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## The World's Most Beautiful Equations?137137

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?"
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## The World's Most Beautiful Equations?

• #### Einstein was onto something... (Score:1)

E = mc ^ 2

Nothing more beautiful then that!
• #### Re:Einstein was onto something... (Score:5, Informative)

on Monday December 26, 2005 @08:06PM (#14341898)
Much overrated as an equation. c is just a constant (and in sensible units c=1) so all it really says is that E=constant*m. This is hardly the stuff of mathematical wet dreams, even if the fact that it's true does have some interest for physicists.
• #### Re:Einstein was onto something... (Score:2)

I've always considered math that has something to say about the real world the most interesting. E=mc^2 isn't very cool mathematically, but it says something very profound about the real world -- the one we live in when not doing abstract math.
• #### the stuff of mathematical wet dreams (Score:2)

the stuff of mathematical wet dreams

The older I get, the more I appreciate Poisson's summation formula:

Sn f(n) = Sn f^(n)

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)

It's not just "a" constant. That the constant of proportionality between (inertial) mass and energy is the square of the speed of light in a vacuum is less than entirely obvious (it becomes slightly more so if you replace "speed of light in a vacuum" with "speed of massless particles", but that's another story).
• #### Re:Einstein was onto something... (Score:2)

It's not an interesting constant at all. If you define your units suitably then c=1.
• #### Re:Einstein was onto something... (Score:2)

I can define my units so that the speed of light in a vacuum is 1. I can also define my units so that 1 unit of mass is equal to 1 unit of energy.

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)

on Monday December 26, 2005 @09:13PM (#14342130)
The Microsoft Equation:

\$ = (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)

on Monday December 26, 2005 @11:46PM (#14342722)
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.
• #### 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)

I'm aware of that. I'm just pointing out some people do use "mass" to mean inertial mass, and if that's what he meant, then his equation is complete and correct as it stands.
• #### Obligatory bad chat-up line equation (Score:3, Funny)

<{fidelcatsro} {at} {gmail.com}> on Monday December 26, 2005 @07:51PM (#14341833) Journal
First thing that sprang into my head when I read the title , those horrible old chat up lines such as :
Me + you = one beautiful equation
Me + you =meyou(Meow)

(-b(+||-)sqrt(b^2-4ac))/2a

or (-b/2a(+||-)sqrt(b^2-4ac))/2a
• #### Much better equation art (Score:4, Informative)

on Monday December 26, 2005 @08:03PM (#14341888)
Check out Bernar Venet [bernarvenet.com]. The web site is a bit crap, a flash plugin or something. But click on 'paintings' and explore. Make sure you find the commutative diagrams [wolfram.com] the size of a house.
• #### Best Equation? (Score:2, Funny)

Man + woman = baby.
• #### Re:Best Equation? (Score:2)

Man and women can often be babies without needing each other...
• #### Re:Best Equation? (Score:2)

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

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

on Monday December 26, 2005 @08:05PM (#14341894)
Definitely different.
• #### Re:e^(i*pi) = -1 (Score:2)

Different ... because it is the best.
• #### Re:e^(i*pi) = -1 (Score:3, Interesting)

I prefer the actual Euler's formula instead of the special case. e^x = cosx+jsinx
• #### Re:e^(i*pi) = -1 (Score:2, Interesting)

e^jx that is. I should learn to preview.

• #### Re:e^(i*pi) = -1 (Score:1)

Certainly unexpected and kinda mind-blowing. I remember the first time I saw that equation I thought "Yeah, right. Pull the other one.".

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)

<`ten.yevadnai' `ta' `nai'> on Monday December 26, 2005 @09:51PM (#14342272)

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)

on Monday December 26, 2005 @08:11PM (#14341912) Journal
I looked at the Four-colour graph and found it .. beautiful.

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)

I looked at the Four-colour graph and found it .. beautiful.

Hail Eris!

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

I thought the same thing. It's pretty cool you can reduce an infinite amount of maps down to just 633. The fact that mathematicans don't like this proof says more about the biases of mathematicians than anything else.
• #### Re:Does it really matter? (Score:1)

The proof of the Four Color theorem has been discussed to death in mathematics circles. There are two main objections to the proof:
1. The proof is too long to be verified by hand. This is a big problem, since we're trained to dissect logical arguments to find flaws. Coders aren't infallible, bugs are inevitable. Did the proof only work in virtue of a bug? A cosmic ray?
2. The proof is really inelegant. They essentially came up with hundreds of short proofs, instead of abstracting away from concrete maps
• #### The jury is still out. (Score:2)

"It's pretty cool you can reduce an infinite amount of maps down to just 633."

