[Copied from http://www.glue.umd.edu/~jasonp/pi-ref.txt on 10/7/98.]
This document is titled: pi-ref.doc
It is being placed into the public domain by its author, Carey
Bloodworth, on August 11, 1996
I had planned on writing a good reference on the history of pi and
how to calculate it, but never really got the time to do it. So,
rather than let it sit on my hard drive, I'm releasing what I have
done, and adding more when I get time, and re-releasing it.
I am not going to give a large number of formulas for the simple
reason that it's hard to do them in ASCII in a readable manner. I'm
also going to 'short change' you on some of the math in the more
complex formulas. You'll just have to take my word for it, or go
hunt up some reference works.
This is more of a reference on the history of pi calculations,
rather than a 'how to' for calculating millions of digits. Although
that would probably be interesting to write, this isn't it.
This document uses the IBM PC's 'high' ASCII characters. In case
the document gets stripped of all 'high' ASCII charcters and they
are converted into regular ASCII characters, I'm also including what
those are, so you can recognise what they should be when you see
them:
(253) ý (125) } is the 'squared' symbol.
(252) ü (124) | is the 'nth' power symbol.
(251) û (123) { is the square root symbol
(247) ÷ (119) w is the 'approximatly equal' symbol.
(246) ö (118) v is the 'divide' symbol
(243) ó (115) s is the 'less than or equal' symbol: <=
(242) ò (114) r is the 'greater than or equal' symbol: >=
(241) ñ (113) q is the 'positive negative' symbol: +-
(240) ð (112) p is the 'congruent' symbol.
(236) ì (108) l is the 'infinity' symbol.
(228) ä (100) d is the 'sumation' symbol.
(172) ¬ ( 44) , is the 'one fourth' symbol.
(171) « ( 43) + is the 'one half' symbol.
(138) Š ( 10) is an 'e' with an accent. Converts into a line feed.
Unfortunately, charcter (227), the 'pi' character can _NOT_ be
entered into this document, because that is used as the 'end of
line' character in the .QWK message packet format that is used by
many BBSs.
When I write a formula and it requires subscripts, I do it as a
programmer would, by using brackets [ and ] to suround the index /
subscript number. It would be nice if you could easily do sub and
super scripts in an ASCII text, but you can't. Some people put the
subscripts and the next line, but that never really looks good.
Some people may prefer to use parenthesis ( and ), but those are
used to group mathematical sections and it would be confusing. And,
like Frank Sinatra,.... "I did it myyyyy WAAAAY".
I'm going to give a brief overview of some of the ways you can
calculate pi, and a history of pi calculations. I obviously don't
have the space to cover everything, but hopefully, I can cover
enough to be interesting. I also recommend that you read Petr (1
'e') Beckmann's book "A history of pi". It's very entertaining and
gives a very good overview of its history. It doesn't give a great
deal of heavy math, but it gives a few formulas that might be of
interest. My copy is dated 1974, so obviously it doesn't have the
latest algorithms, but it's still quite readable. I do strongly
recommend it for anybody who wants to enjoy reading about pi, and a
bit of history of mathematics.
I also want to say that I have never performed any serious
calculation. By 'serious', I mean anything in the millions. RIght
off hand, I think the largest calculation I ever did was 20k digits
using an arctangent formula on an 0.8Mhz 64k 8 bit microcomputer
back in the late 1980's. The program was of course written in
assembler. On the PC, my tinkering has been much more limited. In
fact, I've done more 'experimenting' than actually finishing and
doing some digit hunting. I could do a hundred thousand digits
without too much effort, but I just don't want to mess with any
calculations until I can easily do millions. That shouldn't effect
what I have to say though. I've also taken a break from this for
several years, so I'm certainly not familiar with the current
happenings.
And finally, I'd like to say that there seems to be considerable
discrepency on dates and the number of digits of some of the
calculations. I've tried to be fairly conservative, but since I
don't have access to the original documents, there is no way I can
really be sure about some of this.
Chapter 1: The early years
===========================
The first estimation of 'pi' was, of course, by direct measurement.
You draw a circle and measure the ratio of the diameter and the
circumference. At best though, you can only get maybe 2 decimal
digits. That's certainly enough for most practical purposes, and
certainly all practical purposes in ancient times.
