Your vote for the most mysterious and wonderful of equations

[eratosthenes2]

Figured this would bring interesting responses.

Mine is Newton's third law, only because of how it applies to rockets - the idea that some gun/rocket accelerates forward exactly based on the speed and mass of the bullets/exhaust is pretty weird when u think about it. That's something moving forward exactly at the rate of stuff moving backward. Seems odd.

 

[tiny-tim]

e^iπ = -1 ​

 

[nicksauce]

i^i = e^{\frac{-\pi}{2}} (The principal value of it anyway)

 

[Redbelly98]

C'mon folks, this is a physics thread

I vote for Schrödinger's Equation.

 

[nicksauce]

Physics-wise, I would vote for the least action principle.

 

[Constructe]

The strangest is String theory, especially since it proposes nothing new except there should be a graviton... How is that different than particle physics? As for the rest of the goop, you can't even test for it. Oh... well a sparticle may show up. But once again, others also proposed super heavy particles. So that also proves nothing. 11 degrees of freedom. Gee if you are bored maybe you can make some theory with 297 invisible dimensions that explains why a cat has whiskers.

 

[tiny-tim]

Gee if you are bored maybe you can make some theory with 297 invisible dimensions that explains why a cat has whiskers.

me bored when i can constantly ponder the mysteries of the bowliverse?

my theory is that whiskers have cats …

the cat is merely whiskers' way of producing more whiskers! ​

 

[rbj]

e^iπ = -1 ​

that's the one that first came to my mind, except i was thinking in this form:

e^{i \pi} + 1 = 0

that relates the five most prominent pure numbers together in one equation.

the Additive Identity operator
the Multiplicative Identity operator
the Imaginary unit
the base of natural logarithms
and pi.

 

[Crosson]

that's the one that first came to my mind, except i was thinking in this form:

e^{i \pi} + 1 = 0

If f(x) is an infinitely many times differentiable function, then:

e^{\frac{d}{dx}} f(x) = f(x + 1)

 

[jackiefrost]

euler+gauss+reimann=infinity

 

[pallidin]

My vote is for equations involving pi. I've always been fascinated by the 3.14... relationship.

 

[gel]

e^{i\pi} = -1 ​

nah, that's just a boring identity. What does it have to say about anything?
Besides, this is a physics thread.

After a bit of thought, I suggest the following

<br /> \frac{dQ}{dt}\ge 0<br />

i.e., the second law of thermodynamics. Total entropy of a closed system can only increase over time.

It's certainly mysterious. It doesn't appear in the fundamental laws of physics, at the lowest level, but must be a consequence of them. It seems like you should be able to get around it (e.g. Maxwell's demon) but there's always a catch, and the second law always holds. It's also very important and rules all of our lives.

 

[sp1408]

not really,S=k*ln(w),I just can't understand how they can define the amount of disorder in a system...

 

[Andy Resnick]

I vote for Euler's first law:

\frac{d}{dt}\int \textbf{v} dm = \textbf{f}

Because it covers all of continuum mechanics, including dividing surfaces.

 

[uart]

If f(x) is an infinitely many times differentiable function, then:

e^{\frac{d}{dx}} f(x) = f(x + 1)

Hey Crosson, what is the definition of e^{\frac{d}{dx}} ?

Am I right to assume it's the operator :

[1 + \frac{d}{dx} + \frac{1}{2!} \, \frac{d^2}{dx^2} + ...].

Or does it mean something else?

 

[Topher925]

\frac{Sin x}{n} = 6

+10 cool points for anyone that figures that out.

 

[tiny-tim]

\frac{Sin x}{n} = 6

+10 cool points for anyone that figures that out.

Easy!

special case of …

\frac{Sin^m x}{n^m}\ =\ 6

​

 

[Topher925]

Easy!

special case of …

\frac{Sin^m x}{n^m}\ =\ 6

​

NO! But nice try though.

EDT: Ok, Tim figured it out. Hes now 10 points cooler.

 

[gel]

...I just can't understand how they can define the amount of disorder in a system...

that's the mystery part

 

[gel]

\frac{Sin x}{n} = 6

+10 cool points for anyone that figures that out.

can't believe I actually had to think about it at all...

Spoiler

just cancel the n

can you pop my points in the post? ta:)

 

[George Jones]

1 + 2 + 3 +... = -\frac{1}{12}

 

[humanino]

1 + 2 + 3 +... = -\frac{1}{12}

Ah, that's simple : if you go all the way to infinity to the right, you come back to zero from the left. The exact value depends of course on how fast you go to infinity. Do you mention this because measuring an integer (number of dimensions) is robust ? I guess no

Although it's not an equation strictly speaking, I'll vote for the gauge principle.

 

[Chi Meson]

2+2=5

(for large values of 2)

 

[morphism]

Although it's not an equation either, and (probably?) has nothing to do with physics, but what about the http://planetmath.org/encyclopedia/FundamentalTheoremOfGaloisTheory.html ? It's definitely both mysterious and wonderful. There are many beautiful results in math, but I don't think anything will ever be as satisfying to me as this one.

 

[gel]

1 + 2 + 3 +... = -\frac{1}{12}

Is that \zeta(-1)? Mysterious maybe, not sure about wonderful though.
If I'm allowed pure maths equations, then I've always been partial to the Riemann-Roch theorem

<br /> L(D)-L(K-D)=\mathop{deg}(D)-g+1<br />

The Lefschetz fixed point theorem is also cool, as is the functional equation for the Riemann Zeta function.

Actually these 4 equations are all closely related.

 

[morphism]

If I'm allowed pure maths equations, then I've always been partial to the Riemann-Roch theorem

<br /> L(D)-L(K-D)=\mathop{deg}(D)-g+1<br />

Funny, I was going to mention that, (and the Atiyah-Singer index theorem), but in my opinion the fundamental theorem of Galois theory trumps them both. Although they are certainly very satisfying too.

Another very satisfying result is the commutative Gelfand-Naimark theorem; in equation form:

\mathcal{A} \cong C_0(\sigma(\mathcal{A})).