Can Math Be Beautiful?

April 5, 2017 Francis Finch

To commemorate Mathematics and Statistics Awareness Month, this is our first in a weekly series of posts in April exploring the beauty of numbers.

What do you find beautiful? Have you ever considered that your favorite building, flower or even vegetable may have mathematical significance? Unless you’re a mathematician or have a natural affinity for numbers, you may find it hard to believe that numbers, shapes, equations and formulae have an allure extending far beyond the intellectual realm.

Join us as we explore The Beauty of Numbers and celebrate Mathematics Awareness Month in a series of dedicated posts, you might be surprised at some of the discoveries…

Beauty and the brain

iStock-182213475.jpgBeauty is inherently subjective; people can differ wildly on what they believe to be aesthetically pleasing based on their culture, upbringing, education, personal tastes and more. ‘Beauty is in the eye of the beholder’ is a common phrase used to encapsulate this subjectivity.

However, research has indicated that beauty can be objectively identified and quantified, at least in part, through brain activity. Professor Semir Zeki and Dr Tomohiro Ishizu from the Wellcome Laboratory of Neurobiology at University College London discovered in 2011 that certain pieces of art and music have a measurable impact on the section of the brain related to pleasure and reward.

The researchers asked 21 volunteers to rate excerpts of art and music as either ‘beautiful’, ‘indifferent’ or ‘ugly’. Participants were then asked to view or listen to the examples again while lying in a functional magnetic resonance imaging (fMRI) scanner to measure their brain activity. The experiment revealed that the medial orbitofrontal cortex, a region at the front of the brain, lit up when people saw or heard art and music they had previously rated ‘beautiful’. Brain activity also increased in the caudate nucleus region, which is commonly linked to feelings of romantic love. Conversely, no specific regions of the brain were notably active when ‘ugly’ pieces were repeated.

What is mathematical beauty?

Professor Zeki and Dr Ishizu’s research suggests beauty is an abstract concept in the mind – one that exists irrespective of the source material. It didn’t take long for Professor Zeki to link this discovery to the beauty of mathematics. Three years after publishing his previous research, Professor Zeki performed a similar experiment to test how mathematicians ranked the intellectual beauty of famous formulae.

“To many of us, mathematical formulae appear dry and inaccessible, but to a mathematician an equation can embody the quintessence of beauty.”

“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.” Professor Zeki

Fifteen mathematicians were given 60 formulae and asked to rate them ‘beautiful’, ‘indifferent’ and ‘ugly’ before undergoing an fMRI scan. Again, the medial orbitofrontal cortex lit up among participants who were shown the formulae they had previously described as beautiful. But which ones were deemed the most beautiful and why? According to the results, Leonhard Euler’s identity, the Cauchy-Riemann equations and the Pythagorean identity consistently topped the rankings. Of these, Leonhard Euler’s identity was considered the most beautiful, while Srinivasa Ramanujan’s infinite series was the ‘ugliest’.

While the Wellcome Trust study falls short of identifying what exactly makes a formula beautiful, the Euler identity was favorably compared to Hamlet’s soliloquy. It is also not the first time this particular formula has ranked highly for its aesthetic value; a 1988 issue of scholarly journal ‘Mathematical Intelligencer’ revealed that its readers had voted Euler’s identity as mathematics’ most beautiful theorem. For many, Euler’s identity is admired because it uses three complex numbers (e, pi and i), as well as three basic mathematical operations (addition, multiplication and exposition). The formula also utilizes five core constants: zero, one, pi, i and e. The ability to link seemingly disparate mathematical areas and create complex relationships between them, while maintaining simplicity and succinctness of structure, is a common factor in beautiful formulas. These characteristics also exist in other elegant equations, such as Einstein’s Theory of General Relativity and Pierre de Fermat’s Two Square theorem. Our fascination with the beauty of mathematics isn’t just an intellectual pursuit, however. Mankind frequently discovers mathematical concepts in nature and has drawn inspiration from numbers and formulas for artistic purposes for thousands of years. The search for mathematical beauty is as popular now as it ever was.

The above is an extract from a free eBook, The Beauty of Numbers. Download your copy here.

Image Credit: Philartphace/Shutterstock

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