## The World's Most Beautiful Equations? 137

music4l numb3rs asks:

*"'An exhibition of the world's most beautiful equations...and some of the ugliest ones too' is how the artist Justin Mullins describes his upcoming show in London. He's exhibiting a number of old favourites such as Maxwell's equations and Euler's relation plus some I've not come across such as entanglement. As for ugliness, he points to the four color theorem. My question to contemplate over the holiday period is: what do Slashdot readers think are the most beautiful equations, and the most ugly ones too?"*
## Einstein was onto something... (Score:1)

Nothing more beautiful then that!

## Re:Einstein was onto something... (Score:5, Informative)

## Re:Einstein was onto something... (Score:2)

## the stuff of mathematical wet dreams (Score:2)

the stuff of mathematical wet dreamsThe older I get, the more I appreciate Poisson's summation formula:

Hmmm... looks like the lame-ass SlashDot lameness filters forbid sub's and sup's.

Anyway, it was discovered about 150 years before its time, its [modern] proof is breath-takingly elegant, and in various incarnations [such as "Shannon's Sampling Theorem"], it governs just about every electrical device you interacted with in the twentieth century [or will interact with in the twenty

## Re:Einstein was onto something... (Score:2)

## Re:Einstein was onto something... (Score:2)

## Re:Einstein was onto something... (Score:2)

In either case I will have a "c"; the c in E = m c^2 in the former, for example. *That* c is a dimensionless number-- hence interesting-- and that it is 1 is definitely interesting.

## Re:Einstein was onto something... (Score:5, Funny)

$ = (size of lie x price of product x number of suckers x number of PCs x number of years of great products) - (cost of legal defense + cost of penalties + cost of political contributions + cost of Bill's house + cost of Indian programming labor) + K,

where K = a factor I shall explain but you have to pay me first.

## Re:Einstein was onto something... (Score:3, Informative)

Nothing more beautiful then that!Except that it's only half the equation.

E^2 = (mc^2)^2 + (pc)^2

E = mc^2 only includes the energy contributed by the rest mass.

## Re:Einstein was onto something... (Score:4, Informative)

## Re:Einstein was onto something... (Score:2)

You and the OP are probably using different m's. His equation (E = m c^2) is correct at all energies if m is the inertial mass. Your equation is correct if m is the rest mass.Yeah, well inertial/relativistic mass is simply energy in different units via E=mc^2, so you may as well call it energy use mass to refer to rest mass.

Otherwise you'd have to say photons have mass.

## Re:Einstein was onto something... (Score:2)

## Obligatory bad chat-up line equation (Score:3, Funny)

Me + you = one beautiful equation

Me + you =meyou(Meow)

## Quadratic Formula (Score:1)

## Re:Quadratic Formula (Score:1)

## Much better equation art (Score:4, Informative)

## Best Equation? (Score:2, Funny)

## Re:Best Equation? (Score:2)

## Re:Best Equation? (Score:2)

True. But I can't think of another combination of "stuff" that could be more important for us humans.

## Re:Best Equation? (Score:2)

('!' was referred to as 'bang' when I learned about factorials)

## e^(i*pi) = -1 (Score:3, Insightful)

## Re:e^(i*pi) = -1 (Score:2)

## Re:e^(i*pi) = -1 (Score:3, Interesting)

## Re:e^(i*pi) = -1 (Score:2, Interesting)

## Re:e^(i*pi) = -1 (Score:2)

## Re:e^(i*pi) = -1 (Score:1)

I guess it says more about the relationship between e and pi and not so much about i, right?

## Re:e^(i*pi) = -1 (Score:5, Interesting)

No no no.

e^(i*pi) + 1 = 0

There. Fixed your equation. Now it contains all five principal numbers: e, i, pi, 1, and 0.

## Does it really matter? (Score:3, Interesting)

From an infinate number of maps to 633 maps. The graph its like browsing through freshmeat or Wikipedia and discovering a world of variety and viewpoints. (sorry it reality does not meet some your expectations of a more "beautiful" number such as 0, 1 or 1,000)

Ugly? I find the the simple formulas. Try explaing what these mean to a child without resorting to "Its because its by definition..." (eg. ALEPH ONE) or having to explain some really complex background on the subject (STARBIRTH, what does pi have to do with this? What is with using the Boltzmann constant?).

## Re:Does it really matter? (Score:2)

Hail Eris!

## Re:Does it really matter? (Score:2)

## Re:Does it really matter? (Score:1)

## The jury is still out. (Score:2)

That is about as cool as a programmer starting with an "infinite number of choices" to solve a problem and ending up with a program with 633 if-then-else statements.

