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The World's Most Beautiful Equations?
music4l numb3rs asks: "'An exhibition of the world's most beautiful equations...and some of the ugliest ones too' is how the artist Justin Mullins describes his upcoming show in London. He's exhibiting a number of old favourites such as Maxwell's equations and Euler's relation plus some I've not come across such as entanglement. As for ugliness, he points to the four color theorem. My question to contemplate over the holiday period is: what do Slashdot readers think are the most beautiful equations, and the most ugly ones too?"
Check out Bernar Venet. The web site is a bit crap, a flash plugin or something. But click on 'paintings' and explore. Make sure you find the commutative diagrams the size of a house.
Much overrated as an equation. c is just a constant (and in sensible units c=1) so all it really says is that E=constant*m. This is hardly the stuff of mathematical wet dreams, even if the fact that it's true does have some interest for physicists.
sigma(i=1, n) = (n*(n+1))/2. There's something very elegant about being able to reduce a huge number of operations into three.
p = (2^(n-1)) ((2^n)-1) always struck me as beautiful as well (where p is a perfect number and 2^n - 1 is a Mersenne prime). It just has a sort of symmetry.
We don't see the world as it is, we see it as we are.
-- Anais Nin
Nothing more beautiful then that!
Except that it's only half the equation.
E^2 = (mc^2)^2 + (pc)^2
E = mc^2 only includes the energy contributed by the rest mass.
"Nature doesn't care how smart you are. You can still be wrong." - Richard Feynman
The heat equation is beautiful, as it applies to so many different things (heat, diffusion, options pricing).
u_t = k*u_xx or, more generally, u_t = k*$\Delta$u
Sigh, I wish slashdot supported some sort of LaTeX markup. u_t = k*/_\u
That's the Laplace operator, in case you couldn't tell.
"Nature doesn't care how smart you are. You can still be wrong." - Richard Feynman
You and the OP are probably using different m's. His equation (E = m c^2) is correct at all energies if m is the inertial mass. Your equation is correct if m is the rest mass.
Can't believe no one mentioned Noether's Theorem, so I'll submit it. Proof that the existence of any symmetry in a Lagrangian implies a conserved quantity.
Hence, the fact that force laws do not change with time implies conservation of energy, that they do not change with position implies conservation of linear momentum, and that they do not change with rotation implies conservation of angular momentum. Highly awesome.
exp(pi*sqrt(163)) is only a near integer, not an exact one. See Ramanujam constant.
When you take the square-root of both sides you should allow for a possible change of sign so:
Girls = +/- Evil