Beautiful Equations

William Blake, Newton, oil on canvas.

Einstein once claimed the only physical theories we are willing to accept are the beautiful ones. But what did he mean by beautiful? You thought art is beautiful. Scientists talk about equations being testable, equations being universal. But beautiful? Even the suggestion is disturbing to you. For most of your life they have been incomprehensible hieroglyphics. What do they describe? Do they have any use in the real world other than passing tests?
Start with E=MC2, the most famous of them all. It conjures up a whole lot of thought in your mind, main ones being it has something to do with atomic bombs and of course it's by Einstein. But what does it really mean? And more importantly, how is it beautiful?
The E here stands for energy. What is energy? Simply put, it's the capacity to do things, like lifting something up. M stands for mass which is basically the amount of stuff in a thing. And the C stands for the speed at which light rays propagate through empty space.
The equation makes a fascinating statement. It says that suppose you have some mass, it is possible to convert that mass into energy. So the equation allows you to calculate how much energy is contained in any given mass. It applies to everything: toothpaste, a book, a nail or uranium for that matter. The five symbols explain the link between energy and all matters across the cosmos. This universality is part of its beauty.
In 1933, while giving the Herbert Spencer lectures at Oxford, Einstein said that the endgame of what he does is experience of the natural world. What guided him was mathematical beauty and mathematical simplicity, he said. Laws which govern the universe would have an elegant simplicity and this could be shared by equations. “Equations are for eternity,” he said.
It's like a painter who paints abstracts. When he puts colours and shapes in a visual order, he too, like the scientist who writes equations, tries to arrive at a convincing metaphor for nature. If a piece of art tells you something important about the universe, so does an equation. Equation is a method, a means of encapsulating what all around us is. They are the most important tools of science for pushing the boundaries of knowledge; explain in a profound way the natural order.
William Blake in his coloured monotype Isaac Newton (1795-1805) shows Newton studying a tiny corner of the world with a pair of dividers. Blake despised Newton who he felt reduced the magnificence of existence to cold and mechanistic equations. But is that all Newton did? Of course you know about the proverbial apple falling from the tree which led to the 'discovery' of gravity. But what does it signify?

“Equations are for eternity.” Photo: wikipedia

His equation for gravity (1687) allows you to understand that the same gravity that makes the apple fall from the tree makes the moon move around the earth and planetary bodies move around the solar system. In short, this equation seems to make sense of, well, the universe. It's like he painted the entire universe with this equation. A few decades later it was used to predict the return of Haley's comet. Before Newton people used to believe that these are mysteries man cannot solve. Newton proved otherwise. But he never wanted his equations to be seen as the final word.
In the spring of 1901, Einstein, in Switzerland, was about to start working as a substitute mathematics teacher, the only job he could get at that time. At about the same time, Picasso was trying to get a show in Barcelona of work he had begun in Paris the year before. Broke and heartbroken over his best friend's suicide, Picasso was painting bleak indoor scenes in shades of blue. Einstein was a scientist with the head of an artist. And Picasso was an artist with the head of a scientist. They both wanted to tackle the full consequences of the newest ideas of time, space and dimension.
Our perception of the universe changed with the arrival of Einstein's theory of relativity. In the same decade something else entered our reality—modern art. Many believed Picasso was involved in the same pursuit as Einstein. Weirdly enough, the one place you hear about relativity is at art school. As an art student you will have to absorb the idea that relativity has something to do with cubist painting.
Cubism was seen as something shocking and terrifying when it first came out. It's different from a renaissance painting where you feel like you are looking through a kind of a window onto the world. With cubism the artist is deliberately confusing you as to where things are and indeed what things are. Take a look at “Seated Nude” (1909-10) by Picasso. The space in the room seems to be eating into the side of the woman. And the textures of the room seem to be no different than the textures of the woman. There is this moving around of space and objects. In a way that's deliberately confusing if you are thinking: where is the thing that looks like ordinary reality?

Pablo Picasso, Seated Nude, oil on canvas, 1909-10.

“Cubism was a new kind of beauty that looked a lot like science,” said Matthew Collings, British art critic, writer and artist. But you are not sure if cubism is science. You are troubled by the idea that you have to think of a connection between Einstein's theory of relativity and Picasso's cubism. “Well, there is one in the sense that both worked on the same problem: linking space and time,” said Professor Arthur Miller of University College of London. “In effect, Picasso had done for art in 1907 almost exactly what Einstein had done for physics in 1905. Cubism was very much a scientific research programme. It had an explicit intent to reduce forms to geometry.”
Both Einstein and Picasso were called revolutionaries in their own field. But you are still not convinced if Picasso was at the same level of that revolution.
Einstein was a revolutionary scientist because he took Newtonian science to the next level. In his Special Theory of Relativity, Einstein showed that there is no time and space but there is time-space. They are inseparable and connected by the velocity of light. That was a thought that was impossible to have before. What Einstein was able to do was to raise himself the heights of abstraction so you could glimpse a world beyond appearances. Reality as you know it had changed. Einstein did it with equations and Picasso, with cubism.

One scientist stands out in all this because he took the beauty of equations further than anyone else. His name is Paul Dirac, considered the most important theoretical physicist after Newton. “He called it the principle of mathematical beauty,” writes Dr Graham Fermello in The Strangest Man, a biography of Dirac. “What he meant by that is as the equations get more and more close to nature, they become beautiful. So for Dirac an equation had to be beautiful to stand in front of Nature. So here is a scientist who insisted that science went through a filter of beauty. And by pursuing beauty you end up with truth. It's an idea that's often used metaphorically but Dirac meant it literally.”
If you are feeling a bit exhausted to have your head crammed with all these unfamiliar ideas, here is a simple one.
There is nothing permanent in nature except change. Equations, like art, try to capture that. That is beautiful.