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)

exp(pi*sqrt(163))=262537412640768744

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

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

Help me out... what's interesting about that equation?
• #### Re:The most beautiful equation is... (Score:1)

Among other things... the fact that you have pi on one side of the equation and a rational number on the other.
• #### Re:The most beautiful equation is... (Score:3, Informative)

by Anonymous Coward
exp(pi*sqrt(163)) is only a near integer, not an exact one. See Ramanujam constant [wolfram.com].

S=k log W

equals 1.

• #### I got better. (Score:2)

I got better: e^(i*pi)+1=0

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)

on Monday December 26, 2005 @08:37PM (#14342014)
1 = 2

wait
• #### Re:Mine (Score:2, Funny)

The one I heard was:

"1 = 2, for very large values of 1"
• #### Re:Mine (Score:2)

Any equation with Pi. The more Pi the better. More!

• #### Arithmetic series (Score:4, Informative)

<slashdot@me[ ]quared.com ['tas' in gap]> on Monday December 26, 2005 @08:45PM (#14342046) Homepage
sigma(i=1, n) = (n*(n+1))/2. There's something very elegant about being able to reduce a huge number of operations into three.

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)

Basically on this post [slashdot.org]. Well, that post asked users favorite equations, not necessarily beautiful. That leads to another interesting question - are your favorite equation and your most beautiful equation the same thing? I just finished a semester of Electrity and Magnetism, and I'm a big fan of Maxwell's eqastions now.
• #### I vote for... (Score:3, Insightful)

on Monday December 26, 2005 @09:08PM (#14342115)

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)

I've always liked the chemistry equation:

PV=nRT
• #### Re:Ideal gas law (Score:1)

I prefer (P1.V1)/T1=(P2.V2)/T2.

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)

a = b a^2 = ab a^2 - b^2 = ab - b^2 (a-b)(a+b) = b(a-b) a + b = b b + b = b 2b = b 2 = 1
• #### Re:1 = 2... (Score:1)

Too bad there's all that multiplying by zero in there
• #### Re:1 = 2... (Score:2)

Multiplying by zero isn't that bad.

Dividing by zero, on the other hand...

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

y = r^3/3

If you determine the rate of change in this curve correctly, I think you'll be pleasantly surprised!
• #### Re:The funniest equation (Score:1)

Mod parent up, hardy har har!
• #### RSA Encryption (Score:4, Informative)

on Monday December 26, 2005 @09:25PM (#14342171)
RSA Encryption is based on the general form of Fermat's Theorem:
x**phi(n) = 1 mod(n)
where phi(n) is Euler's Totient function which is the number of integers less than n that are relatively prime to n. The number n is chosen to be the product of two primes, p and q. Even if n is known, it is hard of find p and q. Then phi(n) = (p-1)(q-1) and it is easy to pick a d and an e such that
d * e = 1 mod(phi(n))
You give out n and e as your public key and use n and d as your private key. Public en/decryption is done with:
Y = X**e mod(n)
Private en/decryption is done with:
X = Y**d mod(n)
• #### The most beautiful equation (Score:2, Funny)

by Anonymous Coward
Subtract the clothes
Divide the legs
Multiply
• #### Girls are Evil (Score:5, Funny)

on Monday December 26, 2005 @09:37PM (#14342224) Homepage
A proof more than a formula:

We all know that girls require time and money, so
Girls = Time x Money

We also know that time is money, so
Time = Money

Therefore,
Girls = Money x Money = Money ^ 2

Furthermore, it is commonly known that money is the root of all evil, so
Money = sqrt(Evil)

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

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

When you take the square-root of both sides you should allow for a possible change of sign so:

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

Furthermore, it is commonly known that money is the root of all evil, so

Money = sqrt(Evil)

Sorry but your proof falls over at this stage. Money is not the root of all evil, but "the love of money" is the root of all evil. ref: [biblegateway.com] 1 Timothy 6:10 (KJV). Better translations render it "the root of all kinds of evil as opposed to "all evil" (i.e. not meaning that its the root of blanket universal evil). Thus, your formula fails it in two places at this stage of the proof.

• #### Heat Equation (Score:3, Informative)

on Monday December 26, 2005 @09:44PM (#14342252)
The heat equation is beautiful, as it applies to so many different things (heat, diffusion, options pricing).

