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The World's Most Beautiful Equations?

Cliff posted more than 8 years ago | from the aesthetics-and-mathematical-elegance dept.

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?"

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Einstein was onto something... (1)

N1ghtFalcon (884555) | more than 8 years ago | (#14341826)

E = mc ^ 2

Nothing more beautiful then that!

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

Darius Jedburgh (920018) | more than 8 years ago | (#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... (0)

Anonymous Coward | more than 8 years ago | (#14342084)

True, but you can make similar comments about just about any other equation.

Re:Einstein was onto something... (1)

ceoyoyo (59147) | more than 8 years ago | (#14342739)

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.

Re:Einstein was onto something... (4, Funny)

Walt Dismal (534799) | more than 8 years ago | (#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... (2, Informative)

pyite (140350) | more than 8 years ago | (#14342175)

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... (3, Informative)

Quadraginta (902985) | more than 8 years ago | (#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... (1)

Geoffreyerffoeg (729040) | more than 8 years ago | (#14343041)

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... (1)

Quadraginta (902985) | more than 8 years ago | (#14343059)

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 (3, Funny)

FidelCatsro (861135) | more than 8 years ago | (#14341833)

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)

 

Most Beautiful.. (0)

Anonymous Coward | more than 8 years ago | (#14341838)

a^2+b^2=c^2
or
E=mc^2

Quadratic Formula (1)

NaNO2x (856759) | more than 8 years ago | (#14341855)

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

Re:Quadratic Formula (0)

madsenj37 (612413) | more than 8 years ago | (#14341897)

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

And I ask the slashdot editors... (0, Troll)

phoenix.bam! (642635) | more than 8 years ago | (#14341869)

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.

I agree, but... (2, Insightful)

rbarreira (836272) | more than 8 years ago | (#14342104)

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).

Re:And I ask the slashdot editors... (1)

GoofyBoy (44399) | more than 8 years ago | (#14342105)

Clearly, it would be better if stories started off with the question.

Re:And I ask the slashdot editors... (0)

Anonymous Coward | more than 8 years ago | (#14342146)

And I, on the other hand, prefer Multiple Choice, you insensitive clod!

Re:And I ask the slashdot editors... (1)

jpmkm (160526) | more than 8 years ago | (#14342377)

Everything in italics is what the submitter wrote. Thus, the question was added by the submitter, not the "editor".

Re:And I ask the slashdot editors... (2, Funny)

ConceptJunkie (24823) | more than 8 years ago | (#14342427)

Please discuss.

definition of a derivative (0, Offtopic)

radical_dementia (922403) | more than 8 years ago | (#14341887)

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

Re:definition of a derivative (1)

ClamIAm (926466) | more than 8 years ago | (#14341997)

But is it art?

Re:definition of a derivative (1)

name773 (696972) | more than 8 years ago | (#14342787)

i also appreciate the fundamental theorem of calculus

Much better equation art (3, Informative)

Darius Jedburgh (920018) | more than 8 years ago | (#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? (1, Funny)

pipingguy (566974) | more than 8 years ago | (#14341891)

Man + woman = baby.

Re:Best Equation? (1)

rbarreira (836272) | more than 8 years ago | (#14342201)

Man and women can often be babies without needing each other...

Re:Best Equation? (1)

pipingguy (566974) | more than 8 years ago | (#14342262)


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

Best Equation?-Dry Well. (0)

Anonymous Coward | more than 8 years ago | (#14342484)

Better equation.

(Man + Man) = (Women + Women)

Re:Best Equation? (1)

alicenextdoor (910558) | more than 8 years ago | (#14343854)

(Man) + Woman = !Baby

e^(i*pi) = -1 (2, Insightful)

SpaceLifeForm (228190) | more than 8 years ago | (#14341894)

Definitely different.

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

hcg50a (690062) | more than 8 years ago | (#14341922)

Different ... because it is the best.

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

confusion here (827020) | more than 8 years ago | (#14342097)

I prefer the actual Euler's formula instead of the special case. e^x = cosx+jsinx

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

confusion here (827020) | more than 8 years ago | (#14342162)

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

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

MaskedSlacker (911878) | more than 8 years ago | (#14343748)

Agreed, aside from the typo. And why was this modded down?

