## Book Review: The Golden Ratio

The Golden Ratio:  The Story Of Phi, The World’s Most Astonishing Number, by Mario Livio

Perhaps the most telling moment in this generally enjoyable book was when the author made the statement that a lot of people point to the “fiveness” of the Great Pyramid as being a sign that its architects had some awareness of the golden ratio, also known by the Greek letter phi (Φ) and then cited someone who commented that the Washington Monument was full of 5’s too, which is all the more suspicious when one considers that Washington and a great many of our nation’s founding Fathers were themselves masons and hence the occult significance of five (since phi is related to the square root of five) as well as pentagons and the pentagram-shaped star of our nation’s flag is highly relevant [1].  Rather than making phi less significant, the reverse is true.  This author does his best to rationally celebrate phi and its significance in all kinds of mathematical and natural ways while also critiquing some of the overblown and contrived reflections on phi that exist in writing, and manages to toe a delicate line but nonetheless one that fails to capture the significance of phi to those who are inclined to look for mystical interpretations.  The author seems annoyed at mystics and interested in debunking their influence rather than interested in giving full voice to their interest in this intriguing number.

In a bit more than 250 pages the author manages to discuss in narrative terms the history and significance of phi in a variety of interesting ways.  The author begins with a discussion of the long chain of mathematical insight and reasoning that was required before incommensurate ratios and irrational numbers became an area of mathematical inquiry (1) before looking at the pyramids and the Pythagorean brotherhood for the tentative beginnings of the understanding of the golden ratio (2) and trying to debunk some discussions of the Great Pyramid as being consciously modeled on the golden ratio (3).  After this the author debunks some claims about the Parthenon and points to the efforts of early mathematicians to grasp with irrational numbers (4) and the vital importance of Fibonacci in transmitting Arab and Indian mathematical insights to the European world (5).  After this the author discusses the importance of phi to Renaissance painters and mathematicians (6) as well as the way in which the golden ratio was treasured by some later painters and poets who found in phi an inspiring tribute to human insight and understanding (7).  The author then takes a look at the importance of phi in recreational mathematics that involve tiling, an area of somewhat unusual study that has proven fertile in understanding crystalline structure in recent decades (8) before closing with a discussion of whether God is a mathematician (9) and including some appendices that contain some mathematical discussion relating to phi.

I can see this book being a generally enjoyable read not only for someone like myself but for others who take a more critical attitude towards those who seek to find phi everywhere and so oversell its ubiquity and importance in the world.  Nevertheless, the author manages to demonstrate that Fibonacci sequences and golden rectangles and spirals are of immense importance in all kinds of scientific fields.  Perhaps most interesting to me was the way in which insights from tiling, where different shapes serve to point out the most efficient patterns for filling a flat surface, are relevant at all for the unusual properties of crystalline structures, a seemingly totally unrelated field.  This book is a reminder, if any reminder is necessary, that areas of study that we think of as unrelated are often deeply connected to others if we will only look at them with the perspective of an interested and curious outsider rather than someone who thinks they understand everything already.  The fact that one ratio can pop up in so many different areas of study suggests that it has a great importance that we ought to pay attention to.