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The Midas Mind: Explorations of the Golden Section

Posted by Jenna Capyk on November 14, 2011

The Golden Section, the Golden Proportion, the Golden Mean. What is this curious and literally irrational number? This blog post will what it is, where it shows up, how we use it, and the science and skepticism behind finding it everywhere.

The Golden Section: What is it?

The shortest answer to this question is: a number. Specifically, the golden mean is an irrational number. This does not mean that the number has trouble thinking clearly and making good decisions, but rather that it can’t be written as a fraction, also known as a ratio. Practically what this means is that the golden proportion, like pi, is a number with an endless stream of digits after the decimal place. This also means, of course, that you can impress girls at parties by wowing them with how you’ve memorized it to the n’th decimal place. Also like pi, the golden ratio is associated with a monosyllabic greek letter: phi. For those readers who like actual digits, the “golden section” is around 0.618, while phi represents the inverse of this number, around 1.618. Because this ratio has very special properties these terms are used pretty much interchangeably in general conversation.

Before delving into the properties of this “special” number, I’d like to put out the disclaimer that I am in no way a mathematician but will try my best not to misrepresent the math. There are a lot of ways to describe phi mathematically. A lot of them either involve algebraic equations or visual representations. My favourite way to explain phi is a number describing a continuous, balanced proportionality. For example, it is the only way to cut a piece of string such that the ratio of the length of the smaller piece to the bigger piece is equal to the ratio of the length of the bigger piece to the whole thing before you cut it. Mathematically, this is described by the equation a/b = b/(a+b). It’s also mathematically important for being the only solution to the equation x² = x + 1, and as such the golden section (~0.618) plus one is equal to the inverse of the golden section (~1.618), or phi.  It is also  the limit to the ratio between any two numbers in a Fibionnacci series.

You may, at this point, be asking yourself: who cares how a line is cut up? How would this show up in my life? Although the line example is a good way to explain what the ratio is, this very basic principal can be used to build all kinds of other shapes and forms. For example, the so-called “golden spiral” is the only spiral shape that looks exactly the same no matter how far you zoom in or zoom out on it, and there are also golden rectangles, triangles, pentagons, pentagrams etc. that are intimately related to the golden mean. There are some excellent diagrams in the review article “All that glitters: a review of psychological research on the aesthetics of the golden section” by Christopher D. Green. These basic shapes can in turn, be used to make up more and more complex shapes and things. Even things like people, galaxies, and quasicrystals.

The Golden Mean in the Physical World


The intriguing thing about the Golden Mean is that it is not only very important in many mathematical models, but shows up in a very real and practical sense in many places in the natural world. I don’t know many nautilus’, abalone, or tritons interested in algebra or geometry, but all of these sea creatures grow shells in the characteristic golden spiral. The same spiral can be found in pine cones, pineapples, and sunflower seed growth patterns. Likewise many many plants use pentagonal symmetry, based on the phi-related pentagon. The Fibonacci series mentioned above can also describe population growth patterns of very different types of species (such as bees vs. rabbits) and these numbers have very close mathematical ties to the golden section. Many people have also suggested that humans and other animals exhibit proportions close to the Golden Mean. For example the ratio of the length of the sections of your fingers divided by your knuckles approximates this ratio. These claims are harder to investigate, especially as they involve assumed “ideal ratios” for a species.

Non-biological sciences

Aside from biology, the shapes of the Golden Ratio show up in physical representations in the natural world. For example, tropical hurricanes and spiral galaxies often spiral in a golden or logarithmic spiral. Beaches can form in this shape due to erosive wave action. Many, many places that you wouldn’t necessarily expect to find expressions of deep mathematical laws turn out, in the end, to be ruled by these same principals.


One of these arguably counter-intuitive manifestations of the golden ratio is between molecules that make up quasi-crystalline material. Peak positions in x-ray diffraction patterns of quasi-crystals are related by the golden mean. In fact, this property is an indicator of the quasi-crystalline state. So while regular crystals are made up of evenly spaced molecules, molecules in a quasi-crystal are not spaced at regular intervals, but at intervals that relate to each other by the golden ratio.