Now if it turns out that that is the shortest program possible to solve the given problem then I guess one will have to accept that as "as cool as it gets".

However if the 633 if-then-else statements can be reduced to a few loops and conditionals, or even a one liner th

## The most beautiful equation is... (Score:2)

I never did believe that stuff about beauty and truth...

## Re:The most beautiful equation is... (Score:2)

## Re:The most beautiful equation is... (Score:1)

## Re:The most beautiful equation is... (Score:3, Informative)

## Boltzman (Score:2)

## How about e^(2*(pi*i)) (Score:1)

## I got better. (Score:2)

You got e, pi, i, 0 and 1 all in a simple equation. Hard to beat. And curse Slashcode not allowing a graphical paste-in of the letter...

## Mine (Score:5, Funny)

wait

## Re:Mine (Score:2, Funny)

"1 = 2, for very large values of 1"

## Re:Mine (Score:2)

## Re:Mine (Score:2)

## Arithmetic series (Score:4, Informative)

p = (2^(n-1)) ((2^n)-1) always struck me as beautiful as well (where p is a perfect number and 2^n - 1 is a Mersenne prime). It just has a sort of symmetry.

## This has been asked before... (Score:2, Interesting)

## I vote for... (Score:3, Insightful)

My vote is for the Einstein field equation [wikipedia.org]. Briefly stated: the curvature of spacetime is proportional to its mass/energy content. Very pretty.

## Ideal gas law (Score:1)

PV=nRT

## Re:Ideal gas law (Score:1)

Sure, it's a bit more tedious to use but it looks (to me at least) more elegant and has none of this R crap.

## 1 = 2... (Score:1, Interesting)

## Re:1 = 2... (Score:1)

## Re:1 = 2... (Score:2)

Dividing by zero, on the other hand...

## The funniest equation (Score:1)

If you determine the rate of change in this curve correctly, I think you'll be pleasantly surprised!

## Re:The funniest equation (Score:1)

## RSA Encryption (Score:4, Informative)

phi(n)is Euler's Totient function which is the number of integers less thannthat are relatively prime ton. The numbernis chosen to be the product of two primes,pandq. Even ifnis known, it is hard of findpandq. Thenphi(n) = (p-1)(q-1)and it is easy to pick adand anesuch that You give outnandeas your public key and usenanddas your private key. Public en/decryption is done with: Private en/decryption is done with:## The most beautiful equation (Score:2, Funny)

Subtract the clothes

Divide the legs

Multiply

## Girls are Evil (Score:5, Funny)

We all know that girls require time and money, so

Girls = Time x MoneyWe also know that time is money, so

Time = MoneyTherefore,

Girls = Money x Money = Money ^ 2Furthermore, it is commonly known that money is the root of all evil, so

Money = sqrt(Evil)Therefore,

Girls = (sqrt(Evil))^2 = EvilHence,

Girls = Evil## Re:Girls are Evil (Score:3, Informative)

Girls = +/- Evil## Re:Girls are Evil (Score:2)

ormoney". When you multiply time by money, you ensure that both terms are required to be nonzero in order for the product to be nonzero, so multiplication is correct in this case.## Re:Girls are Evil (Score:2)

Sorry but your proof falls over at this stage.

Moneyis not the root of all evil, but "thelove ofmoney" is the root of all evil. ref: [biblegateway.com] 1 Timothy 6:10 (KJV). Better translations render it "the root of allkinds ofevil as opposed to "allevil" (i.e. not meaning that its the root of blanketuniversalevil). Thus, your formula fails it in two places at this stage of the proof.## Four Constants == Beauty (Score:1, Redundant)

e^(pi *i)) + 1 = 0 [google.com]## Re:Four Constants == Beauty (Score:2)

You know, I originally had five in the title, but I figured some smartass would come along and say there were only four constants on the left hand side so the title should say four.

Some days you can't win for losing.

## Heat Equation (Score:3, Informative)

u_t = k*u_xx or, more generally, u_t = k*$\Delta$u

Sigh, I wish slashdot supported some sort of LaTeX markup. u_t = k*/_\u

That's the Laplace operator, in case you couldn't tell.

## Fundamental Theorem of Calculus (Score:2, Insightful)

## Re:Fundamental Theorem of Calculus (Score:1)

a lot, and Stokes' formula looks just like it.## When I posted this there were 42 comments (Score:4, Funny)

I win!

