Can Equations Be Purely Aesthetic?

[Greg Bernhardt]

The goal is to create the most beautiful or interesting equation aesthetically (pleasing to the eye).
This is not about it's significance.

• Each member is allowed to post one equation.
• The equation can be completely new and made up (doesn't have to be famous).
• Be creative!
• The equation must be valid and true.

To vote for an equation simply click the "like" button. You can vote more than once. The contest will close next Thursday the 31st.

The winner will receive a copy of Carlo Rovelli's new book "Seven Brief Lessons on Physics"
https://www.amazon.com/dp/0241235960/?tag=pfamazon01-20

ps. do not try to register new usernames for more entries or for likes. It's painfully easy to figure these out.

Have fun! Go!

 

[jfizzix]

\psi(x_{1},x_{2})=\frac{1}{\sqrt{2\pi\sigma_{+}\sigma_{-}}}\;e^{-\Big[\frac{(x_{1}+x_{2})^{2}}{8\sigma_{+}^{2}}+\frac{(x_{1}-x_{2})^{2}}{8\sigma_{-}^{2}}\Big]}

This is a popular wavefunction describing pairs of entangled particles, called the Double Gaussian wavefunction. It has a lot of symmetry, is incredibly easy to work with, and to me, is easy on the eyes as far as joint wavefunctions go.

 

[DaTario]

$a_n = \frac{(\frac{1 + \sqrt{5}}{2})^n - (\frac{1 - \sqrt{5}}{2})^n}{ \sqrt{5}}$
The Binet´s Fibonacci number formula, which produces de Fibonacci's sequence 0,1,1,2,3,5,... from powers of the golden ratio.

 

[samalkhaiat]

The goal is to create the most beautiful or interesting equation ascetically (pleasing to the eye).

• Each member is allowed to post one equation.
• The equation can be completely new and made up (doesn't have to be famous).
• Be creative!
• The equation must be valid and true.

To vote for an equation simply click the "like" button. The contest will close next Thursday the 24th.

The winner will receive a copy of Carlo Rovelli's new book "Seven Brief Lessons on Physics"
https://www.amazon.com/dp/0241235960/?tag=pfamazon01-20

ps. do not try to register new usernames for more entries or for likes. It's painfully easy to figure these out.

Have fun! Go!

For all x \geq 0,
\frac{x^{2} + 1}{x + 1} \geq 2 ( \sqrt{2} - 1 ) , and the equality holds for x = \sqrt{2} - 1.
The problem is, I believe, the above algebraic inequality is more useful than Rovelli’s book.

 

[mrspeedybob]

e^iπ+1=0

 

[ProfuselyQuarky]

$a^2+b^2=c^2$
I was going to go with De Moivre's Theorem, but simplicity beautiful and this is a classic. The Pythagorean Theorem is what first made me realize that math is so alluring.

 

[strangerep]

$$\Delta\phi ~\propto\, \oint A \cdot dx$$Cf. The Aharonov-Bohm effect. I.e., the electron phase shift arising in a closed loop around a solenoid. This is also one of the most beautiful things in (theoretical and experimental) quantum mechanics, imho.

 

[mrnike992]

t' = γt

 

[Docscientist]

I am going to do something interesting.
According the law of parallel universes,There can be different worlds with different laws.If in our world Einstein proved E=mc^2,then there's another world where Ainstein proved :
M=ec^2
That's beautiful,right ?

P.S : You can't question it's validity.It's true about the parallel universes and hence questioning whether the equation has any existence has no basis

 

[Esfand Yar Ali]

e=mc2. The most brilliant equation I've ever seen and it kind of made me appreciate mathematics

 

[Student100]

Let's see who gets this first.. $$y=cos (x)+cos (2x) $$

 

[MexChemE]

The continuity equation. An elegant, yet simple way to state one of physics most powerful laws, the conservation of mass.
\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0

 

[ShayanJ]

$\mathcal{R}_{\mu \nu}-\frac 1 2 \mathcal R g_{\mu \nu}=8 \pi \mathcal{T}_{\mu\nu}$
"Matter tells spacetime how to curve...", ooops, only one equation, so only half the sentence!

 

[Laurie K]

\frac {\lambda}{2 \pi} = \frac{\hbar}{m c}

 

[Stella.Physics]

The Schwarzschild solution can be considered as a global solution of the vacuum equations, Tμν = 0, i.e. Rμν = 0
everywhere, even at r = 0. In this case, the Schwarzschild solution is called a black hole, since nothing can escape from the horizon at r = R_S.

https://scontent.fath4-1.fna.fbcdn.net/hphotos-xaf1/v/t1.0-9/11971_10208901219025504_2497836378603654750_n.jpg?oh=ccf2e5340a092da9cae3de641a72c9d2&oe=577C0CF7

 

[256bits]

Just a simple equation, F=ma, and so much physics and technology from it.

 

[Cruz Martinez]

$d(x, y) = \sup \{\frac{\bar{d}_i(x_i, y_i)}{i}\}$

This represents a metric on the topological product $\prod_{i \in \mathbb{N}} X_i$ of a countable family of metrizable topological spaces .
Here $x, y \in \prod_{i \in \mathbb{N}} X_i$, $x_i, y_i \in X_i$, $i \in \mathbb{N}$ and $\bar{d}_i$ is the standard bounded metric on $X_i$.
This is a powerful metrization theorem in topology since it proves that the product of a countable family of metrizable spaces is itself metrizable, and can be used as the starting point for proving (one of my favorite theorems in mathematics) the Urysohn metrization theorem.