Esthetics of the Golden Mean

The case has been argued over and over that objects that reflect the golden ratio are intrinsically pleasing from an aesthetic point of view. Knowledge of the golden mean existed in ancient Egypt and was then passed along to different ancient civilizations. Such historical figures as Euclid and Vetruvius are recorded as referencing the wonderful properties of this number. James Sully included the first English-language use of the term “golden section” in his article on aesthetics in the 1875 edition of the Encyclopedia Britannica. It’s been called the most beautiful ratio, and whether by accident or on purpose, humans have used their knowledge of this number to create works of art of all kinds. Evidence of this number shows up in architecture in ancient Egypt. Leonardo da Vinci illustrated the book De divina proportione by Luca Pacioli di Brogo, and also used this proportion heavily in many of his own works. People have even made claims that the division points in major symphonic works conform to this ratio. In a modern twist to this aesthetic appeal, plastic surgeons, most often cosmetic dentists, have used this proportion to sell their clients the “most perfectly beautiful” smile.

One of the important things to remember in studying such occurrences is the strong human tendency for confirmation bias. If you go looking for the golden proportion, you WILL find it. It then becomes a process of untangling both the numbers and the motivations surrounding phi’s true prevalence in nature and the man-made world. The importance of phi in basic mathematics is hard to refute, and I mentioned several robust examples in the natural world, but what about more tenuous examples? Finding ratios that are “really close”? Finding it in a painting? Is it coincidence? Does it occur more frequently than other ratios? How many other ratios are reflected in the same thing? As far as finding that phi really is a prevalent component of a work of art or other man-made thing, is this on purpose? Did the creator of this work incorporate this ratio explicitly or was it a consequence of an unconscious aesthetic appreciation? What does either outcome say about the intrinsic value of this number to human consciousness?

Research into the true aesthetics of Phi

Our interest in how we, as humans, perceive things expressing the golden ratio is ancient, and scientific research into phi-related psychology is some of the earliest empirical studies conducted in psychology. Many scientific studies have suggested that people really do like things that conform to the golden ratio. Various experiments have shown that when asked to choose their favourite “thing” (such as a single rectangle) from a group of other “things” (such as a bunch of rectangles with different length-to-width ratios) they tend to overwhelmingly prefer the thing which conforms most closely to phi. For example, there have been studies looking at really pared-down manifestations of phi (literally lines and rectangles) all the way up to measuring “landmarks” on human faces, assigning them a number based on how closely they conform to the golden ratio, and then asking people to rate their level of attractiveness.

Lately, a lot of the research has gone into trying to prove that humans don’t have an intrinsic response to the golden ratio. That is, a lot of scientists have been taking a skeptical look at the long-held assumption that people are able to recognize and subconsciously be affected by things conforming to this ratio, and specifically it’s aesthetic value. The alternative hypothesis is that people don’t have an intrinsic ability to pick out and like this ratio, but that as a society we have an understanding of its prevalence and a somewhat superstitious belief that it’s beautiful. According to Dr. Green, the jury is still out. While many groups have gone into this type of research with each opposing hypothesis, there appears to be no strong consensus as to whether people are truly drawn to things with phi proportionality, or whether it’s more or less just a number our society knows about and therefore names a lot. Dr. Green also brings up the point that if the effect is genuine, we are even further from knowing whether it falls on the nature or nurture side of biology; that is, whether this preference is from a physiological or learned psychological part of ourselves.

The Golden Mean and Human Cognition

One of most interesting things about our fascination with the golden ratio is that it represents an interesting experiment in the way that people can look at seemingly significant things. This number shows up legitimately in all kinds of places in math and in nature. This fact can be looked at in two skeptically opposite ways:

1) This has been worked into the fabric of the universe by some cognizant being

2) There is something about the way this number corresponds with relationships between things that can tell us something about the fundamental laws of the universe we live in

With the Golden proportion there is also the extra layer of aesthetics. In this case, we also have to consider that when we do encounter this ratio, we seem to find it pleasing. A non-skeptical view of this might be that we are recognizing the signature of a cosmic creator that made us using this formula. A more skeptical viewpoint might be to ask first IF we intrinsically find it pleasing, and then WHY. Again, it’s the difference between explaining it away with mysticism and questioning whether there is something more to be learned from this observation that we could explore scientifically and really gain some insight into our world.

Encountering something that seems mystical at first glance can be the starting point of an amazing scientific discovery. It’s often the things that don’t fit our current models that we consider magic, when really these represent an opportunity to expand the model through understanding such outliers. Some people argue that this is taking the magic out of life, but the more you understand it, our world is pretty awesome all on its own. No magic required.

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