But, even then, there were the 'digit hunters', those who calculated
the digits of pi beyond any practical use. The first was Archimedes
who discovered that a 'n' sided polygon inscribed inside of a circle
has a perimeter smaller than the circumference of that circle, while
a similar 'n' sided plygon inscribed outside of that circle has a
perimeter larger than the circumference of that circle. Working
through some fairly tricky math gave him the first method for
calculation pi to any accuracy desired. Of course, the bounds (of
the inner : outer polygons) were not extremely accurate and it took
a 'fairly large value of n' polygon to get even a couple of digits.
For example, using a 96 sided polygon and extracting square roots
four times over yielded only 2 decimal places. He was able to
determin that it was somewhere between 223/71 and 22/7. And the
fact that he was working in roman numerals, rather than a positional
numbering system (like the modern Arabic system) didn't help. But
it worked, and that's what counts.
Archimedes polygons remained fairly much state of the art until
1593, when ViŠte published his infinite sequence of algebraic
operations. In practice, his formula is useless, but it is fairly
significant historically, since his method was the first to be given
as an analytical expression of an infinite sequence of algebraic
operations. In otherwords, it was the first real 'formula' for pi.
Achimedes technique was an 'algorithm', where there was a sequence
of steps you followed, but there was no actual formula that could be
written down. ViŠte's was still based on the Archimedian polygon,
but it did point a new direction.
2
pi = -------------------------------------
û« * û(« + «û«) * û(« + «û(«+«û«))...
In 1655, John Wallis came up with:
2*2*4*4*6*6*8*8....
pi = 2 * -------------------
1*3*3*5*5*7*7*9....
This is a fairly significant formula. Its convergence is still
fairly slow, but it requires only simple numbers, unlike Viete's
which required multiple square roots.
(Actually, one of my references says that this _isn't_ the formula
that Wallis came up with. What he came up with was a 4/pi formula
that is equivelant to the above formula.)
And Brouncker transformed that into a continued fraction
4/pi = 1 + 1ý
---
2 + 3ý
---
2 + 5ý
---
2 + 7ý
---
2 + .......
(And later, Leonard Euler showed that this was equivalent to the
Gregory arctangent series, discovered a few years later.)
Chapter 2: Breakthrough
========================
In 1671, Gregory discovered his famous arctangent formula:
arctan (x) = x - (x^3)/3 + (x^5)/5 - (x^7)/7 + .....
And it was this discovery that opened up 'modern' pi calculations.
If you evaluate '1' in it, you get pi/4.
pi=4*(1-1/3+1/5-1/7+....)
However, the formula has extremely slow convergence. If you want 6
digits of precision, you'll have to do a half million terms. If you
want 100 digits, you'll have to go up to 10^50! Even Archimedes
polygon method was better than this. Even going out and drawing a
circle and measuring it, is better than this! But, there is hope!
Newton used the same concept and came up with an arcsin formula:
1 * x^3 1*3*x^5
arcsin (x) = x + -------- + --------- + .....
2 * 3 2*4*5
And if you evaluate arcsin (1/2) with it, you get pi/6. And it
converges much, much faster than arctan(1) with Gregory's series.
(I should point out that this isn't _exactly_ the formula Newton
came up with, but it is what the history books say. According to
Beckmann, Newton actually used:
3*û3 1 1 1 1
pi = ----- + 24( - - ----- - ------ - ------ ....)
4 2 5*2^5 28*2^7 72*2^9
I don't really know, but it doesn't make too much difference, other
than supplying an additional formula.)
The digit hunters then went back to the Gregory series. Abraham
Sharp realized that you didn't actually have to use arctan(1). You
could use a number, such as û(1/3).
pi 1 1 1 1
-- = ---*( 1 - --- + ----- - ----- ... )
6 û3 3*3 3^2*5 3^3*7
As you can see, those extra numbers in the denominator makes it much
faster than if you had just done arctan(1).
In 1706, John Machin had an incredible idea. He used tan B=1/5 and
2tanB 5
tan 2B=--------- = --
1-taný B 12
2tan2B 120
tan 4B=---------- = ---
1-taný 2B 119
tan 4B -1 1
tan(4B-pi/4) = --------- = ---
1+tan 4B 239
and came up with pi/4=4arctan(1/5)-arctan(1/239). This formula
converges much, much more quickly than just arctan(1) does. This
was the formula that started it all! You'll find this pi formula
quoted and mentioned more often than any other. Machin used his
formula and calculated 100 digits easier and faster than anybody
else had ever done.
