pi

Π or π (or in English, "pi") is the 16^th letter of the Greek alphabet.

The upper case form, Π , means:

• Π   product of a series (mathematics)

The lower case form, π , means:

• π   the ratio of a circle's circumference to its diameter, which is 3.14159... (math)
• π   population percentage (statistics)
• π   pion (physics)

As the top of this e2node indicates, pi is also an Everythingite.

What is pi?

Pi is the number equal to the circumference divided by the diameter of a circle. It's an irrational number so most people just use 3.14 or 3.14159. Though many memorize a lot more (hundreds) of digits, nobody is stupid enough to try to multiply by that.

The circumference of a circle is πR2 (R=Radius) or just πD (D=diameter).
The area of a circle is πR².

See A brief history of Pi for, well, the history of pi.

Also the title of a poem which I first saw as an easter egg on Audiofind.com. Sadly I have no idea who the original poet was.

i knelt down next to you
watched you laying there
i put my hand on your forehead
saw your eyes open
it's time to wake up
there are a lot of people who want to meet you

you looked back at me
stared into my eyes
i'll never forget what you said
"you"
"you are the builder"
"the creator of things unseen"

An attempt at a complete list. This is one overworked letter. :)

1. The 16th letter of the Greek alphabet.
2. Ancient Greek term for the circumference of a circle.
3. Modern term, as N-wing mentioned, for the irrational ratio of a circle's diameter to its circumference. This ratio pops up in many other places too.
4. N-wing states population percentage in statistics. I'll take his word for it.
5. Product series when capitalized. As in, Π(1<x<5) of (x-1), x in Z =(2-1)(3-1)(4-1)
6. π(x) is the number of primes below a number x - for example, π(100) = 25 (this, btw, approaches x/ln(x))

Given its general popularity as a concept meaning product, I'm sure there are other uses in various sciences. Will keep on hunting. Just had to add #6, my personal area of obsession.

15.000? HA! here's the first 119.000 decimal places of pi:

3.141592653589793238462643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151 5574857242 4541506959 5082953311 6861727855 8890750983 8175463746 4939319255 0604009277 0167113900 9848824012 8583616035 6370766010 4710181942 9555961989 4676783744 9448255379 7747268471 0404753464 6208046684 2590694912 9331367702 8989152104 7521620569 6602405803 8150193511 2533824300 3558764024 7496473263 9141992726 0426992279 6782354781 6360093417 2164121992 4586315030 2861829745 5570674983 8505494588 5869269956 9092721079 7509302955 3211653449 8720275596 0236480665 4991198818 3479775356 6369807426 5425278625 5181841757 4672890977 7727938000 8164706001 6145249192 1732172147 7235014144 1973568548 1613611573 5255213347 5741849468 4385233239 0739414333 4547762416 8625189835 6948556209 9219222184 2725502542 5688767179 0494601653 4668049886 2723279178 6085784383 8279679766 8145410095 3883786360 9506800642 2512520511 7392984896 0841284886 2694560424 1965285022 2106611863 0674427862 2039194945 0471237137 8696095636 4371917287 4677646575 7396241389 0865832645 9958133904 7802759009 9465764078 9512694683 9835259570 9825822620 5224894077 2671947826 8482601476 9909026401 3639443745 5305068203 4962524517 4939965143 1429809190 6592509372 2169646151 5709858387 4105978859 5977297549 8930161753 9284681382 6868386894 2774155991 8559252459 5395943104 9972524680 8459872736 4469584865 3836736222 6260991246 0805124388 4390451244 1365497627 8079771569 1435997700 1296160894 4169486855 5848406353 4220722258 2848864815 8456028506 0168427394 5226746767 8895252138 5225499546 6672782398 6456596116 3548862305 7745649803 5593634568 1743241125 1507606947 9451096596 0940252288 7971089314 5669136867 2287489405 6010150330 8617928680 9208747609 1782493858 9009714909 6759852613 6554978189 3129784821 6829989487 2265880485 7564014270 4775551323 7964145152 3746234364 5428584447 9526586782 1051141354 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Pi is a programming (better a specification) language. It was developed in 1984 and is used at some German universities (for example at University of Potsdam).
It was made to develop specifications. Specifcations are provisional programs. They are not for making real programs which can be shipped, but to formulate a given problem into an algorithm, implement this algorithm, test it and show it to client to verify, that that is what he wanted. Then you can start making the real program basing on this specification.
The given Pi-Compiler (can be downloaded from http://www.cs.uni-potsdam.de/~owilhelm/Manuskripte/Pi/pi.zip ) translates the given specification into Haskell (in fact the compiler is buggy so there is a pre-compiler which translates the given code, so that the compiler understands it and a post-compiler, so that the Haskell-code is working).

