# PF Contest: Equations as Art

• Greg Bernhardt
Cauchy's Integral Formula:
You're too late. @Samy_A already entered that one. (You've got to check the whole thread before posting an entry... )

I don't know what this means but here it is:

Jeff Rosenbury
Since no one specified that the equation had to be a math equation, have a chemical equation:

diogenesNY, TeethWhitener, mfb and 4 others
Imo the most beautiful equation or rather ineqation is the Fermat's last theorem.

that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than two.

It is extremely simple to understand,
But urges anyone who reads it to prove it!
Beware, it is not as easy as it looks
Also, it holds the world record of being the longest standing problem(365 years) and being proved incorrectly for most number of times!

Simple equation or oxygen molecule? You decide:

O=O

member 587159, diogenesNY, ProfuselyQuarky and 3 others
δS = 0
"S" stands by action( S=∫dtL) ,this is the Least Action Principle.
All the physics lies in this equation. Equations of motion,symmetries and conservations can be extracted from that.
"The equations of analitycal mechanics have a meaning that goes beyond the Newtonian mechanics"-A. Einstein
Ps: This quote is a free translation, i read it in a brazillian book.

ProfuselyQuarky, mfb and Greg Bernhardt
If there's any Bieber and Miley fans out there in physics land... I reckon I'm onto a winner

Titan97, Hornbein and Mondayman
I have always thought that this was pretty elegant. It's very clean and spacious, though I do think the nested radical one is probably my favorite so far.

$$e=\lim _{n\rightarrow \infty }\left(1+\frac{1}{n}\right)^n$$

Demystifier and Greg Bernhardt
Equation of a circle, written as circle.

member 587159, Titan97, couragerolls and 8 others
The only thing prettier than this is the proof of this equality.

Can you prove it without going to 2 dimensions?

Can you prove it without going to 2 dimensions?

Yes, that's definitely possible.

Yes, that's definitely possible.

Give me a clue?

Give me a clue?

Differentiation under the integral sign and limit-integral theorems.

atyy
Generalized Stokes Equation

I love this because it neatly summarizes things you learn in introductory calculus such as FTC, classical Stokes theorem, divergence theorem.
It's also really aesthetically beautiful- the latin d signifying the exterior derivative turns into the greek ∂ signifying the boundary of the manifold

Markus Hanke, Samy_A and Greg Bernhardt
Love=You+I :v

got it all jumbled up...

$H_2O$
If you don't appreciate it, then you don't appreciate life...

Aafia and Greg Bernhardt
I appreciate having more than one molecule of it.

Angel Penaflor, CynicusRex and Greg Bernhardt
S=K×ln ω Where K is Boltzmann's constant. This entropy equation beautifully connects macroscopic quantities to microscopic states. Entropy of universe cannot decrease. So this equation states that randomness of microscopic states keeps on increasing. So randomness in universe is increasing . This leads us to the idea that there must have been a time when entropy of universe was zero and after that time entropy started to increase (Big Bang). If we want to time travel in past , we will have to decrease the entropy of universe which is not possible . Thus in this way entropy equation beautifully denies possibility of reversing the time .

couragerolls
Minimalistic but very pleasing
$$\nabla^2 = \Delta$$

TeethWhitener, Dembadon, Ygggdrasil and 2 others
$H_2O$
If you don't appreciate it, then you don't appreciate life...

Not really an equation...

No Gauss-Bonnet yet??

No Gauss-Bonnet yet??
You still seem to be thinking in terms of beauty-in-meaning, rather than visual beauty.

Actually, it's fascinating how high powered mathematicians like yourself and A. Neumaier (to name just 2 among others in this thread) perceive beauty-in-meaning where others do not, yet have trouble perceiving the visual beauty that others can. Other types of people (e.g., conventional artists) suffer the reverse -- they create beautiful pictures yet cannot even understand how basic percentages work.

Greg Bernhardt
You still seem to be thinking in terms of beauty-in-meaning, rather than visual beauty.

Actually, it's fascinating how high powered mathematicians like yourself and A. Neumaier (to name just 2 among others in this thread) perceive beauty-in-meaning where others do not, yet have trouble perceiving the visual beauty that others can. Other types of people (e.g., conventional artists) suffer the reverse -- they create beautiful pictures yet cannot even understand how basic percentages work.

I'm not a mathematician nor a physicist, only a physics student, but I still feel like micromass.
Also, let's take a look at others. A good percentage of people posted some explanation along with their equations, which means they had some meaning in mind when they posted it.
The most voted equations are the ones by Samy_A and micromass himself, which are both more about the meaning than the looks.
It seems most of the people actually feel the same as micromass about the equations but they either don't know it or don't want to admit it!
Actually this is reasonable. How people feel about an equation gets stronger and stronger as they work with it and learn more about it and its relation with other equations, to the point that this meaning-induced feeling will over-shadow any feeling related to only the looks of an equation. Even about people who don't know the meaning and are just learning it from the explanations given, the meaning is more exciting than the looks!

Sophia and micromass
The most voted equations are the ones by Samy_A and micromass himself, which are both more about the meaning than the looks.
Don't forget the "all 1" formula from @TeethWhitener. It has the most votes currently.
micromass' formula has small and capital pi in it.

Don't forget the "all 1" formula from @TeethWhitener. It has the most votes currently.
micromass' formula has small and capital pi in it.
Oh...missed that one!
Anyway, I think even that equation actually seems more interesting to people than beautiful!

P.S.
Aren't they asymptotic to each other rather than equal?

Both values are the limit of the shown procedure. Those limits are equal.

Don't forget the "all 1" formula from @TeethWhitener.

That one should definitely deserve to be the winner. But let's but honest, would it be as beautiful to somebody who didn't know what division and square roots are? Or to somebody who doesn't realize those are limiting processes? Because it's very hard to suppress this knowledge when judging the formula...

ShayanJ
Not really an equation...

well before me someone had posted some chemical bonds and the process of electrolysis was shown in the Naturwissenschaften box (I love how this word tickles my tongue! And I love even more the fact that I wrote it correctly without looking at it)

Common, the most beautiful formula is this one:

$L = i \bar{\psi}_i \gamma^\mu D_\mu \psi_i -\frac{1}{4} \sum_{b \in \text{adjoint}} \sum_{a\in U(1),SU(2),SU(3)}F^{ab}_{\mu \nu}F^{ab~\mu \nu} + Y^{ij} \bar{\psi}_i H \psi_{j} + + | D_\mu H |^2 + \mu^2 H^\dagger H - \lambda ( H^\dagger H)^2$

Demystifier
∫ex = f(μ)n

Here is one I think is beautiful!

Ygggdrasil and Greg Bernhardt
Now you have to show that it is either generally true in some way or defines something meaningful.

Now you have to show that it is either generally true in some way or defines something meaningful.

I think most people would consider it true. I just don't get why this forum allows it...

∫ex = f(μ)n

Here is one I think is beautiful!
Whereas a lot of these equations require an advanced degree to appreciate, this one is one that middle schoolers might appreciate more