## What about chemistry (Score:2, Interesting)

CH4 + 2O2 --> CO2 + 2H2O

## Re:What about chemistry (Score:2)

## Re:What about chemistry (Score:1)

## Solids (Score:2)

## Sky high pie (Score:1)

not mine, My

Pie are round

## 1+3+3=7 (Score:2, Interesting)

## the Geller formula (Score:1)

## Gauss's Law: (Score:2)

In words, Gauss's law states that "if you add up the surface integral of the displacement vector

Dover a closed surfaceS, what you get is the sum of the total charge enclosed by that surface."I was taught this as a basic theorem in Physics, and thought it interesting as a tool. Then my girlfriend, who was far smarter than I, told me she was learning the same equ

## The Gauss-Bonnet Theorem (Score:1)

The Gauss-Bonnet theorem asserts that the integral of the curvature of a (compact, oriented) surface equals 2 pi times its Euler characteristic, giving an extraordinary beautiful and deep formula.

(This is just one instance of what's called an index-theorem, which usually provide über-beautiful, über-general, über-deep formulas, but tend to be, well, less accessible to the masses...)

There is a semi-ugly rendition of Gauss-Bonnet'd formula into a GIF (Wolfram does GIFs...) here [wolfram.com].

## Re:The Gauss-Bonnet Theorem (Score:2)

## Symmetric ones will win...(?) (Score:2)

I bet that's why the chicks dig me - because I happen to be lucky enough to have 2 equidistant nostrils.

## Re:Symmetric ones will win...(?) (Score:2)

## F=(MV^2)/2 (Score:1)

so simple. so pretty. describes so so much.

## Re:F=(MV^2)/2 (Score:1)

## The beauty is in the proof. (Score:4, Insightful)

On the other hand, an ugly one would be an equation that's long and complex with just as long and complex a proof.

Just my $0.02.

## My favorites: (Score:2)

## Emmy Noether! (Score:5, Informative)

Hence, the fact that force laws do not change with time implies conservation of energy, that they do not change with position implies conservation of linear momentum, and that they do not change with rotation implies conservation of angular momentum. Highly awesome.

## My postulate is pretty ugly (Score:1)

"Infinity does not exist for item x if total volume of x is continuously increaseing faster then the universe."

Dude, did I blow your mind?

## Re:My postulate is pretty ugly (Score:2)

## Im really glad for this post. (Score:1)

## Lagrange's Theorem (Score:3, Interesting)

## Britney Spears (Score:2)

## Newton's Second Law (Score:2)

F= dp/dt## Truth is beauty, so here's some truth (Score:2)

girls = time x money

And eveyrone knows that money is the root of all evil.

money = sqrt(evil)

Finally, it is trivially shown that time is money.

time = money

girls = time x time

time = sqrt(evil)

girls = sqrt(evil)^2

Therefore,

girls = evil

## Heard this one? (Score:2)

So all the functions rush to the exit, just the exponent remains at the bar, unshaken, finishing his beer.

And then The Great Derivative enters the bar.

- I AM THE GREAT DERIVATIVE YOU SHALL BE DIFFERENTIATED.

- Oh, but I'm e^x and I'm not afraid of you, differentiate all you want.

- Oh, yes? And I'm an y derivative, sucker.

## Re:Heard this one? (Score:2)

## Navier-Stokes equations (Score:2)

## Navier-Stokes Equations (Score:2)

These equations are used

all the timein the design and development of almost everything you use (drive, type on, fly, drink, what have you) on a daily basis. One of the biggest "ah-ha" moments I've ever had was, when taking Fluid Mechanics II, we started into the Navier-Stokes equations, and I realized that the equations describing stress-concentrations lookedreeelfamilair. My Intermediate Mechanics of Materials professor confirmed my insight, and that was## Ugly, yet beautiful (Score:2)

This "Theorem" completely categorizes finite simple groups - in effect the "building blocks" of Group Theory. It is one of the great triumphs of 20th century mathematics. It's also in the area of 15000 pages long, and represents the combined efforts of scores of mathematicians who worked on it. It is confidently believed to be correct, but seeing as very few people really understand the majority of this "theorem" in detail, it's their word that it "works".

## Re:definition of a derivative (Score:1)

art?## Re:definition of a derivative (Score:1)

## I agree, but... (Score:3, Insightful)

## Re:And I ask the slashdot editors... (Score:2)

## Re:And I ask the slashdot editors... (Score:2)

## Re:And I ask the slashdot editors... (Score:3, Funny)