Can Equations Be Purely Aesthetic?

[DaTario]

Due to the Hilbert basis theorem, you can often make an infinite set of equations into one equation

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.

 

[.Scott]

The contest will close next Thursday the 24th.

You posted this yesterday, Thursday the 24th.
I think you meant it would close next Thursday, the 31st.

 

[Mondayman]

Good old Pythagorean's Theorem.

a²+b²=c²

So simple, I always imagine its drawn in crayon.At the other end of the spectrum, there's this clunky, unwieldy thing...

 

[couragerolls]

i\hbar\frac{\partial \psi(r,t)}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\psi(r,t)+V(r)\psi(r,t)
with the beautiful \psi and its curves . Ta-da! The nefarious schrodinger equation!

 

[.Scott]

Mandelbrot (in equation form):
For any c; z_0=0; and z_n=z^2_{n-1}+c:
\lim_{n\rightarrow +\infty} {(z_n-z_{n-1})} = 0

 

[CynicusRex]

Then I honestly see no beauty in the equations. It's like showing me a book written in russian (I don't know any russian) and asking me to pick the most beautiful book based on the arrangement of the letters. I'm like: "ok...".

Thats a loss on your part sadly. The beauty of formulas is one of the main reasons I decided to study science this 'late' in life. I have always been astonished by the look of complex formulas, knowing it's all logical and every part is crucial. I even use it as an argument when people ask me why I want to study physics:
"While many men look at cars in admiration, I look at formulas the same way even without understanding them. But now I do want to understand them."

I'll enter the contest by summoning the Devil's curve:

which produces things like:

Both the formula and result are beautiful. Knowing people were doing math, science, etc., so many years ago, inspires me to do the same. It stupefies me how people like Newton were able to do all of their superfly work. It also saddens me that education is helping so little in making math and science inspirational and fun. I talk too much. Awesome contest.

 

[Student100]

Isnt that just a function?

Depends on how you look at it.

 

[ProfuselyQuarky]

Good old Pythagorean's Theorem.

a²+b²=c²

So simple, I always imagine its drawn in crayon.

Hey, that was mine . . .

 

[samalkhaiat]

Reminder the contest is about the aesthetical beauty of an equation, not it's meaning or significance!

I thought one needed to "make up" one "true" and "beautiful" equation and not just "pick up" one known equation!

 

[Greg Bernhardt]

I thought one needed to "make up" one "true" and "beautiful" equation and not just "pick up" one known equation!

Either or :)

 

[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 .

 

[Mattematics]

Minimalistic but very pleasing
\nabla^2 = \Delta

 

[micromass]

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

Not really an equation...

 

[micromass]

No Gauss-Bonnet yet??

 

[strangerep]

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.

 

[ShayanJ]

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!

 

[mfb]

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.

 

[ShayanJ]

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?

 

[mfb]

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

 

[micromass]

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...

 

[ChrisVer]

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