Can Equations Be Purely Aesthetic?

[Mondayman]

Hey, that was mine . . .

Ah Indeed it was! When loading latex my page tends to jump and skip posts.

 

[ProfuselyQuarky]

Ah Indeed it was! When loading latex my page tends to jump and skip posts.

You can take it, if you desire :)

 

[debajyoti]

Cauchy's Integral Formula: If C is a circle in the complex plane with positive orientation and f is holomorphic on an open set containing C and its interior, then, for any point a in the interior of C:

$$ f(a) = \dfrac{1}{2 \pi i} \oint_C \dfrac{f(z)}{z-a} \,dz $$

CREDIT:
1. https://math.dartmouth.edu/archive/m43s12/public_html/notes/class9.pdf
2. https://en.wikipedia.org/wiki/Cauchy's_integral_formula

 

[rickz02]

Dirac equation (in natural units) by one of my Physics heroes:

(i\partialslash-m)\psi = 0

This is one of the simplest equation in quantum field theory yet the most elegant of all. It's very short but it tells a lot everything there is.

 

[rickz02]

Seems like \partialslash is not working, but Dirac equation should look like this:

 

[collinsmark]

For the last several years, I've been partial to the Gaussian integral.

\sqrt{\pi} = \int_{- \infty}^\infty e^{-x^2}dx

• Visually, while one side is edgy with the square root sign, and other the side is very curvy.
• It is a relationship between \pi and e, two of the most important mathematical constants.
• Everybody loves \pi, and e is not far behind.
• One side involves a square root, while the other side has a square.
• Taking an exponent to another exponent is always cool.
• The function e^{-x^2} defines the general shape of the "bell curve" and is extremely important within probability theory and statistics, including the Central Limit Theorem (a profound theorem within probability theory).
• While the equation is strictly mathematical, it does have many applications in physics (the probability/statistics go without saying). The e^{-x^2} bell curve shape is the general "shape" (complex envelope, if you prefer) of a wavefunction that minimizes the Heisenberg uncertainty.
• Edit: and the analytical proof of the Guassian integral is beautiful in its own right, but I'll leave the proof out of this post.

 

[micromass]

For the last several years, I've been partial to the Gaussian integral.

\sqrt{\pi} = \int_{- \infty}^\infty e^{-x^2}dx

• Visually, while one side is edgy with the square root sign, and other the side is very curvy.
• It is a relationship between \pi and e, two of the most important mathematical constants.
• Everybody loves \pi, and e is not far behind.
• One side involves a square root, while the other side has a square.
• Taking an exponent to another exponent is always cool.
• The function e^{-x^2} defines the general shape of the "bell curve" and is extremely important within probability theory and statistics, including the Central Limit Theorem (a profound theorem within probability theory).
• While the equation is strictly mathematical, it does have many applications in physics (the probability/statistics go without saying). The e^{-x^2} bell curve shape is the general "shape" (complex envelope, if you prefer) of a wavefunction that minimizes the Heisenberg uncertainty.
• Edit: and the analytical proof of the Guassian integral is beautiful in its own right, but I'll leave the proof out of this post.

The only thing prettier than this is the proof of this equality.

 

[Jeff Rosenbury]

But we are seeing equations involving different quantities of physics (forces, angular momentum, photographic blurring, etc) the candidate should be asked to present the equation.

Besides the fact that the photo also presents inequalities.

It's a post modern interpretation of the OP's requirements. It's not actually designed to win, but to encourage thinking outside the box.

Like all art, to quote Tom Lehrer, "What you get out of it depends on what you put into it."

 

[micromass]

I like a function that I found on my own, and saved me a ton of headaches in a project. It allows one to reduce expressions with derivatives of delta functions (assuming an integral over u):

$$
F(u)\delta^{(n)}(u) = \sum_{i=0}^n (-1)^{n-i} \left(\frac{n!}{i!(n-i)!}\right) F^{(n-i)}(0) \delta^{(i)}(u) $$
where
$$f^{(i)}(u) \equiv \frac{\partial^i}{\partial u^i} f(u)$$

Honestly I'm not sure if it has a reference anywhere, possibly a hepth original?

Why do you hate binomial coefficients?

 

[Hepth]

Why do you hate binomial coefficients?

haha i actually deleted it as i realized it wasn't so aesthetically pleasing...

And to be honest, it was more cluttered to use the binomial coef in the final draft, as this was a part of a bit more of the appendix, and I wanted it to match the factorials. I had thought about using them all in terms of Gamma functions too, but I just had to stick with one. (in depth, I had a few more relations for derivatives of plus distributions too, and those don't elegantly fall into such a nice form, and you're stuck with factorials)

 

[strangerep]

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... )

 

[Titan97]

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

 

[Ygggdrasil]

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

 

[thephysicsnerd]

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!

 

[Redbelly98]

Simple equation or oxygen molecule? You decide:

O=O

 

[Victor Alencar]

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.

 

[mgkii]

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

 

[RedDelicious]

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

 

[mfb]

Equation of a circle, written as circle.

 

[atyy]

The only thing prettier than this is the proof of this equality.

Can you prove it without going to 2 dimensions?

 

[micromass]

Can you prove it without going to 2 dimensions?

Yes, that's definitely possible.

 

[atyy]

Yes, that's definitely possible.

Give me a clue?

 

[micromass]

Give me a clue?

Differentiation under the integral sign and limit-integral theorems.

 

[AZW]

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

 

[fuzzyfelt]

Regarding aesthetics in quantum mechanical expressions, we had a thread from a tattoo artist in the Quantum Physics forum not too long ago which may be of interest to the readers here.

These may be of interest here, too,
http://www.concinnitasproject.org/portfolio/

 

[Angel Penaflor]

Love=You+I :v

 

[A. Neumaier]

got it all jumbled up...

 

[ChrisVer]

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

 

[mfb]

I appreciate having more than one molecule of it.

 

[SACHIN RAWAT]

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 .