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## The World's Most Beautiful Equations?137

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?"
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## The World's Most Beautiful Equations?

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• #### Much better equation art (Score:4, Informative)

on Monday December 26, 2005 @08:03PM (#14341888)
Check out Bernar Venet [bernarvenet.com]. 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 [wolfram.com] the size of a house.
• #### Re:Einstein was onto something... (Score:5, Informative)

on Monday December 26, 2005 @08:06PM (#14341898)
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.
• #### Arithmetic series (Score:4, Informative)

<slashdot@NOspaM.metasquared.com> on Monday December 26, 2005 @08:45PM (#14342046) Homepage
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.
• #### RSA Encryption (Score:4, Informative)

on Monday December 26, 2005 @09:25PM (#14342171)
RSA Encryption is based on the general form of Fermat's Theorem:
x**phi(n) = 1 mod(n)
where phi(n) is Euler's Totient function which is the number of integers less than n that are relatively prime to n. The number n is chosen to be the product of two primes, p and q. Even if n is known, it is hard of find p and q. Then phi(n) = (p-1)(q-1) and it is easy to pick a d and an e such that
d * e = 1 mod(phi(n))
You give out n and e as your public key and use n and d as your private key. Public en/decryption is done with:
Y = X**e mod(n)
Private en/decryption is done with:
X = Y**d mod(n)
• #### Re:Einstein was onto something... (Score:3, Informative)

on Monday December 26, 2005 @09:26PM (#14342175)
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.
• #### Heat Equation (Score:3, Informative)

on Monday December 26, 2005 @09:44PM (#14342252)
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.
• #### Re:Einstein was onto something... (Score:4, Informative)

on Monday December 26, 2005 @11:46PM (#14342722)
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.
• #### Emmy Noether! (Score:5, Informative)

on Monday December 26, 2005 @11:52PM (#14342754)
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.
• #### Re:The most beautiful equation is... (Score:3, Informative)

by Anonymous Coward on Tuesday December 27, 2005 @03:56AM (#14343510)
exp(pi*sqrt(163)) is only a near integer, not an exact one. See Ramanujam constant [wolfram.com].
• #### Re:Girls are Evil (Score:3, Informative)

on Tuesday December 27, 2005 @10:46AM (#14344611)
When you take the square-root of both sides you should allow for a possible change of sign so:

Girls = +/- Evil

I've noticed several design suggestions in your code.

Working...