Mathematics is visible everywhere in nature, even where we are not expecting it. It can help explain the way galaxies spiral, a seashell curves, patterns replicate, and rivers bend.

Even subjective emotions, like what we find beautiful, can have mathematic explanations.

“Maths is not only seen as beautiful – beauty is also mathematical,” says Dr Thomas Britz, a lecturer in UNSW Science’s School of Mathematics & Statistics. “The two are intertwined.”

Dr Britz works in combinatorics, a field focused on complex counting and puzzle solving. While combinatorics sits within pure mathematics, Dr Britz has always been drawn to the philosophical questions about mathematics.

He also finds beauty in the mathematical process.

“From a personal point of view, maths is just really fun to do. I’ve loved it ever since I was a little kid.

“Sometimes, the beauty and enjoyment of maths is in the concepts, or in the results, or in the explanations. Other times, it’s the thought processes that make your mind turn in nice ways, the emotions that you get, or just working in the flow – like getting lost in a good book.”

Here, Dr Britz shares some of his favourite connections between maths and beauty.

### 1. Symmetry – but with a touch of surprise

In 2018, Dr Britz gave a TEDx talk on the Mathematics of Emotion, where he used recent studies on maths and emotions to touch on how maths might help explain emotions, like beauty.

“Our brains reward us when we recognise patterns, whether this is seeing symmetry, organising parts of a whole, or puzzle-solving,” he says.

“When we spot something deviating from a pattern – when there’s a touch of the unexpected – our brains reward us once again. We feel delight and excitement.”

For example, humans perceive symmetrical faces as beautiful. However, a feature that breaks up the symmetry in a small, interesting or surprising way – such as a beauty spot – adds to the beauty.

“This same idea can be seen in music,” says Dr Britz. “Patterned and ordered sounds with a touch of the unexpected can have added personality, charm and depth.”

Many mathematical concepts exhibit a similar harmony between pattern and surprise, elegance and chaos, truth and mystery.

“The interwovenness of maths and beauty is itself beautiful to me,” says Dr Britz.

### 2. Fractals: infinite and ghostly

Fractals are self-referential patterns that repeat themselves, to some degree, on smaller scales. The closer you look, the more repetitions you will see – like the fronds and leaves of a fern.

“These repeating patterns are everywhere in nature,” says Dr Britz. “In snowflakes, river networks, flowers, trees, lightning strikes – even in our blood vessels.”

Fractals in nature can often only replicate by several layers, but theoretic fractals can be infinite. Many computer-generated simulations have been created as models of infinite fractals.

“You can keep focusing on a fractal, but you'll never get to the end of it,” says Dr Britz.

“Fractals are infinitely deep. They are also infinitely ghostly.

“You might have a whole page full of fractals, but the total area that you've drawn is still zero, because it's just a bunch of infinite lines.”

### 3. Pi: an unknowable truth

Pi (or ‘π’) is a number often first learnt in high school geometry. In simplest terms, it is a number slightly more than 3.

Pi is mostly used when dealing with circles, such as calculating the circumference of a circle using only its diameter. The rule is that, for any circle, the distance around the edge is roughly 3.14 times the distance across the centre of the circle.

But Pi is a lot more than this.

“When you look into other aspects of nature, you will suddenly find Pi everywhere,” says Dr Britz. “Not only is it linked to every circle, but Pi sometimes pops up in formulas that have nothing to do with circles, like in probability and calculus.”

Despite being the most famous number (International Pi Day is held annually on 14 March, 3.14 in American dating), there is a lot of mystery around it.

“We know a lot about Pi, but we really don't know anything about Pi,” says Dr Britz.

“There’s a beauty about it – a beautiful dichotomy or tension.”

Pi is infinite and, by definition, unknowable. No pattern has yet been identified in its decimal points. It’s understood that any combination of numbers, like your phone number or birthday, will appear in Pi somewhere (you can search this via an online lookup tool of the first 200 million digits).

We currently know 50 trillion digits of Pi, a record broken earlier this year. But, as we cannot calculate the exact value of Pi, we can never completely calculate the circumference or area of a circle – although we can get close.

“What’s going on here?” says Dr Britz. “What is it about this strange number that somehow ties all the circles of the world together?

“There's some underlying truth to Pi, but we don’t understand it. This mystique makes it all the more beautiful.”

### 4. A golden and ancient ratio

The Golden Ratio (or ‘ϕ’) is perhaps the most popular mathematical theorem for beauty. It’s considered the most aesthetically pleasing way to proportion an object.

The ratio can be shortened, roughly, to 1.618. When presented geometrically, the ratio creates the Golden Rectangle or the Golden Spiral.

“Throughout history, the ratio was treated as a benchmark for the ideal form, whether in architecture, artwork, or the human body,” says Dr Britz. “It was called the ‘Divine Proportion’.

### 5. A paradox closer to magic

The unknowable nature of maths can make it seem closer to magic.

A famous geometrical theorem called the Banach-Tarski paradox says that if you have a ball in 3D space and split it into a few specific pieces, there is a way to reassemble the parts so that you create two balls.

“This is already interesting, but it gets even weirder,” says Dr Britz.

“When the two new balls are created, they will both be the same size as the first ball.”

Mathematically speaking, this theorem works – it is possible to reassemble the pieces in a way that doubles the balls.

“You can't do this in real life,” says Dr Britz. “But you can do it mathematically.

“That's sort of magic. That is magic.”

“That's sort of magic. That is magic.”

Fractals, the Banach-Tarski paradox and Pi are just the surface of the mathematical concepts he finds beauty in.

“To experience many beautiful parts of maths, you need a lot of background knowledge,” says Dr Britz. “You need a lot of basic – and often very boring – training. It’s a bit like doing a million push ups before playing a sport.

“But it is worth it. I hope that more people get to the fun bit of maths. There is so much more beauty to uncover.”