The 11 Most Beautiful maths Equations

Albert Einstein

Photo: AP

Mathematical equations aren’t just useful — many are quite beautiful. And many scientists admit they are often fond of particular formulas not just for their function, but for their form, and the simple, poetic truths they contain.While certain famous equations, such as Albert Einstein’s E = mc^2, hog most of the public glory, many less familiar formulas have their champions among scientists.

LiveScience asked physicists, astronomers and mathematicians for their favourite equations; here’s what we found:

General relativity

Standard model


Pythagorean theorem

1 = 0.999999999….

This simple equation, which states that the quantity 0.999, followed by an infinite string of nines, is equivalent to one, is the favourite of mathematician Steven Strogatz of Cornell University.

'I love how simple it is -- everyone understands what it says -- yet how provocative it is,' Strogatz said. 'Many people don't believe it could be true. It's also beautifully balanced. The left side represents the beginning of mathematics; the right side represents the mysteries of infinity.'

Special relativity

Einstein makes the list again with his formulas for special relativity, which describes how time and space aren't absolute concepts, but rather are relative depending on the speed of the observer. The equation above shows how time dilates, or slows down, the faster a person is moving in any direction.

'The point is it's really very simple,' said Bill Murray, a particle physicist at the CERN laboratory in Geneva. 'There is nothing there an A-level student cannot do, no complex derivatives and trace algebras. But what it embodies is a whole new way of looking at the world, a whole attitude to reality and our relationship to it. Suddenly, the rigid unchanging cosmos is swept away and replaced with a personal world, related to what you observe. You move from being outside the universe, looking down, to one of the components inside it. But the concepts and the maths can be grasped by anyone that wants to.'

Murray said he preferred the special relativity equations to the more complicated formulas in Einstein's later theory. 'I could never follow the maths of general relativity,' he said.

Euler's equation

This simple formula encapsulates something pure about the nature of spheres:

'It says that if you cut the surface of a sphere up into faces, edges and vertices, and let F be the number of faces, E the number of edges and V the number of vertices, you will always get V -- E + F = 2,' said Colin Adams, a mathematician at Williams College in Massachusetts.

'So, for example, take a tetrahedron, consisting of four triangles, six edges and four vertices,' Adams explained. 'If you blew hard into a tetrahedron with flexible faces, you could round it off into a sphere, so in that sense, a sphere can be cut into four faces, six edges and four vertices. And we see that V -- E + F = 2. Same holds for a pyramid with five faces -- four triangular, and one square -- eight edges and five vertices,' and any other combination of faces, edges and vertices.

'A very cool fact! The combinatorics of the vertices, edges and faces is capturing something very fundamental about the shape of a sphere,' Adams said.

Euler–Lagrange equations and Noether's theorem

The Callan-Symanzik equation

The minimal surface equation

'The minimal surface equation somehow encodes the beautiful soap films that form on wire boundaries when you dip them in soapy water,' said mathematician Frank Morgan of Williams College. 'The fact that the equation is 'nonlinear,' involving powers and products of derivatives, is the coded mathematical hint for the surprising behaviour of soap films. This is in contrast with more familiar linear partial differential equations, such as the heat equation, the wave equation, and the Schrödinger equation of quantum physics.'

The Euler line

Glen Whitney, founder of the Museum of maths in New York, chose another geometrical theorem, this one having to do with the Euler line, named after 18th-century Swiss mathematician and physicist Leonhard Euler.

'Start with any triangle,' Whitney explained. 'Draw the smallest circle that contains the triangle and find its centre. Find the centre of mass of the triangle -- the point where the triangle, if cut out of a piece of paper, would balance on a pin. Draw the three altitudes of the triangle (the lines from each corner perpendicular to the opposite side), and find the point where they all meet. The theorem is that all three of the points you just found always lie on a single straight line, called the 'Euler line' of the triangle.'

Whitney said the theorem encapsulates the beauty and power of mathematics, which often reveals surprising patterns in simple, familiar shapes.

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