# Beauty is in the Eye of the Mathematician

By Maths Doctor Wednesday, September 24, 2014

*This article has been written by Michele Cheng, Project Assistant at Mactrac (Maths Doctor's tutor platform)*

The natural world is simply full of patterns modelled on mathematical perfection.

Of all the subjects that are taken in school, I sometimes feel like maths gets a bad (aesthetic) reputation. Mention beauty, and the first things that come to mind are probably art or music – it's almost never maths. After all, numbers aren't particularly known for being attractive. What does nature have to say about this though?

The natural world is simply full of patterns modelled on mathematical perfection. From the six-fold symmetry of each unique snowflake to Fibonacci spirals that can even be found in a cabbage, let’s face it... maths can be pretty stunning.

## Aesthetic Beauty

**Golden Ratio**

The golden ratio refers to two quantities whose ratio is the same as the ratio of their sum to the larger of the two quantities. For those who like things spelled out in numbers, the golden ratio is equal to 1.61803398875. This divine number can be found everywhere, even in the proportions of world-renowned architectural wonders. For example, the African UNESCO World Heritage town of Kairouan in Tunisia is home to the Great Mosque of Kairouan, and the mosque's design consistently uses the golden ratio according to a geometrical analysis carried out in 2004. The ratio is also used in Leonardo da Vinci's Vitruvian Man – the name was inspired by the works of Roman architect Vitruvius – to represent the ideal proportions of the human body.

**Logarithmic Spiral**

Moving on from ratios, the immediately apparent aesthetic beauty of spirals in the natural world also gives testament to the artistry of math. The logarithmic spiral is a precise curving pattern that can be found in everything from shells (on a smaller scale) to entire galaxies (on a larger scale). What separates a logarithmic spiral from a 'regular' arithmetic spiral is that instead of constant distances, the distance between the turnings increase with each round in geometric progression.

## Theoretical Beauty

**Simple Elegance**

Another way in which math can be perceived as beautiful is by its method of proof – simple and effective translates to mathematical beauty. Take the straight-forward but highly effective Pythagorean theorem (a2 + b2 = c2). With this elegant equation, you can calculate the hypotenuse of a right triangle knowing only the lengths of the other two sides. It doesn't stop there though. American mathematician Elisha Scott Loomis went on to publish over 300 proofs from the Pythagorean theorem in his book, *Pythagorean Proposition*.

## Beautiful Results

In mathematical terms, results drawn from the connection of two (seemingly) very different areas of mathematics can also produce a profound sort of beauty. Take, for example, the fundamental theorem of calculus. This theorem can be divided into two parts: the first states that a function's anti-derivative can be reversed by differentiation, and the second states that the definite integral of a function can be found using one of its anti-derivatives.

While the aesthetic beauty associated with an intricate geometrical pattern may be very visually appealing, mathematical beauty – based on slightly more abstract theorems that are not only sometimes based on invisible theories, but occasionally, entirely imaginary ones – can be less obvious in its attraction. So, what is it that makes maths so beautiful to some people?

Earlier this year, reporter James Gallagher explored this question in the article, “Mathematics: Why the brain sees maths as beauty” for the BBC (https://www.bbc.co.uk/news/science-environment-26151062). He discovered that mathematicians appreciate maths in the same way as many people appreciate art. When describing why Euler's identity (eiπ + 1 = 0) is such a beautiful thing, Professor David Percy from the Institute of Mathematics and its Applications said, “[g]iven that e, pi, and i are incredibly complicated and seemingly unrelated numbers, it is amazing that they are linked by this concise formula.”

Whatever your views on the appeal of numbers, equations, or patterns, one thing is certain – you can hardly look out of your window without seeing its implications. Whether you're looking at looming skyscrapers and other architectural wonders or resting your eyes in a sea of green: there's maths involved there, and it's a beautiful thing.