Terriers are the most beautiful dogs, Jodie Comer is the prettiest actress and Chester is the most aesthetic city in Britain. At least that’s what the math says. Can that be true?
First of all, I would like to make it clear: There is no unanimous scientific definition of beauty. Nevertheless, one often reads about connections between mathematics and aesthetics – especially in connection with symmetries or the golden ratio. The latter is a very specific aspect ratio, which when rounded corresponds to approximately 1.618. The numerical value is an irrational number, i.e. a value with an infinite number of decimal places that never repeat regularly. You often hear that the golden ratio appears everywhere in nature, art and culture. Experts have legitimate doubts about this – even if the golden ratio is very interesting from a mathematical point of view.The claim that people perceive the golden ratio as aesthetic is persistent. According to some media reports, the British insurance comparison portal “money.co.uk” evaluated the proportions of different dog breeds and rated the animals based on their “beauty”. Accordingly, the most beautiful dog breed is the one whose ratios (length of the snout, distance between the eyes, head length and so on) are closest to the golden ratio. First place goes to the Cairn Terrier, second place to the Westie and third place to the Border Collie. The faces of female celebrities were also examined “for perfection,” as the British tabloid “The Sun” reported in July 2023. Accordingly, actress Jodie Comer’s facial features had the most in common with the golden ratio, closely followed by her colleague Zendaya and the model Bella Hadid. The British mortgage advisor “Online Mortgage Advisor” used Google Streetview to determine which British cities have the most buildings with proportions that correspond to the golden ratio. According to the results, Chester is the most beautiful British city, closely followed by London and Belfast.
Many people think mathematics is complicated and boring. In this series we would like to refute this – and present our favorite counterexamples: from bad weather to magical doublings to tax tricks. You can read the articles here.
While this may be entertaining news, such studies have nothing to do with science. »The evidence that the golden ratio is particularly appealing is quite thin. “Psychological studies in which people were shown different rectangles suggest that there is a wide range of preferences,” writes mathematician Chris Budd of the University of Bath in “Plus Magazine.”
Other claims are that the golden ratio is omnipresent in nature. The proportions of the human body (such as height in relation to the height of the belly button) would follow the golden ratio (φ). One would also find φ in the shape of pearl boats, in the arrangement of sunflower seeds, in hurricanes and breaking water waves. However, one should be careful with such statements, warn the three French biologists Teva Vernoux, Christophe Godin and Fabrice Besnard: “Often the numerical value only results from rough rounding, arbitrary exclusions or distortions in sample selection. After all, it’s not very difficult to find a quotient that is around 1.6.« You are not alone with this criticism.
Back to the origins
But how did this rumor even start? To understand this, you have to travel far back in history. The oldest traditions about the golden ratio come from ancient Greece. At that time, the scholar Euclid wondered how a line of length L could be divided into two sections A and B so that the ratio of A to B is equal to the quotient L to A. In other words: The total length L relates to the longer section A as A relates to B.
Golden Ratio | If a is to b as the total length a+b is to a, then the golden ratio has been found as the length ratio.
Since L =A + B, this can be formulated as an equation: (A + B)/A = A/B = φ. The first term can be rewritten as: (A + B)/A = 1 + B/A = 1 + 1/φ. Plugging this back into the first equation gives: φ = 1 + 1/φ, which corresponds to the quadratic formula φ2 − φ − 1 = 0. This can be solved using the p-q formula and you get: φ = ½± √(¼+1) = ½(1 ± √5). Since the original question was about length ratios, you can ignore the negative result, i.e.: φ = ½(1 + √5). So φ is an irrational number that is approximately equal to 1.618.
Euclid had found this value, but the golden ratio became really famous thanks to the mathematician Luca Pacioli, who named his famous book “Divina proportione” (German: “Divine ratio”) after it in the 16th century, illustrated by Leonardo da Vinci. And here, according to physicist Mario Livio, there was a misunderstanding. It is often claimed that Pacioli praised the golden ratio as a way to create pleasant, harmonious shapes, when in reality the mathematician preferred completely different proportions.
