Hands up, who thinks that maths is boring? I am willing to bet that most kids in school certainly believe that it is. Why is that, I wonder?
The evidence that maths is relevant, interesting and indeed even beautiful is all around us. The curling of a leaf, the gradual increase in size of letters as we lift a magnifying lens from the surface of a page, the sound of laughter, a symphony, a recipe… who would have thought that there are mathematical proofs behind each of these phenomena, or that maths actually is the language that can explain everything from birth to growth and decay.
The problem with modern day education is its emphasis on boring formulas and calculations which do not seem to have any significance on our lives. Mathematicians who see their subject as beautiful, creative and eternal, note that everyone should see how fascinating maths can be, if students could tap into their own creativity and to see the many fascinating applications of even the most simple of formulae.
Bear with me, as I take a look at some of the most admired mathematical formulae, and what makes them beautiful.
The experts speak
At the recent World Science Festival, Simon Singh and Marcus du Sautoy explained why, to them, maths is immensely beautiful. To listen to them speak is to understand the passion that people have for maths, once they understand it.
Marcus Sautoy, for instance, says that one of the most, groundbreaking stories he ever heard as a child was “the story that there are infinitely many primes” (Euclid’s infinite primes proof). Primes, notes du Sautoy, are the ‘atoms’ of mathematics; we build all numbers from them, in the sense that any given number can be created by multiplying a specific set of prime numbers. Du Sautoy notes that prior to Euclid’s proof, the question as to whether there was a finite number of prime numbers. What Euclid did, some 2,000 years ago, was to prove that there will always be more primes, however far we continue to count. He said that supposing there was a finite number of primes, by adding one to the latter, that set of numbers ceased to be finite.
Simon Singh notes that what makes Euclid’s proof so beautiful is that while infinity may not exist in the physical world, in the mathematical world, it does. Singh says, “it isn’t about how great Euclid was; it is about how, in four lines of argument, you can grapple with infinity”.
The proof is doubly beautiful because it will last eternally; unlike the case with the sciences, where new findings are constantly replacing previous ones, mathematical truths can never be negated. We could argue that no other subjects allows human beings to come face to face with such astounding universal truths.
Maths can move us
It isn’t just the theoretical beauty of maths that is capable of moving the human mind; a recent study, published in the journal, Frontiers in Human Neuroscience, shows that mathematical beauty can have the same effect on the mind, as the greatest works of art. In the study, researchers presented 15 mathematicians with 60 different mathematical formulae, which they were asked to rate as beautiful, indifferent or ugly.
Findings revealed that the experience of mathematical beauty occurs in the same part of the emotional brain as the experience of beauty derived from other sources, such as a beautiful painting. The more highly the mathematicians rated formulae, the more activity researchers detected in the emotional brain during functional magnetic resonance imaging scans.
Interestingly, many mathematicians agree that specific formulae are beautiful. One of the most oft admired formulae is Euler’s identity, which looks like this:. It’s arguably the most beautiful theorem in existence. E, pi and i, three extremely complicated numbers, are linked by a formula that also utilises three basic operations (addition, multiplication and exponentiation) and five of mathematics’ most vital constants (zero, one, e, pi and i).
Another beautiful theorem is that attributed to Pierre de Fermat, who showed that any prime number that could be divided by four with a remainder of one, was also the sum of two square numbers. One of the reasons for this theorem’s beauty, says Marcus du Sautoy, is that the relationship between prime numbers and squares is unexpected, yet as the theorem unfolds, we can begin to understand the intricate relationship between them.
For du Sautoy, the journey is towards the final proof is one of the most exciting aspects of mathematics; many proofs are like complex symphonies one has to hear over and over again to discover new connections and unexpected flashes of beauty. Du Sautoy notes that this passion and excitement can be found in even the most simple of theorems, and that even the youngest of students should be introduced to the immense beauty of mathematics.
Other equations scientists and mathematicians generally regarded as beautiful include Einstein’s theory of General Relativity, which actually manages to describe the complex idea of space-time. Says Kyle Cranmer, Physicist at New York University, “It is a very elegant equation… it tells you how (space-time and matter and energy) are related – how the presence of the sun warps space-time so that the Earth moves around it in orbit… it also tells you how the universe evolved since the Big Bang and predicts that there should be black holes.” Then there is the Pythagorean Theorem, the simple yet fascinating equation which states that in any right-angle triangle, the square of the length of the hypotenuse, c, equals the sum of the squares of the lengths of the two shorter sides (a and b).
Ultimately, what makes maths beautiful or not is, to a certain degree, subjective. On the other hand, mathematicians often agree that specific qualities lend proof to undeniable beauty. These include the ability of a theorem to connect seemingly unrelated areas (the way Euler’s identity, discussed above, does) and its elegance (a theorem that is surprisingly succinct, for instance, is often considered elegant). While more precise components of mathematical beauty can be difficult to identify, we should, perhaps, recall the words of Bertrand Russell, who described mathematical beauty as:
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.”