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

Ignominious Cow Herd (540061) | more than 8 years ago | (#14342257)

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 (5, Interesting)

iced_773 (857608) | more than 8 years ago | (#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? (2, Interesting)

GoofyBoy (44399) | more than 8 years ago | (#14341912)

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? (1)

Pseudonym (62607) | more than 8 years ago | (#14342056)

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

Hail Eris!

Re:Does it really matter? (1)

Vellmont (569020) | more than 8 years ago | (#14342149)

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? (1)

poopdeville (841677) | more than 8 years ago | (#14342380)

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 and dealing with the abstract case. This is along the lines of showing that a group is Abelian by computing every product of elements. It works, but there better ways to do it.

The jury is still out. (1)

TheLink (130905) | more than 8 years ago | (#14343549)

"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.

The most beautiful equation is... (1)

exp(pi*sqrt(163)) (613870) | more than 8 years ago | (#14341916)

exp(pi*sqrt(163))=262537412640768744

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

Re:The most beautiful equation is... (0)

Anonymous Coward | more than 8 years ago | (#14342565)

That's not bad, but along those lines I think pi^4 + pi^5 = e^6 is much nicer.

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

John Miles (108215) | more than 8 years ago | (#14343168)

Help me out... what's interesting about that equation?

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

Majik Sheff (930627) | more than 8 years ago | (#14343272)

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... (3, Informative)

Anonymous Coward | more than 8 years ago | (#14343510)

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

Boltzman (1)

the eric conspiracy (20178) | more than 8 years ago | (#14341933)

S=k log W

How about e^(2*(pi*i)) (1)

wrathpanda (63661) | more than 8 years ago | (#14341987)

equals 1.

I got better. (1)

FooAtWFU (699187) | more than 8 years ago | (#14342129)

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...

Huh? (0, Offtopic)

The NPS (899303) | more than 8 years ago | (#14341999)

Is anyone here good at math? I'm not.

Mine (5, Funny)

ClamIAm (926466) | more than 8 years ago | (#14342014)

1 = 2

wait

Re:Mine (1)

iLogiK (878892) | more than 8 years ago | (#14343351)

that's correct!
0 = 0
1-1 = 2-2
1*(1-1) = 2*(1-1) |:(1-1)

1 = 2

Arithmetic series (3, Informative)

Metasquares (555685) | more than 8 years ago | (#14342046)

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... (2, Interesting)

emplynx (735511) | more than 8 years ago | (#14342085)

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... (2, Insightful)

Pseudonym (62607) | more than 8 years ago | (#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 (1)

kaos_ (96522) | more than 8 years ago | (#14342140)

I've always liked the chemistry equation:

PV=nRT

Re:Ideal gas law (1)

McTaggart (893466) | more than 8 years ago | (#14343037)

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.

Re:Ideal gas law (0)

Anonymous Coward | more than 8 years ago | (#14343139)

*shudder*

Thank you, and the grandparent, for a flashback to 1st year chemistry which I really didn't need... :p

1 = 2... (1, Interesting)

Luigi30 (656867) | more than 8 years ago | (#14342147)

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... (1)

magicchex (898936) | more than 8 years ago | (#14343054)

Too bad there's all that multiplying by zero in there

Re:1 = 2... (1)

Vaevictis666 (680137) | more than 8 years ago | (#14343143)

Multiplying by zero isn't that bad.

Dividing by zero, on the other hand...

Re:1 = 2... (1)

iLogiK (878892) | more than 8 years ago | (#14343369)

2 minutes ago i posted the proof for 1=2, here [slashdot.org]

The funniest equation (1)

Rude Turnip (49495) | more than 8 years ago | (#14342168)

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 (1)

shobadobs (264600) | more than 8 years ago | (#14342308)

Mod parent up, hardy har har!

