## The most beautiful equation?

There’s an interesting article on the BBC website today that discusses the way mathematicians’ brains appear to perceive “beauty”. A (slightly) more technical version of the story can be found here. According to functional magnetic resonance imaging studies, it seems that beautiful equations excite the same sort of brain activity as beautiful music or art.

The question of why we think equations are beautiful is one that has come up a number of times on this blog. I suspect the answer is a slightly different one for theoretical physicists compared with pure mathematicians. Anyway, I thought it might be fun to invite people offer suggestions through the comments box as to the most beautiful equation along with a brief description of why.

I should set the ball rolling myself, and I will do so with this, the Dirac Equation:

This equation is certainly the most beautiful thing I’ve ever come across in theoretical physics, though I don’t find it easy to articulate precisely why. I think it’s partly because it is such a wonderfully compact fusion of two historic achievements in physics – special relativity and quantum mechanics – but also partly because of the great leaps of the imagination that were needed along the journey to derive it and my consequent admiration for the intellectual struggle involved. I feel it is therefore as much an emotional response to the achievement of another human being – such as one feels when hearing great music or looking at great art – as it is a rational response to the mathematical structure involved. But it’s not just that, of course. The Dirac Equation paved the way to many further developments in particle physics. It seems to encapsulate so much about the behaviour of elementary particles in so few symbols. Some of its beauty also derives from its compactness.

Anyway, feel free to suggest formulae or equations through the comments box, preferably with a brief explanation of why you think they’re so beautiful.

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February 13, 2014 at 12:43 pm

Eulers Identity… just wow!

E=mc^2 packs so many ideas into such a short space though.

February 13, 2014 at 12:45 pm

Although it should actually be E^2 = p^c^2 + m^2c^4 or, in beautiful units, E^2=p^2+m^2

February 16, 2014 at 6:19 pm

Actually, Einstein’s famous equation is just E = M. The c-square is unnecessary; but I guess people need c-square to tell everyone they passed college algebra and they “understand” it.

February 16, 2014 at 6:22 pm

E^2=p^2+m^2 is the full equation in natural units.

February 17, 2014 at 3:38 am

But could we not use “c=1” and have E = M?

February 13, 2014 at 1:04 pm

The Hamiltonian of the Harmonic Oscillator:

p(t)^2/(2 m)+1/2 m omega^2 x(t)^2

Early on, it was impressed upon me that the harmonic oscillator is one of the most beautiful and important systems in all of physics (I think it was Sidney Coleman who said that “The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction.”).

But I’d only just started to absorb the idea when the Hamiltonian was suddenly dropped on me. And the first thing I ever saw in a Hamiltonian formulation was the harmonic oscillator. Still haven’t really gotten over its effect. Indeed, it more or less represents the level I’m currently at in my studies.

I very nearly went with the Friedman equation, but that’s really way above my pay grade.

February 13, 2014 at 1:28 pm

I’d also say Dirac’s equation. Not only for content but also for visual beauty. In fact, its one of my favorite ‘when on the telephone’ scribbles. I think its visually more appealing written with the dot product tho, \gamma \dot \partial. Furthermore, the calligraphy for how to draw a proper \psi also allows for a lot of creativity.

February 13, 2014 at 1:28 pm

I like Kepler’s first law, for its originality, simplicity, impact, and Kepler’s tortuous journey in getting there. It is even expressed in normal language and not as an equation! Kepler deserves to be included in any list of great mathematicians.

February 13, 2014 at 2:03 pm

As a non-physicist and a non-mathematician, I don’t think I will ever experience the beauty of an equation…which I really regret 😦

February 13, 2014 at 4:07 pm

Feynman said that all equations of physics can be written as U=0.

That’s too stark to be beautiful, I’d say. A beautiful equation should hide a lot of detail, but the compression process must still let you glimpse some of the power of the machinery underneath. I’d give the example of Maxwell’s equations, which I found a complicated mess on first encounter. But when I saw them written as a single wave equation for the 4-potential A^{mu}, that just blew me away (still does):

\partial^\nu \partial_\nu A^\mu = \mu_0 J^\mu

February 13, 2014 at 10:47 pm

Hear hear!

February 13, 2014 at 6:37 pm

Einstein’s equation, R_{\mu \nu} – 1/2 R g_{\mu \nu} = 8 \pi T_{\mu \nu} (with \Lambda g_{\mu \nu} included as dark energy on the right). Equating geometry on the left to distribution of matter/energy and its flow on the right is, for me, amazing. I also love the applications of this equations to cosmology, astrophysics, and GPS, and I love the pure maths (e.g., differential geometry) that arises.

February 13, 2014 at 8:45 pm

I like Gauss’s Law for electromagnetism and gravitation. But I don’t know how to write equations here, so I shan’t bother trying to write it.

February 14, 2014 at 11:08 am

As a so-called rocket scientist, for me it’s the rocket equation: Delta V = g Isp ln (m_i/m_f) . With this one first-year level doodle the universe is ours (that, and lots of lots and complicated engineering, but that’s accessory🙂 ).

February 25, 2014 at 10:26 am

Dirac’s equation is probably the equation I would consider to be “the most beautiful” too, although Euler’s equation [identity] comes a close second.

May 30, 2014 at 5:53 pm

Maxwell’s equations, definitely! Quite the ‘waw’ moment in my physics education.

May 31, 2014 at 4:14 pm

[…] As you will observe, this time the (Lemon Drizzle) cake was decorated with the Dirac Equation (which I consider to be the most beautiful equation in physics).. […]

November 16, 2015 at 12:11 am

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