Later that same year, William Jones, introduced the 'pi' symbol. It
appears that he used it as an abreviation for 'periphery'. Nobody
really paid any attention to it though, since he was fairly
insignificant in the world of mathematics.
Leonard Euler came up with quite a few pi related items:
1/1ý + 1/2ý + 1/3ý + ... = piý/6
1/1ý + 1/3ý + 1/5ý + ... = piý/8
1/1ý - 1/2ý + 1/3ý - ... = piý/12
He also liked the idea that Machin had and developed:
arctan 1/p = arctan 1/(p+q) + arctan q/(pý + pq + 1)
and
arctan x/y = arctan (ax-y) / (ay+x) + arctan (b-a) / (ab+1) +
arctan (c-b) / (cb+1) + .....
And these methods give rise to an infinite number of arctangent
relationships! What Machin had done that was so stupendous was now
mearly common and mundane.
He also found a faster converging arctangent formula
2 2*4 2*4*6
arctan x = (y/x) (1 + -y + ---y^2 + -----y^3 + .....)
3 3*5 3*5*7
where y = (xý)/(1+xý)
This one converges faster, but also requires more effort for many of
the numbers. Using pi/4=5arctan(1/7)+2arctan(3/79) and his own
arctangent formula, he was able to calculate 20 digits in less than
an hour.
And at this point, we've truly reached the start of Digit Hunting...
Chapter 3: Digit hunting
=========================
With the Gregory and Euler arctangents, and Eulers method for
arctangent relationship, calculating digits of pi became almost
fashionable! It was no longer the province of a few dedicated
mathematicians who didn't have a life and couldn't get a date, but
something that you and a few friends could do on a rainy evening.
There were other methods besides the arctangent relationships, but
not too many. And most of those that were used were of little use
or historical significance.
I'm going to give a few of the more common arctangent relationships,
plus their 'measure' of effort required for them. (More on the
'measure' later.) There are of course, an infinite number of
relationships, although only a few are actually useful, and only a
few of those are presented.
pi/4= arctan(1) (infinite)
pi/4= arctan(1/2) + arctan(1/3) (5.4)
pi/4= arctan(1/2) + arctan(1/5) + arctan(1/8) (5.8)
pi/4= 2arctan(1/3) + arctan(1/7) (3.27)
pi/4= 4arctan(1/5) - arctan(1/239) (1.85)
pi/4= 4arctan(1/5) - arctan(1/70) + arctan(1/99) (2.47)
pi/4= 5arctan(1/7) + 2arctan(3/79) (1.88)
pi/4= 6arctan(1/8) + 2arctan(1/57) + arctan(1/239) (2.09)
pi/4= 8arctan(1/10) - arctan(1/239) - 4arctan(1/515) (1.28)
pi/4=12arctan(1/18) + 8arctan(1/57) - 5arctan(1/239) (1.78) (1.21)
The 'measure' is just the sum of the reciprocal's of the Log of the
numbers. For example, Machin's 4arctan(1/5)-arctan(1/239) would be
1/Log(5) + 1/Log(239) = 1.85.
There are many, many things that can influence the actual amount of
work that is done by the calculator. For example, some numbers are
easier to deal with than others. Such as '10' in our base 10
system. If you were using a computer, some power of '2' (such as 8)
would be better. Some numbers also have a pattern that can help,
such as the '99' in one of the formula's above.
It is also influenced by the arctan formula used. In some cases it
may actually be better to use Euler's than Gregory's.
For example, in the 12arctan(1/18)+8arctan(1/57)-5arctan(1/239)
formula, if you look at the arctan(1/18) and arctan(1/57) term using
Euler's arctangent, you see a very interesting coincidence.
1 2 2*4
arctan(1/18)= 18(--- + ------- + --------- + ... )
325 3*325^2 3*5*325^3
1 2 2*4
arctan(1/57)= 57(---- + -------- + ---------- + ... )
3250 3*3250^2 3*5*3250^3
As you can see, each term is just divided by extra 0s. You don't
even have to copy them, you can add them and mentally shift the
decimal point the required number of places. This effectively
removes the arctan(1/57) term from the measure, and reduces it to
only (1.21)
There used to be a lot of discussion as to which method was best,
and even that varied depending on the circumstances. Probably the
'best' of the ones I've listed would be the (1/18) formula, and
using the (1/8) as a check. The (1/10) one would also be worth
considering. Doing (1/18) with Euler's has the benefit of
automatically calculating the (1/57), which can also be used in the
(1/8) formula. And the (1/239) is also used in both. So the (1/8)
formula really only has a measure of 0.90. So, it can be computed
and checked for only slightly more work than Machin's formula can
just compute it.
pi/4= 6arctan(1/8) + 2arctan(1/57) + arctan(1/239) (0.90)
pi/4= 8arctan(1/10)- arctan(1/239) - 4arctan(1/515) (1.31)
pi/4=12arctan(1/18)+ 8arctan(1/57) - 5arctan(1/239) (1.21)
And with these tools, pi calculations exploded!
