Book review: The Golden Ratio by Mario Livio

A new subgenre of mathematical books seems to have evolved over the past few years, in which the author chooses some special number and describes its history and properties.  Books have been written about the numbers pi (more than once), e, i, and even a rather obscure number known as Euler's constant (which is not the same thing as e, though many people confuse the two).  The latest addition to this genre is a book by astronomer Mario Livio about a number known as the Golden Ratio.

The Golden Ratio, also referred to by the Greek letter phi, can be defined in many ways, but the classical way is as follows:  Consider a line segment divided into two parts, such that the ratio of the larger part to the smaller is the same as the ratio of the entire segment to the larger part.  There is a unique way to make this division.  The ratio in question is the Golden Ratio, and has the approximate value of 1.618.  The true value cannot be represented by a finite number of decimal digits.

The Golden Ratio has many remarkable properties, which have fascinated mathematicians for thousands of years.  Chapter by chapter, Livio's book discusses a selection of these properties.  One of the most beautiful (at least to me) is that the golden ratio can be expressed as the continuous fraction 1 + 1/(1 + 1/(1 + 1/(1 + 1/(...)))).  (The dots represent an infinite repetition of the same pattern.)  Other numbers can be represented by using numbers other than 1 in this formula, but only the Golden Ratio is represented by a continued fraction with all 1's.  Livio explains how this property makes phi the number the most difficult to approximate by a rational fraction.

Other remarkable properties of the Golden Ratio's have to do with its relationship to the famous Fibonacci sequence.  This sequence of numbers is built up by starting with a pair of 1's, then forming each subsequent number by adding the previous two.  The sequence begins 1, 1, 2, 3, 5, 8, 13, and continues indefinitely from there.  The Fibonacci numbers themselves have many fascinating properties, and many of these have to do with the Golden Ratio.  For instance, if you divide successive terms of the Fibonacci sequence, the results more and more closely approximate the Golden Ratio.

Besides its numeric properties, phi has a host of important geometric properties as well.  Many have to do with proportions in the pentagon and the star-shaped pentagram.  The Greeks were well aware of many of these properties.  Later in history, relationships were also discovered with other more complex geometric figures, such as the logarithmic spiral.  Surprising discoveries of phi's geometric properties have continued into the last hundred years.  It appears yet again in the study of such up-to-date topics as quasicrystals and fractals.

The Golden Ratio's remarkable geometric traits has led to a kind of cottage industry of looking for it in areas such as art and architecture.  It is often claimed that the most aesthetically attractive rectangle is the one where the ratio of the sides is phi (this is likely the reason that it is referred to as the "Golden Ratio").  There isn't really a lot of strong empirical evidence to back this claim up, but it is a very popular one (Livio cites some evidence to the contrary).  As the Golden Ratio was definitely known to the Greeks, it is often sought in their classical works.  Going even further back, people look for it in ancient Egyptian architecture.

Almost inevitably these claims tend to the very grandiose--if the Egyptians used the Golden Ratio in their architecture, naturally they must have done so in their most prominent works, such as the pyramids.  For example, consider the structure of the Great Pyramid.  From the midpoint of any side, you can draw a line to the tip of the pyramid, and another line to the point on the ground directly below the tip.  One of the "Golden Numberist" claims is that the ratio of these two lengths is exactly the Golden Ratio.  If so, it may well have been built into the pyramid deliberately, as if it were a different height the lines would have a different ratio.  Thus if the claim is true, it strongly suggests that the Egyptians knew about the Golden Ratio.  However, Livio questions the evidence for this claim.  It is purportedly based on a passage from the Greek historian Herodotus.  Livio cites research that this supposed quote has been misread.  However this has not stopped enthusiasts from repeating the claim endlessly--the strong fascination of the Golden Ratio keeps the myths alive.

