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The World's Most Beautiful Equations?
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?"
E = mc ^ 2
Nothing more beautiful then that!
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)
The only things certain in war are Propaganda and Death. You can never be sure which is which though
a^2+b^2=c^2
or
E=mc^2
(-b(+||-)sqrt(b^2-4ac))/2a
Utinam me logica falsa tuam philosophiam totam suffodiant.
Do you really have to leave a question on the tail of each story to get slashdotters to post? Do you not think we'd post what we think even without your third grade Discussion Time questions used to force us to speak? I'm fairly certain that the readers here have no problem figuring out what to talk about in relation to the story. Adding those questions to the end of many entries is obnoxious and frankly, quite insulting. We don't need your prompt to speak, so please start cutting them out.
the derivative of a function f at a is f'(a) = lim:h->0 (f(a+h) - f(a))/h thats pretty much the basis of calculus
Check out Bernar Venet. 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 the size of a house.
Man + woman = baby.
Definitely different.
You are being MICROattacked, from various angles, in a SOFT manner.
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?).
The surprise isn't how often we make bad choices; the surprise is how seldom they defeat us.
I never did believe that stuff about beauty and truth...
Doesn't it make you feel good to know that our freedoms are protected by politicans, lawyers and journalists.
S=k log W
equals 1.
Is anyone here good at math? I'm not.
1 = 2
wait
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.
Basically on this post. 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.
-Tim
I agree, but in this case this is a Ask Slashdot, so it's normal that a question will be presented. By the way, the question wasn't added by the editors (same reason).
The AACS key is NOT 0xF606EEFD628B1CA427BEA93A9CA9773F
My vote is for the Einstein field equation. Briefly stated: the curvature of spacetime is proportional to its mass/energy content. Very pretty.
sub f{($f)=@_;print"$f(q{$f});";}f(q{sub f{($f)=@_;print"$f(q{$f});";}f});
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...
The World Wide Web is dying. Soon, we shall have only the Internet.
I've always liked the chemistry equation:
PV=nRT
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
503 Sig Unavailable
The Signature could not be accessed. Please try again later or contact the administrator
y = r^3/3
If you determine the rate of change in this curve correctly, I think you'll be pleasantly surprised!
Bill Clinton: Pimp we can believe in. - The Shirt!!!
We don't see the world as it is, we see it as we are.
-- Anais Nin
"what do Slashdot readers think are the most beautiful equations, and the most ugly ones too?"
[Beautiful]
(Time + Effort) = (Exchangable Value + Intrinsic Value)
[More beautiful]
(Time + Effort) + Compensation = Everyone's Happy.
[Ugly]
(Time + Effort) / copyright violations = Market Dilution.
[Uglier]
(Time + Effort) - (Middle finger to Artist) = (F***K You! I'm becoming an Electrician) = (Empty Stocking for Consumer)
Add the bed
Subtract the clothes
Divide the legs
Multiply
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
(e^(pi * i)) + 1 = 0
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.
"Nature doesn't care how smart you are. You can still be wrong." - Richard Feynman
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.
Omnes stulti sunt.
42
I win!
Lump lingered last in line for brains, and the ones she got were sorta rotten and insane.
Combustion of propane and oxygen.
CH4 + 2O2 --> CO2 + 2H2O
"It is not a dream, It is a simple feat of scientific electrical engineering, only expensive...blind faint-hearted, doub
V - E - L + 2(F - S + G) = 0
Pi r square
not mine, My
Pie are round
We are Dead Stars looking back Up at the Sky
Sorry if already said, but: 1+3+3=7
- Agilo
U=RI
lucm, indeed.
Better equation.
(Man + Man) = (Women + Women)
Can't even paste the surface integral symbol into /.'s HTML restrictor ... see http://cnx.rice.edu/content/m1005/latest/ 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 equation in Calc II, and that it could be proven using regular calculus (and had been proven, in fact, by Gauss, hence the name). I was stunned. Took me a week to come down off the glow.
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.
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.
Don't think that a small group of dedicated individuals can't change the world. It's the only thing that ever has.
F=(MV^2)/2
so simple. so pretty. describes so so much.
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.
This sig rocks the casbah.
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.
History will be kind to me, for I intend to write it - Sir Winston Churchill
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 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?
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 to understand but it's implications intrigue me greatly. Anyway, I sincerely want to thank the poster for showing me the way to many things I may ponder over. Thank you, and I wish many nights of brain-strain upon you as well.
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.
Some of these are nice: http://britneyspears.ac/lasers.htm
F = dp/dt
"Oh drat these computers, they're so naughty and so complex, I could pinch them." --Marvin the Martian
Sum(n=1..Infinity, 1/n^2) = Pi^2/6
There are no karma whores, only moderation johns
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
It's always a long day... 86400 doesn't fit into a short.
"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 then that would be a lot cooler.
I figure the mathematicians are looking for a far "better compression" than 633 conditionals.
That said, I do wonder whether the mathematicians and physicists will ever be able to compress the laws of the universe to a single theorem.
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.
Anagram("United States of America") == "Dine out, taste a Mac, fries"
34:24:34
x = sqrt(1+sqrt(1+sqrt(1...))) -- continue forever.
If you square both sides, you can remove the first squareroot:
x^2 = 1+sqrt(1+sqrt(1...))
Because the other value on the right side has an infinite number of squareroots itt is almost equal to x itself. Therefore, we can write:
x^2 = 1 + x
And that is the equation that defines the golden proportion. Find r1 and r2:
r = (1 ± sqrt(5))/2
Discard the negative vlues and you get
r1 = (1 + sqrt(5))/2. This is approximately close to 1.6180339887...
The golden mean is quite absurd
It's not your ordinary surd.
If you invert it (this is fun!)
You'll get itself, reduced by one;
But if increased by unity,
This yields its square, take it from me.
- Paul S. Bruckmann
Arkanoid
gethostbyintuition()... why not?
In algebra, this is one of the first things one proofs after introducing the second operation and its eigenvalue. As to ugliness: I have read neither, but I believe the proof of the four-colour theorem is easy compared to the proof that describes all simple groups, or to some of the stuff in Russell and Whitehead's Principia Mathematica (according to the Wikipedia, it states "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." on page 362)
The most beautiful equation is ......
0 0000000000
income - spending = 1000000000000000000000000000000000000000000000000
the stuff of mathematical wet dreams
The 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-first century].
fork + shoe = spleen
A dozen, A gross, A score.
Plus three times,
the squareroot of four.
Divided by seven,
Plus five times eleven.
Equals nine square plus zero no more.
Which equates to
(12+144+20+3*(sqrt(4)))/7+5*11 = 9^2+0
I thought of Hamilton's formula for quaternions
before I thought of the ideal gas laws.
i^2 = j^2 = k^2 = ijk = -1
As for the quadratic equation, it may be useful
but I find it ugly. But if you want really ugly,
take a look at the cubic and quartic equations.
You write your nine symphonies, then you die.
Smart man + smart woman = romance smart man + dumb woman = affair dumb man + smart woman = marriage dumb man + dumb woman = pregnant
http://en.wikipedia.org/wiki/Navier-Stokes/
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 when it all clicked.
That, and when viewed together using the Einstein summation convention, they're just plain pretty. What's funny is, I used to hate them.
n.b. Yes, I am a Mechanical Engineering geek.
Classification of Finite Simple Groups and here
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".
b=36
w=24
h=36
Table-ized A.I.