The most beautiful formulae in maths & science

[rohanprabhu]

I just though i'd make up a list of a few of the really beautiful formulas in mathematics.

One in my list is:

<br /> e^{i\pi} + 1 = 0<br />

and definately, the Pythagoras theorem... as for physics.. the energy-mass equivalence is another beautiful formula:

<br /> E = mc^{2}<br />

and.. the Bohr's formula for quantized angular momentum:

<br /> L = \frac{nh}{2\pi}<br />

I also think that the Dirac equation is a beautiful result as it predicted the existence of a positron before it's actual discovery..

any others u can think of??

 

[robphy]

There was a Physics World article on this:
http://physicsworld.com/cws/article/print/20407
which was the topic of a New York Times article:
http://www.nytimes.com/2004/10/24/weekinreview/24chan.html

 

[lisab]

You already have my favorite- Euler's.

There's a lot of ugly ones, too .

 

[Jimmy Snyder]

No object is so beautiful that, under certain conditions, it will not look ugly. - Oscar Wilde.

http://img401.imageshack.us/img401/2388/homeworkallforcesbh1.jpg

 

[lisab]

No object is so beautiful that, under certain conditions, it will not look ugly. - Oscar Wilde.

http://img401.imageshack.us/img401/2388/homeworkallforcesbh1.jpg

[/URL]

 

[George Jones]

No object is so beautiful that, under certain conditions, it will not look ugly. - Oscar Wilde.

http://img401.imageshack.us/img401/2388/homeworkallforcesbh1.jpg

[/URL]

There's a sign mistake for the term a third of the way through the line that's a quarter of the way down.

Dark matter found!

 

[lisab]

"...sans gravity..."!

Well duh...wouldn't want to complicate it!

 

[Werg22]

A formula doesn't strike beautiful, an idea does though.

 

[tacosareveryyum]

What book is that from? You've got me interested now.

 

[Mk]

Probably a joke, note the "Excercise #1.1.1.1.1.1.a"

 

[robphy]

http://nuclear.ucdavis.edu/~tgutierr/files/stmL1.html

http://nuclear.ucdavis.edu/~tgutierr/files/sml2.pdf

 

[Kurdt]

The Schrödinger equation is pretty nice.

 

[Schrodinger's Dog]

For sheer actual physical good looks, and usefulness I'd say the Laplacian looks quite nice.

\displaystyle \Delta = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}

 

[humanino]

Well, if I were a genius indian mathematician I'd say \sum_{n=1}^{\infty }n=-\frac{1}{12}

 

[gravenewworld]

Maybe my eyes are just bad, but that isn't an equation, it is meaningless. one of my favs.-

 

[Pythagorean]

For sheer actual physical good looks, and usefulness I'd say the Laplacian looks quite nice.

\displaystyle \Delta = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}

shouldn't it be turned around, squared, and acting on something on the left side?

 

[gravenewworld]

This one also helped the US win WWII and also has been called one of the most important advances in science in the 20th centurty

N2 + 3H2---------> 2NH3 (Fe/K2O/Al2O3/250 atm/500 C)

 

[CaptainQuasar]

I've always been partial to the http://en.wikipedia.org/wiki/Leibniz_formula_for_pi" :

\huge \mathbf{\pi} = 4 \sum_{n=0}^\infty \frac{(-1)^n}{2n + 1}​

 

[humanino]

Maybe my eyes are just bad, but that isn't an equation, it is meaningless.

They were telling the same to Fourier.

Probably one of the most important and beautiful mathematical object is the Riemann zeta function.
Since you like to copy equations from wikipedia, maybe you can check it out for real from time to time. It is instructive.

EDIT BTW, I was referring to Ramanujan[/color] who was maybe one of the most prolific for genuinely beautiful formulae

 

[gravenewworld]

They were telling the same to Fourier.

Probably one of the most important and beautiful mathematical object is the Riemann zeta function.
Since you like to copy equations from wikipedia, maybe you can check it out for real from time to time. It is instructive.

EDITBTW, I was referring to Ramanujan[/color] who was maybe one of the most prolific for guenuinely beautiful formulae

No, I said it because what was posted did not contain any equal sign, and hence it is impossible for it to be an equation. At least the riemann zeta function has an equal sign.