Euler calculated 20 digits. Baron Georg Von Vega calculated 140
digits. Rutherford calculated 208 places. Lehmann made it to 261
digits. Tseng Chi-hung did 100 digits in a month. Clausen went to
248 digits. And on, and on, and on....
There are only two notable digit hunters of this period.
Strassnitzky had Johann Dase, a calculating prodigy, calculate 200
digits using pi/4=arctan(1/2)+arctan(1/5)+arctang(1/8). He did most
of those calculations in his head and in under two months! Dase
then started calculating the first milion natural logarithms to 7
digits, a table of hyperbolic functions, and all the factors of the
numbers from 7 million to 10 million. Gauss urged Hamburg Academy
of Sciences to financially support Dase during his factoring. This
is the first known incident of paying for 'computer' time. (Prior
to the electronic calculator/computer, humans such as Dase, were
known as computers.)
And the other, was the infamous William Shanks! You've probably
heard the story of how he calculated 707 digits but screwed up after
507 digits... Well, as Paul Harvey would say, this is the Rest of
the Story.
In 1853, Shanks published 530 digits of pi. It agreed with his
mentor's calculation of 441 digits. It is now known that 527 of the
530 digits he calculated were correct. The last few digits being
incorrect is a normal round off / truncation error and is to be
expected. It may also have been a genuine error. I don't have his
original work, so I can't say anything for sure except that it is
normal for the last few digits to be wrong because of round off and
truncation errors.
Later than same year, he published his result to 607 digits. He
also gave all the details of his 530 digit effort. But, somewhere
during the editing, he introduces some errors in the 460-462 and
513-515 places. It also appears that he does not correct his
previous error at 527 digits, so it is quite likely it was a genuine
error rather than a truncation error.
These errors persist in his first paper of 1873, where he gave 707
digits. His second paper that year finally corrects his previous
errors, but there ends up being a _single_ typographical error in
the 326th digit!
Most stories have it that Shanks first published in 1873, and that
he did all 707 digits at once, and that he spent 30 years of his
life doing it, and that there were errors in it but nobody knew
about it, and all sorts of things. Well, much of it isn't true. He
only did 530 digits first, and that was verified to at least 441
digits. It took him less than a year to continue to 607 digits.
And although there were 20 years between his first two publications
and his last two, he certainly didn't spend the entire time
calculating just 100 digits more. And he was aware of the errors in
the first three publishings and did correct all of them in the final
one, but was foiled by a single, simple typographical error during
the printing process itself. And it's fairly likely he even knew
about that but didn't want to go to the trouble of republishing the
whole thing.
Unfortunately though, his bad fortune was the rule, rather than the
exception! Hand calculations were notoriously unreliable, and it
was never considered official until the results were independantly
checked. Depending on how he did his work, he may have had to do
more than 3 million individual digit operations. There is a lot of
opportunity for error. (It's extrememly difficult to estimate
exactly how many operations it took, because there are several ways
he could have done this. And you have to consider the divisions,
multiplications, additions, subtractions, carry/borrow propogation,
etc. Regardless though, it was a lot.)
Shanks' error has gotten all the bad publicity, but errors have
occured at almost every attempt at breaking the digit record.
That's why no calculation was considered an official record until it
was independantly verified.
Shanks record stood until the early 1940s for the simple reason that
nobody else wanted to spend that much time doing all those hand
calculations. Part of the reason is that the transcadentalism and
irrationality of pi had been proven, and there was no longer any
reason to compute large quantities of digits in an effort to find
out more about the nature of pi. It even appears that the hobbiest
calculations slowed, because most references mention only 200-400
digit calculations for that time period. Almost like everybody was
waiting for the next breakthrough....
In 1945, Ferguson used a desk calculator and reached 620 digits. It
was when trying to verify his results against Shanks that the errors
finally became widely known. Even though he was unable to verify
his work, he continued calculating, eventually reaching 808 places.