 

[Aafia]

an equation in physics giving the kinetic energy of a photoelectron emitted from a metal as a result of the absorption of a radiation quantum: Ek=hν−ω

 

[Physics-Tutor]

when studying the different areas in Physics, have you ever wondered why waves, cycles and oscillations appear to dominate our universe, from the distribution of matter in the CMB down to description of particles, and even further down for some (string theory).
And it's not all, how come we, as conscious beings, find beauty when we combine them in symmetrical ways (music, combination of colors like in paintings, cyclic arrangements...)?

 

[micromass]

\prod_{p~\text{prime}} \frac{1}{1 - p^{-2}} = \frac{\pi^2}{6}

Why is this beautiful? Well, on the left hand side, you have objects from arithmetic. Prime numbers, which are essentially determined by numbers divisible only through 1 and itself. On the right-hand side, we have an object from geometry. The number pi, which gives us the circumference and area of a circle. At first sight, these two are very elementary objects with no relation. But then we obtain this very strange, elusive and beautiful relation between such two objects.
Contrary to $e^{\pi i} +1 = 0$ (which is essentially a definition and cannot be checked heuristically), this one can be checked heuristically. Just take as many primes as you want and calculate the product to see you get closer and closer to $\pi$.

 

[Patrick Sossoumihen]

$a_n = \frac{(\frac{1 + \sqrt{5}}{2})^n - (\frac{1 - \sqrt{5}}{2})^n}{ \sqrt{5}}$
The Binet´s Fibonacci number formula, which produces de Fibonacci's sequence 0,1,1,2,3,5,... from powers of the golden ratio.

Praise to the Golden Ratio.

 

[Patrick Sossoumihen]

e^iπ+1=0

This one is famous i think so.

 

[Patrick Sossoumihen]

Let's see who gets this first.. $$y=cos (x)+cos (2x) $$

Isnt that just a function?

 

[Patrick Sossoumihen]

\prod_{p~\text{prime}} \frac{1}{1 - p^{-2}} = \frac{\pi^2}{6}

Why is this beautiful? Well, on the left hand side, you have objects from arithmetic. Prime numbers, which are essentially determined by numbers divisible only through 1 and itself. On the right-hand side, we have an object from geometry. The number pi, which gives us the circumference and area of a circle. At first sight, these two are very elementary objects with no relation. But then we obtain this very strange, elusive and beautiful relation between such two objects.
Contrary to $e^{\pi i} +1 = 0$ (which is essentially a definition and cannot be checked heuristically), this one can be checked heuristically. Just take as many primes as you want and calculate the product to see you get closer and closer to $\pi$.

If you re trying to solve the Riemann Hypothesis, do it before i die.

 

[Physics-Tutor]

\prod_{p~\text{prime}} \frac{1}{1 - p^{-2}} = \frac{\pi^2}{6}

Why is this beautiful? Well, on the left hand side, you have objects from arithmetic. Prime numbers, which are essentially determined by numbers divisible only through 1 and itself. On the right-hand side, we have an object from geometry. The number pi, which gives us the circumference and area of a circle. At first sight, these two are very elementary objects with no relation. But then we obtain this very strange, elusive and beautiful relation between such two objects.
Contrary to $e^{\pi i} +1 = 0$ (which is essentially a definition and cannot be checked heuristically), this one can be checked heuristically. Just take as many primes as you want and calculate the product to see you get closer and closer to $\pi$.

NEW
Ooooh, I really like this one... gives me the shivers :-)!

 

[Samy_A]

Complex Analysis is pure poetry.

One poem:

$\displaystyle f(a)=\frac{1}{2\pi i}\oint_{\gamma} \frac{f(z)}{z-a}dz$

 

[TheQuietOne]

-R^2y+y^3-2Rxz-2x^2z+yz^2=0

I have always loved the mobius srip, but still can't emagine how much it helped the world of science!

 

[micromass]

Complex Analysis is pure poetry.

One poem:

$\displaystyle f(a)=\frac{1}{2\pi i}\oint_{\gamma} \frac{f(z)}{z-a}dz$

I would have voted for you if you did Stokes theorem which is the more general version (in some sense).

 

[Samy_A]

I would have voted for you if you did Stokes theorem which is the more general version (in some sense).

Damn, that was my other finalist.

 

[kith]

The first things which come to my mind when I think about beauty in physics are the connection between gravity and geometry in GR and the connection between symmetries and conserved quantities which is given by Noether's theorem. Aesthetically, I really like the bra-ket notation of Dirac. Trying to combine physics and aesthetics, I end up with the Schrödinger equation:

i \hbar \frac{d}{dt} |\psi\rangle = H |\psi\rangle

 

[kith]

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.

 

[Greg Bernhardt]

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

 

[TeethWhitener]

Well, if it's just aesthetics and not significance, then how about this:
1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}}} = \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}

 

[Greg Bernhardt]

Well, if it's just aesthetics and not significance, then how about this:

Very cool!

 

[TeethWhitener]

Very cool!

http://mathworld.wolfram.com/GoldenRatio.html
It's funny because it's probably more significant to aesthetics than to math/physics.

 

[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