The Golden Ratio

For centuries, claims both scientific and pseudoscientific have been made for the golden ratio.

by Brian Dunning

Filed under Ancient Mysteries, General Science

Skeptoid #325
August 28, 2012
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Also available in Russian

Fibonacci spirals in a sunflower
Fibonacci spirals in the
seeds of a sunflower
Photo: Wikimedia

You've probably heard that for thousands of years, mankind has been fascinated by the golden ratio, which is the value of 1 to φ (phi, pronounced 'fee' in the United States and its native Greek, and 'fye' in many other countries). φ is about 1.618. It's an irrational number that goes on forever, and is defined as:


A golden rectangle, whose sides are proportional to the golden ratio, is just a bit more squat than your high-definition TV screen. This rectangle is said to be the most aesthetically pleasing, and moreover, to be found throughout nature defining the proportions of all sorts of living creatures including the ideal human face. Great composers, artists, and architects are said to have based their work upon this ratio. Players of financial markets create formulas that rely upon it. Throughout your home, you'll find many objects whose shape is tantalizing similar to a golden rectangle: books, appliances, electric outlet covers, playing cards, paintings, windows. People have found it in the structure of DNA and the arrangement of molecules in crystals. Most famously of all, the Greek Parthenon, the preeminent icon of architecture, is said by some to be based almost entirely on the golden ratio. This ratio is believed by many to be so ubiquitous in both nature and design that it's also been called the divine proportion.

φ and the golden ratio are best known by their unique mathematical and geometrical properties. If you take a rectangle whose sides are proportional to the golden ratio, you can cut a square off one end of it, and the resulting small rectangle that remains is of the exact same proportions as the original. You can cut a square off of that and you'll get a still smaller golden ratio rectangle, and you can do this ad infinitum. That's its basic geometric property.

The essential mathematical property of φ is that it is the ratio of successive numbers in the Fibonacci series. The Fibonacci series consists of values equal to the sum of the two preceding values: 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. 5+8=13, 8+13=21, 13+21=34, and on and on. As this series continues infinitely, the ratio of each number to the previous one gets closer and closer to φ; but as the series is infinite, it never quite gets there. So φ is the limit of the ratio of that sequence. By the 40th value in the Fibonacci series, which is 102,334,155, φ is accurate to 15 decimal places.

Here is perhaps the most interesting manifestation of phi in nature. It has to do with efficient packing. When you look straight down at a tree from above, the tree is most efficient if as many leaves as possible are visible and not shaded by other leaves. As a stem grows, it follows a genetic formula to know how often to produce a leaf and at what angle from the preceding leaf. If it produced each leaf at intervals such as 1/4 turn, every leaf would end up shaded by the fourth leaf above it. In fact, no matter what integer fraction of a full turn we use, we end up eventually repeating a pattern and shading leaves. So evolution eventually resulted in a more efficient genetic instruction: φ. Produce φ leaves per turn — just over 137.5° between successive leaves — and no two leaves will ever shade each other. This angle is called the golden angle.

We see a very specific and obvious example of this pattern in the seeds in the center of a sunflower. Florets, which mature into seeds, grow from the center of the sunflower, and each new one pushes the existing seeds outward. The result of each new seed growing a golden angle away from its predecessor is a sunflower with seeds packed as efficiently as possible within the circular head of the flower, no matter how the large the flower gets. This type of packing produces visibly criss-crossing spiral patterns going both directions around the head. Not coincidentally, at any size of flower, the number of clockwise spiral arms and counter-clockwise spiral arms are always two consecutive Fibonacci numbers. Seeds in a pine cone, and other similar structures from the plant world, follow the same blueprint.

This tendency for ratios based on φ to eliminate repetition has engineering applications. One of the most familiar is in the design of sound rooms for listening to music or watching movies, rooms in which we want to cancel out standing audio waves and resonances. Audio engineers refer to the golden room ratio, which establishes the basic ideal dimensions of a sound room to be 10 × 16 × 26. The height of the room 10 × φ ≈ 16, giving the length of the room, and 16 × φ ≈ 26 which gives the width of the room. Any diagonal straight-line path traveled inside a golden rectangle will reflect infinitely without ever repeating its course, and so sound waves inside a room of such dimensions are always dispersed as efficiently as possible.

Despite many books and articles claiming otherwise, the exact history of man's understanding of the golden ratio is not known. Around 500 BCE, the Greek mathematician Pythagoras established the Pythagorean school of thought, the symbol for which was a pentagram. When you inscribe a star inside a pentagon, the ratio of all the line segments is the golden ratio; so although it seems he must have known about the ratio, Pythagoras himself left no writings that tell us for certain. A more concrete early definition comes from Euclid who established the golden ratio in his book Elements around 300 BCE, calling it the extreme and mean ratio. It seems almost certain that they would have had to have known about the Fibonacci series; however it wasn't until around 1200 that Leonardo Fibonacci described the famous sequence that now bears his name, but none of Fibonacci's writings show that he ever made its connection with φ or the golden ratio. Today the concepts and relationships are well understood, and they are now common mathematical devices.

