidas, who is believed to have made frequent use of this ratio in his work.
Phi is found throughout the universe in the most astonishing places. These include the relationship of the sides of a pentagon and a pentagram to each other. Also, the arrangement of petals around the center of a rose or of the branches of a tree around its center is dictated by this ratio. Mollusk shells and galaxies are in the shape of logarithmic spirals, whose structure is dictated by this ratio. The structure of many crystals has been found to be determined by this ratio. Phi is also believed to be the most aesthetically pleasing proportion, and many works of art are believed to feature this ratio in their composition and structure. Even the ups and downs of the stock market are believed to occur in a relationship dictated by the ratio phi.
Mario Livio examines all of these instances thoroughly, starting from the ancient history of mathematics and going all the way to the present. He concludes that phi is indeed at the heart of all of these natural phenomena. He then goes on to give a thorough examination of the instances of phi in the arts. His conclusion here is that there is little if any evidence to support the claim that phi is as prevalent in the arts as many people want to believe. There is no automatic canon by which aesthetically pleasing works of art can be generated. In fact, most enduring works of art are the result of their creators’ attempts to break away from the aesthetic canons of their day.
The final chapter is about the idea that mathematics is beautiful because it is surprising. In this chapter Livio looks at the idea, in a soon-to-be-published scientific book, that all of physics can be modeled by computer programs instead of mathematical equations. He then concludes that mathematics is not based entirely on fixed universal laws. Instead it is based on man-made axioms at the most basic level; however, once those basic axioms were chosen they had consequences which affected the development of the entire mathematical system we know today. For example, we say that 1 + 1 = 2 because we live in a universe of discrete objects. But suppose we were all blobs of liquid living in a liquid world. Then one blob and another blob would merge together to form one bigger blob. In such a world we might say that 1 + 1 = 1, and our entire mathematical system would be radically different. In my opinion, these ideas are strictly speculative, yet they are intriguing possibilities.
