# The most beautiful formulae in maths & science

Ivan Seeking
Staff Emeritus
Gold Member
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!!!

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One in my list is:

$$e^{i\pi} + 1 = 0$$
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
Staff Emeritus
Gold Member
I have to add the Cantor-Bernstein Theorem as one of the biggest surprises. Now that one can cost a guy some sleep.

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.

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
Staff Emeritus
Gold Member
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.

CaptainQuasar said:
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.

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.

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
Staff Emeritus
Gold Member
Hardy congratulations for Hardy.

jimmysnyder said:
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.
:rofl:

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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.

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

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.

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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.

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).

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$$\neg (p \wedge \neg p)$$

$$\neg (p \wedge \neg p)$$
:rofl:
(This entire sentence is a false statement), OR (you will give me one million dollars). :tongue2:
I wonder how many people will read the previous sentence...

(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.)

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.

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

Beautiful.
:rofl:
I am not arguing anymore :rofl:

:rofl:
I am not arguing anymore :rofl:
I'm just glad I don't understand the terminology behind logical arguments, because I don't have 1 million to give. :tongue2:

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

$\Delta A \Delta B \geq \frac{1}{2}\left|\langle[A,B]\rangle\right|$

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$\Delta A \Delta B \geq \frac{1}{2}\left|\langle[A,B]\rangle\right|$
Oh it's merely Cauchy–Schwarz inequality :tongue2:

Mathematics Is Beautiful

Treatment for cancer

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

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

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