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)

I was always partial to the fundamental theorem of calc... pretty profound (tangents and integrals are opposites) but, unlike for example Maxwell's equations, it is VERY easy to understand and prove.
• #### Re:Fundamental Theorem of Calculus (Score:1)

You should look into Stokes' Theorem. The FToC generalizes a lot, and Stokes' formula looks just like it.
• #### When I posted this there were 42 comments (Score:4, Funny)

on Monday December 26, 2005 @09:51PM (#14342267) Journal
42

I win!
• #### What about chemistry (Score:2, Interesting)

Combustion of propane and oxygen.
CH4 + 2O2 --> CO2 + 2H2O

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

hey, while the equations of how C2H5OH interact with neuroproteins may not be as pretty, the effects are definitely more spectacular.
• #### Solids (Score:2)

V - E - L + 2(F - S + G) = 0
• #### Sky high pie (Score:1)

Pi r square
not mine, My
Pie are round
• #### 1+3+3=7 (Score:2, Interesting)

Sorry if already said, but: 1+3+3=7

U=RI
• #### Gauss's Law: (Score:2)

Can't even paste the surface integral symbol into /.'s HTML restrictor ... see http://cnx.rice.edu/content/m1005/latest/ [rice.edu] for a decent formatting.

In words, Gauss's law states that "if you add up the surface integral of the displacement vector D over a closed surface S , 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)

Damn straight. Gauss-Bonnet connects a topological property (one that doesn't change with deformation) to the surface's curvature - which obviously varies under deformation! That's one of the more beautiful results I've seen in mathematics. It's also generally handy in differential geometry.
• #### Symmetric ones will win...(?) (Score:2)

Recent studies have shown that symmetry is the most visualy appealing.

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)

equidistant from each other?
• #### F=(MV^2)/2 (Score:1)

F=(MV^2)/2

so simple. so pretty. describes so so much.
• #### Re:F=(MV^2)/2 (Score:1)

Except that it's wrong. (mv^2)/2 is kinetic ENERGY, not Force.
• #### The beauty is in the proof. (Score:4, Insightful)

on Monday December 26, 2005 @11:27PM (#14342649) Journal
Not to be a humbug, but isn't the beauty of an equation in it's proof? I mean, mathematically, the difference between 2^(3*4)=4096 and e^(pi*i)=-1 isn't a whole lot. The proof, however, for e^(pi*i)=-1 is real mind-bender that culminates in a simple, beautiful little equation. It's that culmination that makes it beautiful, not the equation itself.

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)

Gauss's Law Green's/Stokes Theorem Eulers formula (Of Course) The Wave Equation (And Schrodingers Equation) Gauss's Law is one of the coolest equations I have ever used, unfortunatly it is pretty useless in all but the simplest of circumstances.
• #### Emmy Noether! (Score:5, Informative)

on Monday December 26, 2005 @11:52PM (#14342754)
Can't believe no one mentioned Noether's Theorem, so I'll submit it. Proof that the existence of any symmetry in a Lagrangian implies a conserved quantity.

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)

My infinity postulate is pretty ugly.
"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)

nope, and pass on the joint.
• #### Im really glad for this post. (Score:1)

I had a chance to look into a several concepts I haven't previously learned about. For example aleph numbers, which I'll admit only caught my eye because the word "aleph" had been used in several science fiction pieces I have enjoyed. Mathematical concepts relating to infinity can get pretty thought provoking and this is certainly one of them. I cant explain it after only ten minutes of absorption, so I highly recommend doing some learning for yourselves. Godel's Theorem, I am also struggling desparatly
• #### Lagrange's Theorem (Score:3, Interesting)

on Tuesday December 27, 2005 @01:30AM (#14343087)
Not an equation, but I find Lagranges Theorem (If H is a subgroup of G, then the order of H divides the order of G) to be beautiful in that it is not very obvious at first why this should be true.
• #### Britney Spears (Score:2)

Some of these are nice: http://britneyspears.ac/lasers.htm [britneyspears.ac]

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

Girls cost time and money.
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)

Various mathematical functions sit in the bar, drinking. Suddenly x^2 bursts in and yells: The Great Derivative is coming! Run or you'll be differentiated!!!
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)

The correct punchline is "But I'm dx/dy and you're nothing to me!"

• #### Navier-Stokes Equations (Score:2)

I'm surprised no one has mentioned these [wikipedia.org] yet.

These equations are used all the time in 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 looked reeel familair. My Intermediate Mechanics of Materials professor confirmed my insight, and that was

• #### Ugly, yet beautiful (Score:2)

Classification of Finite Simple Groups [wikipedia.org] and here [wolfram.com]

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".

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