How to write a Pi-specification (please make yourself familiar with specifications before reading the following):
As stated in specification it consists of the export, the import, the common parameters and the body. In the first three there are only the signatures of the operations and types ( in import and common parameters are these which are not in this module (specification) ). A signature is the first line of an operation (ex. in C int add (int x, int y) ).
To show you how this all works I show you a short specification of the natural numbers + 0 ({0,1,...}):

this is what is exported.

cem NAT

type view specification


export


type Nat


constructor null : -> Nat
constructor successor: Nat -> Nat
operation null : -> Nat
operation successor: Nat -> Nat
operation add : Nat,Nat -> Nat
operation one : -> Nat
operation two : -> Nat

The import is empty as we only import one specification, but this is also needed to be exported

import

Now follow the common parameters

common parameters
type Logic

operation true : -> Logic
operation false : -> Logic
constructor true : -> Logic
constructor false : -> Logic

Now the body

body

construction of type Nat is internal

constructor null : -> Nat
constructor successor: Nat -> Nat


operation add : Nat,Nat -> Nat
variables x,y : Nat
equations 
        add(x,null) = x;
        add(x,successor(y)) = successor(add(x,y))


operation one : -> Nat
equations one = successor(null)


operation two : -> Nat
equations two = successor(one)

A constructor is the basis on which a type is constructed.

Please Note: Most of the modules are German so it can be hard for non-german speakers to understand

The world's longest mnemonic for pi is a short story by Michael Keith entitled Cadaeic Cadenza. This record was previously held by Poe, E., Near a Raven (encoding the first 740 digits of pi) by the same author.

Cadaeic Cadenza encodes the first 3835 digits of pi, while simultaneously being an interesting read -- not a simple feat.

I think I should elaborate on what others have said about solving for pi. You can approximate pi/4 as accurately as your heart desires by the formula 1 - 1/3 + 1/5 - 1/7 + 1/9 ... always alternating adding and subtracting, and adding two to the denominator. Note that when you are on the adding phase you will always be slightly above pi/4, and on the subtracting phase you will always be just a little below pi/4. But if you just decide how many digits you need pi to, you can run the equation until the digits you will actually use stop changing, and are therefor accurate. Then multiply your result by 4 (because remember, this is solving for pi/4) and ta-da, pi. Here look, I wrote this C++ program to find pi:

#include "iostream.h"

int main(int argc, char* argv)
{
	long double pi=1;
	int denominator = 3;
	bool alt = 0;
	long double divisor;

	while (true)
	{
		divisor = 1 / (long double)denominator;
		cout << pi*4 << " ";

		if (alt == 0)
		{
			pi = pi - divisor;
			alt = 1;
		}
		else
		{
			pi = pi + divisor;
			alt = 0;
		}

		denominator += 2;
	}
	return 0;
}

I'm no expert coder, so there is probably a way to get more digits in the output that I'm to stupid to know about, I'll update here and say so if I ever figure it out. -grin- But in the mean time, play with that.

Pi

~Wislawa Szymborska~

The admirable number pi:
Three point one four one.
All the following digits are also initial,
Five nine two because it never ends.
It can't be comprehended six five three five at a glance,
Eight nine by calculation,
Seven nine or imagination,
Not even three two three eight by wit, that is, by comparison
Four six to anything else
Two six four three in the world.
The longest snake on earth calls it quits at about forty feet.
Likewise, snakes of myth and legend, though they may hold out a bit longer.
The pageant of digits comprising the number pi
Doesn't stop at the page's edge.
It goes on across the table, through the air,
Over a wall, a leaf, a bird's nest, clouds, straight into the sky,
Through all the bottomless, bloated heavens.
Oh how brief - a mouse tail, a pigtail - is the tail of a comet!
How feeble the star's ray, bent by bumping up against space!
While here we have two three fifteen three hundred nineteen
My phone number your shirt size the year
Nineteen hundred and seventy-three the sixth floor
The number of inhabitants sixty-five cents
Hip measurement two fingers a charade, a code,
In which we find hail to thee, blithe spirit, bird thou never wert
Alongside ladies and gentlemen, no cause for alarm,
As well as heaven and earth shall pass away,
But not the number pi, oh no, nothing doing,
It keeps right on with its rather remarkable five,
Its uncommonly fine eight,
Its far from final seven,
Nudging, always nudging a sluggish eternity
To continue.

starting with the following function:

f(x)= -1     x < 0 
    = 0      x = 0
    = 1      x > 0 

notice that f(x) is odd and as such, it's fourier series expansion will only have sin() coeficients:

     +inf
f(x)= Σ an * sin(2*n*Pi*x/L)
     n=1

where

  an - fourier series coeficients yet to be determined
  L - length of the interval


an (the fourier series coeficients) are calculated as:

an=2/L * integral(a, a+L, f(x)*sin(2*n*pi*x/L), x)


if we calculate the expansion from a=-1 to a+L=1 we get (L=2):

an=2/2*integral(-1,1,f(x)*sin(2*n*pi*x), x)

noticing that f(x) and sin() are odd functions, their product will be even, and as such we can say that

integral(-1,1, ..., x)=2*integral(0,1, ..., x)

then:

an =2*integral(-1,1,f(x)*sin(2*n*pi*x), x)

   = 2/(n*pi) * (1-(-1)^n)
   = 0           n even
     4/(n*pi)    n odd

f(x) then becomes:

      +inf
f(x) = Σ 2/(n*pi) * (1-(-1)^n) * sin(2*n*Pi*x/2)
      n=1

changing variables n=2k+1 (getting rid of the even terms (which are 0)):

      +inf
f(x) = Σ 4/((2k+1)*pi) * sin((2k+1)*Pi*x)
      k=0

if we get the value of f(x)at x=1/2 (which we know to be 1):

   +inf
1 = Σ 4/((2k+1)*pi) * sin(k*pi+Pi/2)
   k=0

   +inf
  = Σ 4/((2k+1)*pi) * (-1)^k
   k=0

divide by 4/pi:

      +inf
pi/4 = Σ (-1)^k/(2k+1)
      k=0

     = 1 - 1/3 + 1/5 - 1/7 + ....

just thought it'd be interesting to know how to actually get the result.

note though that this converges very slowly. Check the pi metanode for faster ways of calculating pi.

Here's yet another way to derive pi:

Set xo to 2*sqrt(3).
Set yo to 3.

Define xn as:

2 * xn-1 * yn-1
xn-1 + yn-1

Define yn as sqrt(xn * yn-1).

The limit for both xn and yn as n approaches infinity will be pi.

When working with vectors, Π (uppercase) is also often used as a symbol for a plane.

For example:

Π: r.(4 5 1) = 7

is the equation of the plane Π

In number theory, the function pi(x) is defined to be the number of primes less than or equal to x. For example, pi(3)=2 since 2 and 3 are prime.

There have been many attempts to approximate the value of pi(x) for any given x. The logarithmic integral is one approximation of pi(x); indeed, in the limit (as x approaches infinity), the ratio pi(x)/Li(x) approaches one. This is one of the underlying results of the prime number theorem. Another famous approximation is given by Riemann and is based the logarithmic integral and the Mobius numbers:

  ∞
-----
\
 \    u(n)    /  1/n \
 /   -----*li | x    |
/      n      \      /
-----
 n=1

Where u(n) is the nth Mobius number. For most values of x, Riemann's estimate is among the best. But, whenever Li(x)<pi(x) (see Skewes number), Riemann's estimate is worse than many others.

For related information, see the Node of Prime, prime number and the Riemann zeta function. Note that since there are an infinite number of primes, pi(x) returns the value of every natural number at least once.

According to http://mathworld.wolfram.com/PrimeCountingFunction.html, the upper bound for pi(x) is (2x-6)/ln(x), where ln(x) is the natural logarithm of x.

An interesting point: To the best scientific guesses we have, the Universe .. Everything, ;-) .. shouldn't be more than a few Googol (10^100) kilometers wide, and it's probably a sphere or hypersphere. So you would use Pi to calculate its area or how much stuff is inside of it.

The thing is that to calculate anything to a Googol with significant digits, you only need 100 digits of Pi. So we could calculate the number of quarks (small atomic particles) in the entire Universe with, giving ourselves a thousand times more head-room than we think we need, 100,000 digits of Pi.

Here is the first 119,000 digits of Pi. Just imagine - with this number we could calculate perfectly the amount of atomic particles in Creation, if we had Its diameter (or some related precise measurement).

So, if this is the case, why do we have supercomputers working at this minute on Pi's digits well into the Billions? And why do people have it memorized to all those digits?

π

Released in 1998. Written and directed by Darren Aronofsky. Rated R for language and disturbing imagery.

CAST:

• Sean Gullette - Max Cohen
• Mark Margolis - Sol Robeson
• Ben Shenkman - Lenny Meyer
• Pamela Hart - Marcy Dawson
• Stephen Pearlman - Rabbi Cohen
• Samia Shoaib - Devi
• Ajay Naidu - Farrouhk
• Kristyn Mae-Anne Lao - Jenna
• Joanne Gordon - Mrs. Ovadia

Max Cohen, the protagonist, is a brilliant mathematician who makes his living by finding patterns.However, Max's brilliance comes at a steep price. He has debilitating migraine headaches, constant nosebleeds, and violent, scary-as-hell hallucinations that are the result of seizures.