Real aesthetics or just a rounding error?
The golden ratio achieved a real breakthrough outside of the natural sciences in the 1860s through the work of the German doctor and psychologist Gustav Theodor Fechner. He presented subjects with rectangles with different aspect ratios and asked them which ones they found most pleasant. 76 percent chose three rectangles with aspect ratios of 1.75; 1.62 and 1.50 – with most opting for the “golden rectangle” with a ratio of 1.62.
Driven by this, Fechner examined every rectangle he could get his hands on: window frames, book covers, picture frames, and so on. He found the golden ratio everywhere and saw this as evidence that people find these proportions particularly aesthetic. The British psychologist Ian McManus came to similar conclusions to Fechner in 1980, but clarified: “Whether the golden ratio per se is in contrast to other ratios such as 1.5; 1.6 or 1.75 is unclear.”
Golden rectangle | If a and b have the golden ratio as their length ratio, it is a golden rectangle.
The role of the golden ratio in the attractiveness of faces has not yet been fully clarified. In 1990, the psychologist Judith H. Langlois and her colleagues examined which faces were attractive to people. Portraits that performed best in this experiment were those that were superimposed from 16 or 32 real images. Apparently, the more average (in the true sense of the word) a face, the more pleasant we find it. Some people thought they could find the golden ratio in exactly these average portraits. It is often said that beautiful faces have proportions that are close to φ. But the same applies here: The golden ratio φ is a precise value – so it cannot be judged whether people actually find a ratio of 1.6 or 1.55 or 1.63 more appealing.
The golden ratio rarely appears in nature
And φ is not found nearly as often in nature as some people carelessly claim. The spiral-shaped shell of pearl boats is often cited as an example. To understand how this is related to φ, you have to take a closer look at the golden rectangle. This has side lengths L and A, where L⁄A = φ.
The rectangle can be divided into a square with side length A and a rectangle with side lengths A and B – A and B also have the size ratio φ. The AB rectangle can also be divided into a square with side length B and a smaller rectangle. You can always carry on like this. In the end, smaller and smaller squares and rectangles arise within the golden rectangle, which have an aspect ratio of φ.
False spiral | The blue curve is often referred to as the “Fibonacci spiral”. However, these are just circular arcs strung together, not a spiral.
Now you can draw a quarter circle in each square. This creates a spiral curve. And this is exactly what is often associated with spirals that appear in nature: from pearl boats to hurricanes to breaking waves. But that is wrong. Strictly speaking, there is no spiral in the golden rectangle, just circular arcs lined up next to each other. At the transitions (i.e. whenever a circular arc meets a smaller one) they do not form a smooth curve, but are slightly bent. You rarely find such unevenness in nature. As it turns out, almost all natural spirals are so-called “logarithmic spirals.”
The most irrational of all irrational numbers
Nevertheless, the golden ratio is quite interesting from a scientific point of view. For example, φ is described by some mathematicians as the most “irrational” of all irrational numbers. When I first heard this I was surprised. Because the property “rational” or “irrational” is not a continuous spectrum. Either a value can be represented as a fraction of two whole numbers – or not.
In fact, not all irrational numbers can be approximated equally well using a “simple” fraction. Let’s take the circular number π for example: A good approximation for the irrational value is 22⁄7, which only deviates from π in the third decimal place. But the fraction 355⁄113 is even more convincing. The fraction only differs from π in the sixth decimal place.
However, there are no such simple approximations for the golden ratio. If you want to get to φ through fractions, the whole numbers appearing in the fractions grow rapidly. As it turns out, the golden ratio is very difficult to approximate using rational numbers – and is therefore one of the most irrational numbers of all from a mathematical perspective. So perhaps it’s better to talk about it in order to highlight the peculiarities of the golden ratio. But that probably generates less attention than cute dogs or attractive actresses.