(808 was chosen to provide comparable precision to what 'e' was
known at.) You could suggest that Ferguson simply compare his
calculations to two parts of Shanks' that were correct, but that is
somewhat dishonest, because you simply don't know for _certain_
about his calculations.
Since it was not possible to verify Ferguson's work against Shanks,
J. W. Wrench and Levi B. Smith undertook a calculation to 808
digits. They were able to verify 710 digits with Ferguson's value.
Upon further work by both parties, an error at the 723rd place
showed up in Wrench's work. By 1948, after performing corrections,
they were able to verify all 808 digits and this became the official
record for the time.
Apparently Wrench had enjoyed the work, because he continued and
eventually reached 1,120 digits and was in the process of verifying
it, when... the 'computer' ENIAC managed to calculate 2,037 in 70
hours. Contrary to popular belief, it wasn't Shanks record that
fell, but Smith's and J. W. Wrench's of 808 confirmed digits (or of
Wrench's 1,120 unverified digits, depending on which you consider to
be the record.)
And then of course, arctangents were used in all of the computer
calculations for the next 35 years. See the pi time line at the end
of the document for a brief run down of who did what. There isn't
really anything too interesting, except that I should point out that
even with electronic computers, errors were still quite common, and
as usual, it wasn't official until it was independantly verified.
In the case of Gosper in 1985, and Bailey in 1986, they both found
faults in the hardware itself that had to be fixed or worked around,
or in the case of transient failures, calculations reran.
Chapter 4: Modern Breakthrough: Eugene Salamin
===============================================
In 1972, Eugene Salamin came up with an algorithm that doubles the
number of correct digits at each iteration. This was a phenomenal
improvement. Previous methods required hundreds of millions of
operations, but Salamin's AGM (Arithmetic Geometric Mean) would only
require a few hundred.
Let A[0]=1, B[0]=1/û2
Then iterate from 1 to 'n'.
A[n]=(A[n-1] + B[n-1])/2
B[n]=û(A[n-1]*B[n-1])
C[n]ý=A[n]ý - B[n]ý
n
PI[n]=4A[n+1]ý / (1-( ä (2^(j+1))*C[j]ý))
j=1
There is an actual error calculation, but it comes out to slightly
more than double on each iteration. I think it results in about 17
million correct digits, instead of 16 million if it actually
doubled. PI16 generates 178,000 digits. PI19 to over a million.
PI22 is 10 million, and PI26 to 200 million.
This results in a vast reduction in the number of operations
required. The old pi/4=12arctan(1/18)+8arctan(1/57)-5arctan(1/239)
would require about 105,000 full precision operations for 100,000
digits. This would only require 112 full precision operations!
Although this method was discoverd by Salamin in 1972, and was item
143 in the famous MIT Hakmem memo, it wasn't formally published
until 1976. At that time, it was discovered that Brent had also
discovered it at about the same time, and they both discovered that
they had rediscovered a method given by Gauss 150 years previously.
And by using Strassen's discovery of just a few years previous, you
could multiply extremely large numbers is O(n log n log log n) time.
This is of course vastly better than the O(ný) time of traditional
methods. By having fast multiplication, you can also easily do
division, by computing its reciprocal, and square roots, by Newton's
method.
Even though this formula was published in 1976, it appears it wasn't
used in a serious calculation until 1982, when Tamura and Kanada
used it to calculate 4 million decimals.
This method is no longer used, but it opened up a whole new source
of pi formulas, such as those by the Borwein's.
Chapter 5: The Borweins
========================
Jonathan and Peter Borwein found Salamin's method to be quite
interesting. After several years work, they derived a general
method for higher order equations. (See their book: Pi and the AGM
- A study in Analytic Number Theory and Computational Complexity,
Wiley, N.Y., 1987).
Where as Salamin's only doubled, they derived formulas where the
correct number of digits increased by 4 and by 5 times! Thirteen
iterations would have been enough for more than 1,000,000,000 digits
of pi. However, some of their formulas were only mathematically
interesting, meaning that they actually required too much work for
each iteration.
Also, based on work by Ramanujan (1887-1920), and their own work
with modular equations, they derived a number of series where each
iteration added 'x' number of digits at each time. One of them
generates 25 more digits with each term, although that particular
one isn't really practical.
A)
Let a[0]=6-4û2 and y[0]=û2-1.
