Euler’s Equation: A Thing of Great Beauty

A picture showing the equation for Euler's Identity: e^{i Pi} + 1 = 0.Euler’s Identity

Please take a moment to enjoy this thing of great beauty…  Simple to look at.  Yet incredibly profound.  Does it get any better than this? 

The likes of Euler’s identity or the Pythagorean identity are rarely mentioned in the same breath as the great works of Mozart, Shakespeare or Van Gogh.  The untrained eye may not see much beauty in Euler’s identity, but in a study in the journal Frontiers in Human Neuroscience about the experience of mathematical beauty and its neural correlates, in which 15 mathematicians were asked to rate 60 formulae, it was the formula of choice for mathematicians!

“To many of us, mathematical formulae appear dry and inaccessible but to a mathematician an equation can embody the quintessence of beauty,” says Professor Semir Zeki of University College London (UCL), the lead author of the study.  “The beauty of a formula may result from simplicity, symmetry, elegance or the expression of an immutable truth.  For Plato, the abstract quality of mathematics expressed the ultimate pinnacle of beauty.”


A Neurobiological Basis to Beauty

In the study, mathematicians were shown “ugly” and “beautiful” equations while in a brain scanner at University College London.  The more beautiful they rated the formula, the greater the surge in activity detected during the fMRI (functional Magnetic Resonance Imaging) scans.

Brain scans demonstrates a complex string of numbers and letters in mathematical formulae can evoke the same sense of beauty as artistic masterpieces and music from the greatest composers. 

The same emotional brain centres used to appreciate art were being activated by “beautiful” maths.

Prof Zeki, thinks: “A large number of areas of the brain are involved when viewing equations, but when one looks at a formula rated as beautiful it activates the emotional brain – the medial orbito-frontal cortex – like looking at a great painting or listening to a piece of music.”

The researchers suggest there may be a neurobiological basis to beauty.


A picture showing trigonometric equations.Is Euler’s Identity the Fairest Equation of Them All?

For Professor David Percy of the Institute of Mathematics and its Applications, it is a personal favourite: “It is a real classic and you can do no better than that.  At first you don’t realise the implications, it’s a gradual impact, perhaps as you would with a piece of music, and then suddenly it becomes amazing as you realise its full potential.  Beauty is a source of inspiration and gives you the enthusiasm to find out about things.”

Euler’s identity comprises the five most important mathematical constants: 0 (an additive identity), 1 (a multiplicative identity), e and π (the two most common transcendental numbers) and i (the fundamental imaginary number).

Euler’s identity also comprises the three most basic arithmetic operations: addition, multiplication and exponentiation.

Given that e, π and i are incredibly complicated and seemingly unrelated numbers, it is amazing that they are linked by such a concise formula.  Additionally to being profound in nature, the Euler formula also possesses an aesthetic quality that is both simple and elegant.

Several great physicists have spoken about the importance of seeking elegance and beauty in mathematics.  For example, British mathematician and physicist Banesh Hoffman said of his friend Albert Einstein:  “The essence of Einstein’s profundity lay in his simplicity; and the essence of his science lay in his artistry – his phenomenal sense of beauty.”

According to Prof Zeki: “Neuroscience can’t tell you what beauty is, but if you find it beautiful the medial orbito-frontal cortex is likely to be involved.”

You can find beauty in anything.