 

[CaptainQuasar]

Yeah, but I think gravenewworld is right, Ramanujan wouldn't write [post=1636702]what you posted there[/post], which currently shows an infinite summation that doesn't converge on a limit being equal to -1/12. And oblique name dropping is more of a vice than referencing Wikipedia, btw.

⚛​

 

[gravenewworld]

wait wait wait. I accidentally deleted what I was referring to. When I said "that isn't an equation, it is meaningless" I was referring to what jimmy posted. As far as I can see there is no equals sign in it anywhere.

 

[humanino]

Ahhhh, sorry for the confusion... Indeed, the "Standard model lagrangian (density) = " is missing in this, plus it is partly in french (sans gravity) so it really does not make sens

 

[Math Jeans]

Not much of a formula, but more of a process. However, its hard to deny that they are beautiful.

http://en.wikipedia.org/wiki/Fourier_transform"

 

[Cyrus]

1=0.999999

[/tex]

 

[Ivan Seeking]

deBroglie's hypothesis has always made my knees weak, and who doesn't love Heisenberg?

The more fundamental the statement, the more I like it.

One set of equations that really struck me upon first encounter were the Lienard-Wiechert potentials for a moving charge; and in particular the direction vector for the electric field as measured by an observer. In spite of the fact that the retarded potential is calculated, the resulting electric field vector points from the present position of the charge! Amazing!

 

[Poop-Loops]

One in my list is:

<br /> e^{i\pi} + 1 = 0<br />

This one is just scary. I'm in my second year of using complex notation and being comfortable with it, but I always took it at face value. I never tried to actually think "Wait, WTF? How?"

And actually, I just went to Wikipedia to figure it out and now I understand it. So thanks for inspiring me to do so. :)

Well, I understood the formula itself, but I didn't quite understand Euler's general formula.

 

[Ivan Seeking]

I have to add the Cantor-Bernstein Theorem as one of the biggest surprises. Now that one can cost a guy some sleep.

 

[Jimmy Snyder]

wait wait wait. I accidentally deleted what I was referring to. When I said "that isn't an equation, it is meaningless" I was referring to what jimmy posted. As far as I can see there is no equals sign in it anywhere.

The text implies there is an 'L = ' at the beginning or an '= L' at the end.

 

[Schrodinger's Dog]

shouldn't it be turned around, squared, and acting on something on the left side?

It can be represented by either \nabla^2 or \Delta and technically it should be \Delta f but I left it out because I am in fact quite mad/evil.

 

[George Jones]

Yeah, but I think gravenewworld is right, Ramanujan wouldn't write [post=1636702]what you posted there[/post], which currently shows an infinite summation that doesn't converge on a limit being equal to -1/12.

Before I checked humanino's link, I thought to myself, "This seems like exactly the type of thing that Ramanujan would write." Checking the link "confirms" that he did write this.

And oblique name dropping is more of a vice than referencing Wikipedia, btw.

Oblique writing often is meant to be taken in good-natured way as a puzzle for the reader to solve.

Let me have a go at it!

A Swiss mathematician (and an all-time great), after whom an equation is named that has a special case given above, also wrote (hundreds of years ago) the same equation that humanino did. And the Indian genius's English mentor wrote a whole book on such things.

 

[Jimmy Snyder]

Oblique writing often is meant to be taken in good-natured way as a puzzle for the reader to solve.

Let me have a go at it!

A Swiss mathematician (and an all-time great), after whom an equation is named that has a special case given above, also wrote (hundreds of years ago) the same equation that humanino did. And the Indian genius's English mentor wrote a whole book on such things.

Spoiler for obliquophobes: hilite the blanks with your mouse to see the hidden text below.
The English mentor's name is well known: Hardy. He wrote a book on Ramanujan, but I'm guessing that's not what you mean by 'such things', so I'll guess it's his book on number theory. As for the equation of which humanino's equation is a special case, I think it's a special case of the Riemann zeta function, but that's not Swiss, it's Greek, unless you count the German part.
The special case is covered extensively in a problem in Zwiebach's book 'A First Course in String Theory'. The follow on book 'A Last Recourse in String Theory' has not yet been written. He uses the equation to prove that the world has 26 dimensions. In my opinion, this does not reflect well on Ramanujan.

