PF Contest: Equations as Art

[ShayanJ]

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

Hard to tell the difference, and actually to be honest, the feeling is more about the things you know about that equation. A very good example is the Cauchy's integral theorem(mentioned by Samy_A). What I(and he and certainly many others) feel about that equation is not at all related to how it looks, e.g. that it has a $\pi$ downstairs and the integral sign has a circle in its middle. This equation is a reminder of all the beauty in complex analysis (So I should double what Samy_A said "Complex Analysis is pure poetry") and that's why he chose it and I voted for it.
Otherwise you should show the equations to people who don't know what they mean and ask them which is more beautiful!

 

[micromass]

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

But the beauty IS the meaning!

 

[Greg Bernhardt]

But the beauty IS the meaning!

Not this time Think of these equations on a painting shown to people who don't know math.

 

[micromass]

Not this time Think of these equations on a painting shown to people who don't know math.

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...".

 

[Greg Bernhardt]

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...".

Then don't participate. I don't know math, but I find the equations interesting and beautiful like calligraphy. Be a designer not a mathematician.

 

[ShayanJ]

Then don't participate. I don't know math, but I find the equations interesting and beautiful like calligraphy. Be a designer not a mathematician.

Why don't you participate?

 

[Greg Bernhardt]

Why don't you participate?

I'm the one giving out the prize! Let's please get back to the contest. This thread is going to be very muddled.

 

[Isaac0427]

$$\frac{b \pm \sqrt{b^2 - 4ac}}{2a} + x = 0$$
I find that rearranging the quadratic formula gives a very appealing result.

 

[Amrator]

No idea what it means, but it's pretty.
$φ(λ_i)=z_i$

 

[Ssnow]

The pre-quantization condition on $M$ (manifold):

$$\int_{\Sigma}\,\omega \,\in\, 2\pi \hbar\cdot \mathbb{Z}$$

where $\Sigma$ closed 2-surface in $M$ and $\omega$ the symplectic structure, is very nice!

 

[axmls]

The wave equation is quite aesthetically pleasing: $$\nabla^2 y = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2} $$

 

[Jeff Rosenbury]

An equation is an equality containing one or more variables. Here's an odd one I "borrowed" from an advertisement:

 

[DaTario]

An equation is an equality containing one or more variables. Here's an odd one I "borrowed" from an advertisement:

This should be taken as an infinite set of equations... It is not fair...

 

[QuantumQuest]

Perturbation Hamiltonian (low intensity limit) for the interaction of radiation with atomic systems (semiclassical treatment):
$$H' (t)= -\frac{e}{mc}Ap -\frac{e}{mc}SB$$

where $$\text{m = particle mass}\\
\text{e = particle charge}\\
\text{S = particle spin}\\
\text{A = vector potential}\\
\text{B =} \nabla \times\text{A the magnetic field}$$

 

[micromass]

This should be taken as an infinite set of equations... It is not fair...

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

 

[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)