Which Equation Deserves the Title of the World's Most Beautiful?

[Or Entity?]

Lets elect World's most beautiful equation!

Two categories:

1.Mathematics
2.Physics

My personal favourites would be:

1. e^{i\pi}+1=0 (Do i need to give an argument?)

2. E=mc^{2} (I know its mainstream.. but i doesent get much more simple and general than this!)

 

[Ed Aboud]

<br /> i\hbar\frac{\partial\Psi}{\partial t} = \frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V\Psi<br />

 

[vanesch]

I'd say: 1 = 0. From this one, you can derive everything

 

[saman]

Maxwell's equations of EM?

\nabla \cdot D= \rho
\nabla \cdot B=0
\nabla \times E=- \partial B/ \partial t
\nabla \times H=J+ \partial D/ \partial t

 

[andrewm]

p = \frac{h}{\lambda}

 

[robphy]

<br /> i\hbar\frac{\partial\Psi}{\partial t} = \frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V\Psi<br />

I think you are missing a minus sign.

 

[joeyar]

1/. cos²(x) + sin²(x) = 1
2/. ω² = k/m

 

[Borek]

2. E=mc^{2} (I know its mainstream.. but i doesent get much more simple and general than this!)

Isn't it only an approximation?

 

[redargon]

F_D=½ρv²AC_D

Some engineering fudgeamatics

 

[Binhjuventus™]

For me, it's:
i²=-1

 

[joeyar]

Isn't it only an approximation?

It's true http://nz.news.yahoo.com/a/-/world/5160859/einsteins-emc2-proven-103-years/

 

[Borek]

As far as I remember mc² is only a first term of the power series. Next terms are smaller by at least c² factor (or even c⁴, my memory fails me here), so they can be safely ignored, but E=mc² is still only an approximation.

 

[carapauzinho]

As far as I remember mc² is only a first term of the power series. Next terms are smaller by at least c² factor (or even c⁴, my memory fails me here), so they can be safely ignored, but E=mc² is still only an approximation.

I think you refer this equation:

E²=m²c⁴+p²c²

 

[Borek]

No, I refer to the way equation is derived.

http://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf

Neglecting magnitudes of fourth and higher orders we may place

 

[robphy]

As far as I remember mc² is only a first term of the power series. Next terms are smaller by at least c² factor (or even c⁴, my memory fails me here), so they can be safely ignored, but E=mc² is still only an approximation.

You may referring to the Taylor expansion (with respect to the velocity) for the relativistic energy:
http://en.wikipedia.org/wiki/Kinetic_energy
expressed as
E_{rel}=m_{rel}c^2=m_0c^2 \frac{1}{\sqrt{1-(v/c)^2}}=m_0c^2\bigg(1+\frac{1}{2}(v/c)^2+\frac{3}{8}(v/c)^4+\ldots\bigg) \approx m_0c^2 \bigg( 1+\frac{1}{2}(v/c)^2 \bigg)\mbox{[for small (v/c)]}

The rest energy E_0=m_0c^2 is a Lorentz invariant, and E_{rel} and m_{rel} are observer-dependent quantities.

From a special-relativistic viewpoint, these are exact relations.

From a Newtonian-physics viewpoint, one often refers to some of these terms as "relativistic corrections".

 

[Naty1]

E= ir !

 

[Ed Aboud]

My bad

i\hbar\frac{\partial\Psi}{\partial t} = - \frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V\Psi

 

[Friesiangirl]

Well, it may be hard to say. I am a junior in high school, so my physics knowledge is limited, however, I am fond of

i^2 = -1

Imaginary x Imaginary = Real. Seems silly but awesome.

Hayley

 

[M Grandin]

Maxwell`s velocity distribution formula

f(v) = 4 \pi [ m / 2 \pi k T ]^(3/2) v^2 e^(-m v^2 / 2 k T )

The elegance because he derived this formula just by logical reasoning
almost without calculus or any kind of information. Intellectual wizardy.

 

[Loren Booda]

zeta(s)=1+(2^-s)+(3^-s)+(4^-s)+...

- the zeta function from which the Riemann hypothesis derives.

 

[optics123]

E=mc^2.

 

[Max™]

 

[jobyts]

0! = 1

I don't know if it is beautiful; but it is pretty weird.