 

[tribdog]

Does the equation have to be difficult to be beautiful? Can't the most useful equation in the world be beautiful? a squared plus b squared equals c squared.

 

[George Jones]

Hardy congratulations for Hardy.

I'll guess it's his book on number theory.

Nope.

Also, "a special case given above" and "humanino's equation" don't necessarily refer to the same thing.

He uses the equation to prove that the world has 26 dimensions. In my opinion, this does not reflect well on Ramanujan.

 

[Schrodinger's Dog]

Does the equation have to be difficult to be beautiful? Can't the most useful equation in the world be beautiful? a squared plus b squared equals c squared.

Not at all, the most simple are often the most beautiful.

e^{i\pi}+1=0

Being a perfect example. Wonderfully simple, and extraordinarily convenient.

 

[tribdog]

Not at all, the most simple are often the most beautiful.

e^{i\pi}+1=0

Being a perfect example. Wonderfully simple, and extraordinarily convenient.

never used it

 

[Schrodinger's Dog]

never used it

It can be used to derive every trigonometric function. So it's pretty up there. It also contains 9 basic rules of maths in one. exponents, imaginary numbers, addition and so on.

 

[Poop-Loops]

God, this is almost like poetry. On the surface, you just go "WTF? This is crap.", but when you look deeper, you get all the stuff Schrodinger's Dog came up with.

 

[Jimmy Snyder]

Nope.

In that case, I owe you a number of pure apologies for the inequalities in the qualities of my response. Here is another wild guess: Divergent series?

Also, "a special case given above" and "humanino's equation" don't necessarily refer to the same thing.

Indeed. In that case the mathematician in question is: Euler. (that is Leonhard, not his younger brother Houston).

 

[Moridin]

\neg (p \wedge \neg p)

 

[humanino]

\neg (p \wedge \neg p)

(This entire sentence is a false statement), OR (you will give me one million dollars).
I wonder how many people will read the previous sentence... [/size]

 

[Jimmy Snyder]

(This entire sentence is a false statement), OR (you will give me one million dollars).

This is more beautiful: (This entire sentence is a true statement), AND (you will give me one million dollars.)

 

[humanino]

This is more beautiful: (This entire sentence is a true statement), AND (you will give me one million dollars.)

Ah but it does not work ! For your entire statement to be

• true, both parts must be true. Fine, the only way is that we all give you one million.
• But there is nothing wrong with the entire statement being false ! Either part can be wrong, or both...

Now with

(This entire sentence is a false statement), OR (you will give me one million dollars).

we have :

• fasle only is both are false, but then first part makes self contradictory
• true if at least one is true, but then first part cannot be true, so second part must be true

Not sure I can find a better formulation right now... Am I out of my mind ?

EDIT
In the end, the all thing goes down to the old "paradox"

• This sentence is true can be true or false, no problem
• This sentence is false is self-contradictory and cannot be evaluated true or false consistently.

 

[Jimmy Snyder]

Fine, the only way is that we all give you one million.

Beautiful.

 

[humanino]

Beautiful.

I am not arguing anymore

 

[Schrodinger's Dog]

I am not arguing anymore

I'm just glad I don't understand the terminology behind logical arguments, because I don't have 1 million to give.

I disagree on the grounds that I have no clue what you are going on about.

 

[tkjacobsen]

What about:

<br /> \Delta A \Delta B \geq \frac{1}{2}\left|\langle[A,B]\rangle\right|<br />

 

[humanino]

What about:

<br /> \Delta A \Delta B \geq \frac{1}{2}\left|\langle[A,B]\rangle\right|<br />

Oh it's merely Cauchy–Schwarz inequality

Mathematics Is Beautiful

 

[gravenewworld]

Treatment for cancer

http://www2.mrc-lmb.cam.ac.uk/personal/sl/Html/Graphics/Syntheses1.gif

http://www2.mrc-lmb.cam.ac.uk/personal/sl/Html/Graphics/Syntheses2.gif

 

[Schrodinger's Dog]

Now you see gravenewworld not only is that beautifully written and a lovely little chemical equation. But it's beautiful on many levels.