What makes math beautiful

But the word beauty never defined a sunset, nor a flower, or nature in any traditional sense. Dirac was talking quantum theory and gravity. The beauty lay in the mathematics. What does it mean for maths to be beautiful? It is not about the appearance of the symbols on the page.

That, at best, is secondary. Maths becomes beautiful through the power and elegance of its arguments and formulae; through the bridges it builds between previously unconnected worlds. When it surprises. For those who learn the language, maths has the same capacity for beauty as art, music, a full blanket of stars on the darkest night.

Brain scans of mathematicians show that gazing at formulae considered beautiful by the beholder elicits activity in the same emotional region as great art and music. The more beautiful the formula, the greater the activity in the medial orbito-frontal cortex. Ask mathematicians about the most beautiful equation and one crops up time and again.

It is neat and compact even to the naive eye. But the beauty comes from a deeper understanding: here the five most important mathematical constants are brought together. The beauty of other formulae may be more obvious. Maggie Aderin-Pocock , the space scientist, declared it the most beautiful equation and she is in good company. Because it comes to life. Now energy will have mass and mass can be put into energy.

These four symbols capture a complete world. Finally, the British analyst G. Watson , on viewing certain formulas of Ramanujan , describes. We do not wish to leave the idea that only long-dead scientists express such opinions. Some very apposite quotes by Fields Medalist William Thurston have been collected by his son Dylan. In February , a team of British researchers, including two neurobiologists, a physicist and a mathematician, published a groundbreaking study in Frontiers in Human Neuroscience on the human experience of mathematical beauty.

This study is nicely summarized in a BBC Science report. These researchers employed functional magnetic resonance imaging fMRI to display the activity of brains of 16 mathematicians, at a postgraduate or postdoctoral level, as they viewed formulas that they had previously judged as beautiful, so-so or ugly.

The results of this analysis showed that beautiful formulas stimulated activity in same field, namely field A1 of the medial orbito-frontal cortex mOFC , as other researchers have identified as the seat of experience of beauty from other sources. This is an entirely satisfactory result. It gives an experimental validation of the mathematicians' intuition that they are experiencing the same qualitative states qualia as are experienced in other modalities from architecture and sculpture, to poetry and music.

So what exactly is the source of mathematical beauty? All aesthetic responses seem in part to come from identifying simplicity in complexity, pattern in chaos, structure in stasis. In the arts, "beauty" can be accounted for, at least in part, by well-understood harmonies, distributions of colors or other factors.

But what about mathematics? Aesthetic responses, as Santayana in The Sense of Beauty has argued, require a certain distance:. When we have before us a fine map, in which the line of the coast, now rocky, now sandy, is clearly indicated, together with the winding of the rivers, the elevations of the land, and the distribution of the population, we have the simultaneous suggestion of so many facts, the sense of mastery over so much reality, that we gaze at it with delight, and need no practical motive to keep us studying it, perhaps for hours altogether.

A map is not naturally thought of as an aesthetic object And yet, let the tints of it be a little subtle, let the lines be a little delicate, and the masses of the land and sea somewhat balanced, and we really have a beautiful thing; a thing the charm of which consists almost entirely in its meaning, but which nevertheless pleases us in the same way as a picture or a graphic symbol might please.

Give the symbol a little intrinsic worth of form, line and color, and it attracts like a magnet all the values of things it is known to symbolize. It becomes beautiful in its expressiveness. This captures the aesthetic in mathematics: balancing form and content, syntax and semantics, utility and autonomy. The more beautiful they rated the formula, the greater the surge in activity detected during the fMRI functional magnetic resonance imaging scans.

To the untrained eye there may not be much beauty in Euler's identity, but in the study it was the formula of choice for mathematicians.

He told the BBC: "It is a real classic and you can do no better than that. He said beauty was a source of "inspiration and gives you the enthusiasm to find out about things".

Mathematician and professor for the public understanding of science, Marcus du Sautoy, said he "absolutely" found beauty in maths and it "motivates every mathematician". He said he loved a "small thing [mathematician Pierre de] Fermat did". He showed that any prime number that could be divided by four with a remainder of one was also the sum of two square numbers. So 41 is a prime, can be divided by four with one left over and is 25 five squared plus 16 four squared.

He said it was the journey not the final proof that was exciting "like in a piece of music it's not enough to play the final chord". He said this beauty of maths was missing from schools and yet amazing things could be shown with even primary school mathematical ability. In the study, mathematicians rated Srinivasa Ramanujan's infinite series and Riemann's functional equation as the ugliest of the formulae.