This fascination extends to many other works of art and architecture.  As might be expected, Leonardo da Vinci's works have been the focus of such study.  Livio discusses several claims about the appearance of the Golden Ratio in da Vinci's works, which basically amount to finding some rectangle bounding a part of a da Vinci painting, measuring its sides, and determining that they are in the Golden Ratio.  For instance, the dimensions of two different versions of "Madonna on the Rocks" are close to the Golden Ratio to within a couple of percentage points:  1.64 and 1.58.  However such raw data is equally consistent with da Vinci using the simpler ratio of 1.6 too.  Another example is an unfinished painting of St. Jerome.  Supposedly the rectangle bounding the figure of St. Jerome uses the Golden Ratio.  However the rectangle has to be drawn somewhat fancifully, missing part of the Saint's figure, so it is dubious that it has anything significant to do with the painting's composition.  In the final example, a sketch of an old man's head, the rectangles are explicitly drawn into the figure by da Vinci himself, but there are many of them, most not having anything to do with the Golden Ratio.  Even for the ones that are close to Golden Rectangles, the lines are thick enough to leave some doubt about the precision of the ratio.

I find it interesting that people find it so compelling that artists might use the Golden Ratio in their work.  Sometimes it's said (without much justification) that the rectangle whose sides bear the Golden Ratio is the "most beautiful".  Given this, it's not even necessary that the artist deliberately build the ratio into their composition; they might just be doing it from a superior aesthetic sense.  However I find even this to be a dubious point of view.  Does anybody really believe that Leonardo's genius is reducible simply to simple numeric relationships?  Finding the Golden Ratio everywhere doesn't do justice to artists, and I sense that Livio believes the same.  Perhaps this is why he spends so much time on the aesthetic aspects of the Golden Ratio.

Though the occurrence of the Golden Ratio in art and architecture may be dubious, it indubitably appears in nature.  The logarithmic spiral, intimately related to the Golden Ratio, is famously manifest in the shell of the chambered nautilus.  Within its shell, this creature grows at a constant rate.  This results in the spiral, due to its special properties of self-similarity.  For similar reasons, spiral structures are seen in the florets of sunflowers.  Not only that, but the floret spirals come in Fibonacci numbers.

Unlike the claims that the Golden Ratio exists in works of art, its occurrences in nature are unambiguously factual relationships, backed up by clear explanations for why they are true.  Livio is much more impressed by these relationships than the putative artistic manifestations of the Golden Ratio.  This leads him at the end of the book to speculate expansively about the relationships between mathematics and nature.  Such questions have been asked by many, but with few definitive answers.  In fact, the title of one famous paper by physicist Eugene Wigner was entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".  Livio is as mystified by how often mathematics appears in natural phenomena as Wigner, but attempts to provide some provisional answers.  I am not really convinced by any of them.  In the end, I wonder if any answer would turn out to be truly fruitful, but the continual asking of the question may well be.

Personally, I think the basic reason the Golden Ratio seems so unavoidable is simply that it has so many simple, fundamental properties.  This is what leads it to crop up so frequently.  Additionally, these properties span so many areas of mathematics.  Not only is the Golden Ratio geometrically fruitful, it is deeply visible in algebra, number theory, and other branches.  Anywhere these branches of mathematics have connection to the real world, the Golden Ratio is likely to be found, often in very surprising ways.  There seems a convergence or consilience of mathematical objects that's hard to just chalk up to happenstance or our human prejudices.  Among Livio's speculations late in the book is whether an alien civilization, unfamiliar with human geometry, would know about the Golden Ratio.  I myself suspect (and Livio seems to indicate he does too) that it might.  The Golden Ratio is so pervasive that any intelligent life capable of any kind of mathematics at all might well discover it.  If an alien civilization had sophisticated mathematics but no analogue of the Golden Ratio, I wonder if we could even understand them.

All speculation aside, I don't think there's any reason to attribute any special mystery to the Golden Ratio.  It's true that it's a very remarkable number, but this is due to mathematics, not mystery.  Livio is skeptical about expansive claims that people have made about the Golden Ratio, particularly in art and architecture, and I believe his skepticism is appropriate.  Whether the Golden Ratio has some kind of aesthetic quality, its peculiar mathematical properties are quite beautiful enough.