Inspiration, power, simplicity - but above all, beauty

Four scientists select a favourite equation and explain its appeal:

• E=mc²
Peter Galison, Harvard University

In May 1905, while working as a Swiss patent officer, the 26-year-old Albert Einstein finished his theory of relativity, transforming ideas of space and time.

That summer, reflecting on what he had done, Einstein understood that the principle of relativity, taken with older physics, had a remarkable consequence. "Mass must be a direct measure of the energy contained in a body; light carries mass with it. The consideration is amusing and seductive; but for all I know, God Almighty might be laughing at the whole matter and might have been leading me around by the nose."

But Einstein saw that any form of energy had mass. Heat or spin a frying pan and it weighs more. The equivalence went the other way, too. Annihilating a tiny amount of mass would produce an enormous amount of energy.

In 1939, Einstein's insight explained how the nuclear fission of a uranium nucleus could produce so much energy. And in 1945, the conversion of a mass of uranium no greater than a pencil's eraser incinerated Hiroshima.

Our ambitions for science, our dreams of understanding and our nightmares of destruction packed into a few scribbles of the penI Perhaps in E=mc2 we find a beauty more like that of the ancient Greeks than like the tranquillity of a summer evening: admiration, inspiration, terror.

• The renormalisation group equation
Frank Wilczek, Massachusetts Institute of Technology

Imagine that astronomers found a planet that looked like Earth, with elephants walking on it. Imagine, too, that we did not know how far away the planet was.

Is it possible that when we went to visit we would find that everything there was just like on Earth but twice as big?

No, because the bones of a double-sized elephant could not support its weight even on Earth, much less in the increased gravity of double-Earth.

To get a double world that works like ours, you have to change the equations of physics in a very particular way. The renormalisation group equation tells us how. It is really a "magnifying equation", telling us how to construct the set of equations that describe a re-scaled world.

The virtue of this is that sometimes re-scaled objects or theories (not necessarily entire worlds) have revealing properties. The equation gives us marvellous insights into a huge range of phenomena including fractals, phase transitions, and the fundamental theory of quarks and gluons.

To me, the special beauty of the equation is its conceptual purity. It is an equation about equations; an invitation to imagine.

• Schrödinger's wave equation
Arthur I. Miller, University College London

An erotic Christmas holiday in 1925 provided the catalyst for the 38-year-old Viennese physicist Erwin Schrödinger's discovery of one of the most famous equations in 20th-century physics.

Aesthetics was always a driver because Schrodinger was "disgusted" by the atomic theory formulated by German physicist Werner Heisenberg. While Heisenberg's approach lacked visualisation and assumed that atomic processes occurred discontinuously, Schrödinger's equation represented electrons as waves moving continuously through space, like waves on a placid sea. It was beautiful. It would do for the atomic world what Isaac Newton's theory did for the macroscopic world.

Easier to use, Schrödinger's became the favoured method. Having proven that the two theories were equivalent, Schrodinger suggested to "properly use the singular". Heisenberg went ballistic, describing Schrodinger's theory as "disgusting".

Dramatically, Schrödinger's theory, suitably reinterpreted by Heisenberg, became the central one used everyday by scientists. Its coverage is breathtaking, from atomic physics to molecular chemistry and superconductivity. But right to the end Schrodinger preferred not to deal with ambiguity and some of his concerns are still unanswered. Richard Feynman put it well. "I think I can safely say that nobody understands quantum mechanics."

• The Molina-Rowland chemical equations and the ozone layer
Aisling Irwin, science writer

We commonly seek beauty among the great equations of mathematics and physics. But it is three lines of chemistry whose beauty and austerity appeal to me.

The language of chemistry has been honed to depict the essential elements and certainties that lie behind the myriad manifestations of everyday matter. The Molina-Rowland equations use the economy of chemical symbolism to depict one small but potent cycle of events occurring far above us in the stratosphere. The equations'

simplicity betrays nothing of their origins. The events they represent happen among a fug of others - a camouflage of thousands of substances that waft at the whim of changing temperatures, rising and falling pressures, and a host of other rhythms.

Within those seething interactions, Mario Molina and Sherry Rowland found, in 1974, one sequence of reactions of concern to humanity: the destruction of the ozone layer by chlorofluorocarbons. Their work was like sculpture as they chipped away at a mass of irrelevant matter to reveal the simplicity below.

The result was a message of beauty: simple and powerful enough to mobilise nations against the menace of ozone destruction, although no one was to witness it for another decade.