The appearance of the golden ratio in the natural world has led, almost inevitably, to its adoption and co-opting by many alternative researchers in just about every discipline. Perhaps the best known pseudoscientific claim about the golden ratio is that the Greek Parthenon, the famous columned temple atop the Acropolis in Athens, is designed around this ratio. Many are the amateurs who have superimposed golden rectangles all over images of the Parthenon, claiming to have found a match. But if you've ever studied such images, you've seen that it never quite fits, at least not any better than any other rectangle you might try. That's because there's no credible historical or documentary evidence that the Parthenon's designers, who worked more than a century before Euclid was even born, ever used the golden ratio in any way, or even knew of its existence.

Another pseudoscientific claim is that the golden ratio is found throughout the human body. Volumes of nonsense have been written claiming that all sorts of arbitrary body measurements betray the golden ratio. The width of the shoulders compared to the height of the head; the height of the navel relative to the height of the whole body; the length of the forearm compared to the distance from the head to the fingertips; and so on, and so on, and so on. Obviously these measurements are different on everyone; there is probably not a single living human for whom these many claims are true. Moreover, it's completely arbitrary. Give me any number, any ratio, or any shape, and I can just as easily come up with an equally long list of body features that's equally accurate.

The example of a book is another good one. The height and width of a common book is determined by convenience; we want it to be conveniently proportioned when closed (not too tall) and open (not too wide). Some cite 1:φ as the ideal book shape, but this is wrong. A book sized so that it's the same proportions whether open or closed is 1:√2, not 1:φ. φ is significantly larger than √2. In the paper industry, 1:√2 is called the Lichtenberg ratio.

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The simple fact is that rectangles that are neither too square nor too narrow are the most attractive, and often the most usable in design. The golden ratio falls within the range of generally pleasing rectangles, but so does √2, √3, and a lot of other numbers. There is no need to claim that a single perfect ratio exists that is best for all applications.

A good way to tell real manifestations of the golden ratio from made-up or assumed ones is whether it serves a purpose that could not also be served by a similar number. The sunflower's employment of the golden angle is for a very specific purpose, and it absolutely requires the number φ. An example of a mistaken manifestation is the claim that each of your finger joints is longer than the next by the golden ratio. Not only is this measurably wrong, but it would provide no specific benefit for this to be the case, and thus no such thing evolved. The benefit of improved accuracy at increasingly small scales of manipulation means it's useful to have progressively smaller finger segments, so that's what we have. But there's no need for that to be the golden ratio, or even all exactly the same, so they're not.

Another pseudoscientific example from nature is that of the spiraling seashell. A golden spiral is one which becomes φ times wider with each quarter turn, and it's often said that the nautilus shell follows this. Not true. A golden spiral is only one of an infinite number of possible logarithmic spirals. It's beneficial for the nautilus to be able to maintain the same shape as it grows, and this purpose is served by any approximately logarithmic spiral. Having its spiral's growth factor be based on φ would confer no additional benefit.

φ, the golden ratio, and the Fibonacci series are mathematically interesting and do have natural manifestations. That doesn't mean everything, or even anything else, is based on them. The popularity and "big name" of the "divine proportion" has been the real driver of its pseudoscientific assignment to just about anything and everything. Those whose brains' pattern-matching software is in overdrive have probably heard of the golden ratio, and so it's the one they think of whenever they see a rectangle, or a great work of art (like the Mona Lisa, which is not based on the golden ratio), or patterns in the stock market (which don't exist at all, let alone at the golden ratio), or in the numerology of the Bible (unless any other number is allowed to be considered just as significant). Not every claim about the golden ratio is the result of hyperactive pattern matching, but most are. At a minimum, such a claim is always a good tipoff that you should be skeptical.

Brian Dunning

© 2012 Skeptoid Media Copyright information

References & Further Reading

Devlin, K. "The Myth that Will Not Go Away." Devlin's Angle. The Mathematical Association of America, 1 May 2007. Web. 22 Aug. 2012. <>

Green, C. "All That Glitters: A Review of Psychological Research on the Aesthetics of the Golden Section." Perception. 1 Aug. 1995, Volume 24, Number 8: 937-968.

Livio, M. The Golden Ratio: The Story of Phi, The World's Most Astonishing Number. New York: Broadway Books, 2003.

Lucht, M. "Fee, Phi, Flim, Flam." The Skeptic. 1 Jul. 2007, Volume 27, Number 2: 27-31.

Markowsky, G. "Misconceptions about the Golden Ratio." The College Mathematics Journal. 1 Jan. 1992, Volume 23, Number 1: 2-19.

Tung, K. Topics in Mathematical Modeling. Princeton: Princeton University Press, 2007.