RSA Encryption (3, Informative)

DrJimbo (594231) | more than 8 years ago | (#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)

A valuable experience. (0)

Anonymous Coward | more than 8 years ago | (#14342172)

"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)

The most beautiful equation (2, Funny)

Anonymous Coward | more than 8 years ago | (#14342186)

Add the bed
Subtract the clothes
Divide the legs
Multiply

Girls are Evil (5, Funny)

DeltaHat (645840) | more than 8 years ago | (#14342224)

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

Four Constants == Beauty (0, Redundant)

rickwood (450707) | more than 8 years ago | (#14342228)

(e^(pi * i)) + 1 = 0 [google.com]

Re:Four Constants == Beauty (0)

Anonymous Coward | more than 8 years ago | (#14343291)

(e^(pi * i)) + 1 = 0 But that's five constants...

Heat Equation (2, Informative)

pyite (140350) | more than 8 years ago | (#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 (2, Insightful)

sinclair44 (728189) | more than 8 years ago | (#14342256)

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 (1)

poopdeville (841677) | more than 8 years ago | (#14342407)

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 (4, Funny)

Ignominious Cow Herd (540061) | more than 8 years ago | (#14342267)

42

I win!

What about chemistry (2, Interesting)

hvnerd (903682) | more than 8 years ago | (#14342326)

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

Re:What about chemistry (0)

Anonymous Coward | more than 8 years ago | (#14342550)

CH4 is methane, not propane.

Re:What about chemistry (1)

hvnerd (903682) | more than 8 years ago | (#14343097)

Correct, I was thinking about propane at that moment. Combustion of methane and oxygen.

Re:What about chemistry (1)

Vo0k (760020) | more than 8 years ago | (#14343528)

hey, while the equations of how C2H5OH interact with neuroproteins may not be as pretty, the effects are definitely more spectacular.

Solids (1)

shobadobs (264600) | more than 8 years ago | (#14342336)

V - E - L + 2(F - S + G) = 0

Sky high pie (1)

wooferhound (546132) | more than 8 years ago | (#14342371)

Pi r square
not mine, My
Pie are round

1+3+3=7 (2, Interesting)

Agilo (727098) | more than 8 years ago | (#14342390)

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

the Geller formula (1)

lucm (889690) | more than 8 years ago | (#14342395)

U=RI

Gauss's Law: (1)

rpresser (610529) | more than 8 years ago | (#14342502)

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 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 (1)

msuarezalvarez (667058) | more than 8 years ago | (#14342525)

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] .

Symmetric ones will win...(?) (1)

bergeron76 (176351) | more than 8 years ago | (#14342560)

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.

F=(MV^2)/2 (1)

jkerman (74317) | more than 8 years ago | (#14342641)

F=(MV^2)/2

so simple. so pretty. describes so so much.

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

Hikaru79 (832891) | more than 8 years ago | (#14342782)

Except that it's wrong. (mv^2)/2 is kinetic ENERGY, not Force.

The beauty is in the proof. (3, Insightful)

Vorondil28 (864578) | more than 8 years ago | (#14342649)

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: (1)

Vilim (615798) | more than 8 years ago | (#14342721)

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! (5, Informative)

Quadraginta (902985) | more than 8 years ago | (#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 (1)

DanThe1Man (46872) | more than 8 years ago | (#14342802)

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 (1)

Vo0k (760020) | more than 8 years ago | (#14343520)

nope, and pass on the joint.

Im really glad for this post. (1)

guardianfox (853748) | more than 8 years ago | (#14342914)

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.

Lagrange's Theorem (3, Interesting)

siwelwerd (869956) | more than 8 years ago | (#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.

Newton's Second Law (1)

Spock the Baptist (455355) | more than 8 years ago | (#14343158)

F = dp/dt

my vote (1)

The Clockwork Troll (655321) | more than 8 years ago | (#14343483)

Sum(n=1..Infinity, 1/n^2) = Pi^2/6

Truth is beauty, so here's some truth (1)

Curien (267780) | more than 8 years ago | (#14343501)

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? (1)

Vo0k (760020) | more than 8 years ago | (#14343562)

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.

Most Beautiful... (1)

tooth (111958) | more than 8 years ago | (#14343771)

34:24:34

Fibonnaci (1)

arkanoid.dk (895391) | more than 8 years ago | (#14343786)

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
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