 

[jackmell]

The equation relating primes to zeta zeros:

\psi(x)=-\frac{1}{2\pi i}\mathop\int\limits_{c-i\infty}^{c+i\infty} \frac{\zeta&#039;(s)}{\zeta(s)}\frac{x^s}{x}ds

 

[Max™]

The equation relating primes to zeta zeros:

\psi(x)=-\frac{1}{2\pi i}\mathop\int\limits_{c-i\infty}^{c+i\infty} \frac{\zeta&#039;(s)}{\zeta(s)}\frac{x^s}{x}ds

Agreed as well.

I favorited this just because it's one of my favorite studies of zeta so far.

http://arxiv.org/PS_cache/math/pdf/0309/0309433v1.pdf

X-RAY OF RIEMANN’S ZETA-FUNCTION
J. ARIAS-DE-REYNA
1. Introduction
This paper is the result of the effort to give the students of the subject
Analytic Number Theory an idea of the complexity of the behaviour of the
Riemann zeta-function. I tried to make them see with their own eyes the
mystery contained in its apparently simple definition.
There are precedents for the figures we are about to present. In the
tables of Jahnke-Emde [9] we can find pictures of the zeta-function and
some other graphs where we can see some of the lines we draw. In the
dissertation of A. Utzinger [21], directed by Speiser, the lines Re (s) = 0
and Im(s) = 0 are drawn on the rectangle (−9, 10) × (0, 29).
Besides, Speiser’s paper contains some very interesting ideas. He proves
that the Riemann Hypothesis is equivalent to the fact that the non trivial
zeros of ′(s) are on the right of the critical line. He proves this claim
using an entirely geometric reasoning that is on the borderline between the
proved and the admissible. Afterwards rigorous proofs of this statement
have been given.

 

[The legend]

Both of your choices, Entity, are my favourites! I just love those equations and the first one, i understood only a few days ago!

So for me, they are the best equations!

 

[DaveC426913]

Does the Mandelbrot Set count as an equation? It's really more of an algorithm I suppose.

 

[DR13]

2+2=5

 

[Max™]

2+2=5

How Orwellian of you.

 

[The legend]

2+2=5

 

[Chi Meson]

2+2=5

That is only true for large values of 2.

 

[Upisoft]

pi=3.0

 

[n.karthick]

For me the Fourier transform equation is the most beautiful
<br /> <br /> F(\omega)=\int_{-\infty}^{\infty} f(t) e^{i \omega t} dt

 

[Max™]

That is only true for large values of 2.

As long as the error bars are sufficiently broad it could work as well.

 

[hotvette]

I like:

x²+y²=1
y = x^x

 

[lisab]

pi=3.0

 

[Pinčiukas]

1782¹²+1841¹²=1922¹²

 

[Dembadon]

pi=3.0

Upisoft hits R for 9000 damage (.14159 Overkill)
R dies

 

[G037H3]

Upisoft hits R for 9000 damage (.14159 Overkill)
R dies

not over 9K, notice

 

[Upisoft]

Upisoft hits R for 9000 damage (.14159 Overkill)
R dies

They will fix it in the next version...

 

[xeniosm]

the very simple (and suppressed!) standard model lagrangian

 

[chaoseverlasting]

f(x)=\frac{a_0}{2} + \sum_{n=1}^\infty \, [a_n \cos(nx) + b_n \sin(nx)]

I'm surprised no one has put this one up yet!

 

[elfboy]

my identity

a single equation? i don't know too many to think of

 

[micromass]

The most beautiful "equation" is

\langle a,b~\vert~o(a)=2, o(b)=3, o(ab)=29, o((ab)^4(abb)^2)=50, o(a((ab)^4(abb)^2)^{25})=5, o(ab^{abababababb})=34\rangle

Extra points for anybody who says what I've written down here

 

[Containment]

Ohm's law gets my vote.

 

[Pengwuino]

G_{\mu \nu} = 8T_{\mu \nu}

Where I have set c = G = \pi = 1

 

[FtlIsAwesome]

I'd say: 1 = 0. From this one, you can derive everything

What about i>u ?

 

[Pattonias]

I find derivatives beautiful even though I also hate them.

 

[Pythagorean]

Math:

Euler's Equation
e^{\pm i \theta} = \cos(\theta) + i \sin(\theta)

because it demonstrates the meaning of i as a transform/operator, rather than sqrt(-1)

Physics:

Continuity Equation (for conserved \phi):
\frac{\delta \phi}{\delta t} + \nabla \cdot f = 0

because it relates the physical meaning of the partial derivative to that of the total derivative.

Biology:

\Delta G = -n F \Delta E

Energy for work is the number of moles of stored energy from electric charge.