Max believes that everything in life can be reduced to underlying patterns, or its basic math (kind of like Lawrence Waterhouse). His theory is as follows:

1. Mathematics is the language of nature;
2. Everything around us can be represented and understood by numbers;
3. If you graph these numbers, patterns emerge. Therefore: there are patterns everywhere in nature.

The project that he is currently working on is to find a pattern in the stock market. He has built a supercomputer called Euclid (a little in-joke, as Euclid, in Elements of Geometry, made certain observations that resulted in the discovery of π) to help him find this pattern.

Max is a loner who avoids the people around him to extremes. The only person he interacts with is his math professor, Sol Robeson. According to Max, Sol "nearly made it." What Max means is that Sol almost discovered an underlying pattern in the digits of π, before having a nervous breakdown. They play Go together, and Sol advises Max to stop with this obsessive search. He tells Max that some things are not governed by patterns. "Life isn't just mathematics." Max refuses to believe him.

While he does not initiate personal contact with anyone, there are people on the fringes of Max's life. Devi is a young woman who lives in the apartment next door to his, with her boyfriend Farrouhk. She often brings over meals (leaving them by the door), sensing that he neglects to eat. Farrouhk is overpossesive, tending towards abusive, usually misinterpreting Devi's attempts at kindness as Max's seduction of Devi. Often it is Farrouhk and Devi together who find Max after he has collapsed from one of his seziures. Mrs. Ovadia is Max's landlord. After one seizure, she is called because Devi and Farrouhk are unable to get into Max's apartment. She is furious because Max has installed too many additional locks, and Euclid is probably not within regulations on what tennants are allowed to have in their apartments. Jenna is a little girl who lives in his apartement building and always tests Max with obscure math problems, to see if he can do them faster than her calculator.

Max's research with Euclid does not go unnoticed. A young woman named Marcy Dawson leaves cryptic messages on his answering machine, and arranges to bump into him "accidentally." She claims to be from a Wall Street firm that wants to hire him. While sitting in a diner, he meets a Hasidic Jew named Lenny Meyer, who tells him that the Torah is also numerical, and that people have been trying to find the underlying code of it as well.

As a result of these chance meetings, Max is a desirable commodity. This furthers his paranoia and heightens his stress, resulting in more migraines and hallucinations. At one point, he sees a human brain pulsing on the floor of a New York City subway car. It also drives his research. Unfortunately, Euclid keeps spitting out a 216 digit bug. This bug has been found before, by Sol. It is what caused his nervous breakdown. 216 is also believed, by Kabalists, to be the number of letters in the ultimate name of God. So everyone wants this number, which truly exists only in Max's mind.

Pi is a short film (by feature length standards), clocking in around 79 minutes. The film was a first for director Darren Aronofksy, of Requiem for a Dream fame. Despite its length, this film can stay with you for a long time, as it is a veritable mindfuck of a film. Shot in grainy, high contrast black and white, with noir-style voiceovers, this film is a true psychological thriller. As the film progresses, it becomes harder to distinguish Max's hallucinations from reality. His paranoia overwhelms the viewer. The ending is sad, powerful, and abrupt. All in all, a great film.

Because I believe in giving credit where credit is due (and this node took some research):
Tim Dirk's list of the greatest films of 1998 on www.filmsite.org
Hollywood.Com's review and cast list www.hollywood.com/movies/detail/movie/178105
Feomante's Horror Homepage - Pi review by E.C. McMullen Jr. - www.feoamante.com/Movies/PQR/pi.html
Pi Movie Review by Michael A. Maynez http://www.casenet.com/michael/pi.htm
Pi's official website: www.pithemovie.com

Pi (?), n. [See Pica, Pie magpie, service-book.] (Print.)

A mass of type confusedly mixed or unsorted. [Written also pie.]

 

© Webster 1913

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Pi, v. t. [imp. & p. p. Pied (?); p. pr. & vb. n. Pieing (?).] (Print.)

To put into a mixed and disordered condition, as type; to mix and disarrange the type of; as, to pi a form. [Written also pie.]

 

© Webster 1913

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Pi (pI), n. [Gr. pi^.]

1.

A Greek letter (Π, π) corresponding to the Roman letter P.

2.

Specif.: (Math.) The letter Π, π, as used to denote the number or quotient approximately expressing the ratio of the circumference of a circle to its diameter; also, the quotient or the ratio itself. The value of the quotient pi, to eight decimal places, is 3.14159265. The quotient pi cannot be expressed as a root of an algebraic equation; and from this fact follows the impossibility of the quadrature of the circle by purely algebraic processes, or by the aid of a ruler and compass.

 

© Webster 1913