Reference this article:
Dunning, B. "The Golden Ratio." Skeptoid Podcast. Skeptoid Media, 28 Aug 2012. Web. 8 Oct 2015. <>


10 most recent comments | Show all 54 comments

It's also popular with stock chart technicians for support/resistance/pullback levels.

In college I looked at it musically and it creates a memorable and repeating tune when applied to major scale.

Shane, Tennessee
January 27, 2013 10:18pm

have once heard that human binocular vision is roughly golden ratio, i understand that the loose approximation is too inaccurate to compare to a mathematical ratio, variance and all. But also lends the question did, do we perceive this, because it is resonate with the nature of our vision/perception. open to thoughts pro or con, always thought it was its own distinct physical ratio that merited further inquiry is all

joe, colorado
February 3, 2013 9:53pm

No, we just do not see things very well. I have Macky to thank for looking at this.

The number of things we can see because our eyes and the processing centers of our brain and the fact we observe things without reference is astounding.

Sometimes a camera or telescope helps for comparison. Sometimes a second camera or telescope helps. These things are dirt cheap.

I really have to compliment Macky for bringing up one matter and we playing with our eyes and brains seeing a whole lot of visual artefacts.

Playing with your head is a simple thing. Playing with your family and friends heads and looking it up is another.

There always is an explanation for things if you are bothered to look it up.

Mud, Sin City
April 30, 2013 4:14am

As I commentated on another thread - the Stradivarius thread - I had a violin maker friend who made every instrument in accordance with golden section proportions. This included sound post positioning, the ratio of string length above and below the bridge, neck length, and the actual proportions of the body themselves.

He had a great deal of success and believed that the proportions were common in some historical violin making.

He also considered the setting of the violin to be very critical and that most violinists hadn't a clue how to do it. One major difference between a Strad and a cheaper violin, he said, is that if you pay millions of dollars for an instrument, you just might be keen to set it up properly - sound post position, bridge positioning and shape etc

Whether or not Golden proportions help - he was convinced they did - the real magic was in the hands of the maker who completed the task and the final adjustments. I am sure that applies equally to everything from architecture to furniture. Sometimes I think that in attributing success to the golden section the creator of a masterpiece is perhaps a little too modest - or his critic a little too coldly scientific

Beauty like love is one of lifes greatest mysteries. Numbers can't explain it.

Phil, Sydney Australia
August 22, 2013 11:15pm

It's true that the appearance of phi in plant growth patterns leads to efficient packing, but the argument Brian presents does not fully explain why. Any other irrational number of leaves per turn would also prevent any pair of leaves from exactly overlapping. Phi is uniquely valuable because it creates a radial low-dispersion sequence — no two leaves exactly overlap, *and* neighboring leaves in the sequence are as far apart from each other as possible. (There are plants that don't grow like this. In Japanese maples, leaves grow in pairs oriented 180° from each other, and oriented 90° to the pair immediately below, which achieves a similar effect well enough.)

In music, golden ratio relations appear frequently enough in the lengths of different sub-passages in different pieces of music; but there's no strong application for it in the acoustics of harmonies, which have more to do with interactions between notes in concurrent overtone series on different fundamentals — strictly rational pitch relationships, with some compromises for more recent (1700s+) musical styles and practicalities of different instruments imperfect reproductions of the overtone series.

Jeremy, Asheville, NC
August 23, 2013 1:58pm

I remember the "golden ratio" as a rule for growth: take what you have now and add to it what you had before. Begin with any number. The ratio of two consecutive iterations approaches phi. I have no idea why this "growth rule" ratio approaches phi after many iterations, or why it doesn't matter what number you start with as long as it isn't zero. I conjecture that when the rule can no longer be applied, the thing that is "growing" stops growing and begins dying. Good thing, too, or the world would be a lot more crowded than it already is.

Howard B. Evans, Jr., Dayton/Ohio
August 28, 2013 6:17pm

I always thought it was a variant of the placebo effect, and looking for other ratios also work. Some figures for the golden ratio would be 1.8, not 1.618...
The golden spiral is not logarithmic, for the golden spiral comprises of arcs that are a quarter of a circle.
Other rectangles form similar spirals, but the one using the rectangle with sides in the golden ratio is the only one to have circles; the others use ellipses.

Bill, Canberra
October 20, 2013 12:35am

So those ideal sound room dimensions are 10 wide by 16 long by 26 high? I've never been in a room with such dimensions, although I have seen many reverberation algorithms in hardware and software reverb generators using the same ratio.

Chris Smart, Ancaster
November 15, 2013 11:22am

pentagram -- the sign of the school
if i remember my geometry classes from the past. there is no way to construct a pentagram using the straight line and compass -- comments/corrections

ckey, nm
July 29, 2014 11:17pm

ckey, there is indeed a way to construct a Pentagram using a straight line and a compass.

Dmytro Polovinkin, Lviv
September 26, 2014 2:15am

Make a comment about this episode of Skeptoid (please try to keep it brief & to the point).

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