Wilson - Eulers pioneering equation: the most beautiful theorem in mathematics

Mathematics, rightly viewed,

possesses not only truth,

but supreme beauty a beauty cold

and austere, like that of sculpture

Bertrand Russell

In mathematics a theorem is a result, derived from more basic principles, thats been proved to be true: examples include the well-known Pythagorean theorem on right-angled triangles, and Euclids theorem that the list of prime numbers continues for ever. Many theorems are relatively simple to prove; others, such as Fermats last theorem, may take many years, or even centuries. Some theorems have short and straightforward proofs; others may involve much tedious and lengthy analysis of special cases. In general, mathematicians tend to prefer proofs that are efficient, ingenious, surprising, or elegant even beautiful.

So whats the most beautiful theorem in mathematics? Over many years mathematicians have conducted surveys to answer this question. The Mathematical Intelligencer, a quarterly mathematics journal, carried out such a poll in 1988 in which twenty-four theorems were listed and readers were invited to award each a score for beauty. While there were many worthy competitors, the winner was Eulers equation.

An equation is a statement with an equals sign in it, such as 1 + 1 = 2, Einsteins equation , or an algebraic equation such as . In 2004 the popular monthly magazine Physics World polled its readers to find The greatest equations ever, and even among physicists Eulers equation came a close second to the winning entry, Maxwells equations for electromagnetism. Lagging far behind these two front-runners were such classical equations as the Pythagorean theorem, Einsteins equation, Newtons second law of motion () linking force and acceleration, and Boltzmanns law of entropy (). In their responses to the polls findings the participants described Eulers equation as mind-blowing, filled with cosmic beauty, and the most profound mathematical statement ever written.

In 1933, at the age of only 14, the future Nobel Prize-winning physicist Richard P. Feynman described Eulers equation in similar terms as The most remarkable formula in math, and in later years he referred to its close relative Eulers identity in the same way, adding This is our jewel. Waxing even more enthusiastically, the Stanford mathematician Keith Devlin was moved to claim that

Like a Shakespearian sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Eulers equation reaches down into the very depths of existence.

Can this beauty be tested? In 2014, in what was described as an equation beauty contest, two neuroscientists used MRI scanning to test the brain activity of several mathematicians while viewing each of fifteen equations that theyd earlier described as beautiful, indifferent, or ugly. They found, as BBC News put it, that the same emotional brain centres used to appreciate art were being activated by beautiful maths.