1-(1-y[n]^4)^¬
y[n+1] = --------------
1+(1-y[n]^4)^¬
a[n+1] = (1+y[n+1])^4 * a[n] - (2^(2n+3)) * y[n+1] * (1+y[n+1]+y[n+1]ý)
Then: 0 < a[n] - 1/pi < 16*4ü*e^(-2*4ü*pi), which means a[n]
converges to 1/pi quartically (4 times.)
B)
Let s[0]=5*(û5-2) and a[0]=1/2.
25
s[n+1] = --------------------
(z + x/z +1)ý * s[n]
x=5/s[n]
y=(x-1)ý+7
z=(«(y+û(yý-4*x^3)))^(1/5)
s[n]ý-5
a[n+1]=s[n]ý*a[n] - 5ü * (------- + û(s[n]*(s[n]ý-2*s[n]+5)))
2
then 0 < a[n] - 1/pi < 16*5ü*e^(-5ü*pi), which means a[n] converges
to 1/pi quintically (5 times.)
They also mention, and independantly derive, an old Ramanujan
formula:
C)
1 ì 2n 3 42n+5
- = ä ( ) ---------
pi n=0 n 2^(12n+4)
The also found a series that adds 25 digits for each iteration:
D)
ì
__ (-1)ü(6n)![212,175,710,912û61+1,657145,277,365+
1 < \ n(13,773,980,892,672û61+107578,229,802,750)]
-- = 12 > ---------------------------------------------------------
pi <__/ (n)!^3(3n)![5,280(236,674+30,303û61)]^(3n+3/2)
n=0
Although this one really isn't practical for an actual computation.
Chapter 6: Bill Gosper
=======================
While the Borwein's were working on the Ramanujan formulas, Bill
Gosper, one of the original hackers, converted the Ramanujan
formula:
ì
__
1 û8 < \ (4n)! (1103 +26390n)
--- = ------- > ---------------------
pi 9801 <__/ (n!)^4 396^(4n)
n=0
into a continued fraction and generated 17 million digits on a small
work station, running LISP. At the time, it was a record. I should
further mention that his tiny workstation had only a small fraction
of the power of the super computers that the calculations had
previously been done on. It was far less powerful than what most
people have at home and play games on!
(Even though this isn't related to pi, I should point out the
difference between a classic Hacker, and today's cracker, which many
idiots and newbies call 'hackers'. A classic Hacker was a computer
on legs. They almost thought in binary. They were the ones that
could code for 24 hours straight. They were the ones who drank Jolt
cola. (Not that there was actually Jolt back then, but....) These
guys were the source of phrases like: "It's a real hack" (meaning
it's a real beaut of a piece of code) and "It's a kludge" (meaning a
disgustingly ugly piece of code that works, but isn't 'elegant', it
just simply works for all the wrong reasons), and "It's a quick
hack" (meaning a piece of code that works but was thrown together
very quickly.) They were the ones that were the 'midnight Hackers',
back in the 70's when doors to mainframe computer rooms at
universities were left unlocked and some students took the
opportunity to literally spend all night exploring what a computer
was and how to do things. These were the guys who developed
networking, BBSs, virtual reality, and many of the things taken for
granted today. They learned for the sheer joy of learning and
pushing the boundaries of what was and was not possible. The
Hackers _were_ the computer revolution.
A cracker is somebody who has the goal of doing deliberate damage.
They may have similar skills, but a Hacker would generally only
cause damage by accident. Crackers are scum, and it irks me
immensely to hear them called 'hackers'.)
There were a few interesting things about Gosper's computation.
First, when he decided to use that particular formula, there was no
proof that it actually converged to pi! Ramanujan never gave the
math behind his work, and the Borweins had not yet been able to
prove it, because there was some very heavy math that needed to be
worked through. It appears that Ramanujan simply observed the
equations were converging to the 1103 in the formula, and then
_assumed_ it must actually be 1103. (Ramanujan was _not_ known for
rigor in his math, or for providing any proofs or intermediate math
in his formulas.) The math of the Borwein's proof was such that
after he had computed 10 million digits, and verified them against a
known calculation, his computation became part of the proof.
Basically it was like, if you have two integers differing by less
than one, then they have to be the same integer.
The second intersting thing is that he chose to use continued
fractions to do his calculations. Most calculations before and
since were done by direct calculation to the disired precision.
Before you did any calculations, you had to decide how many digits
you wanted, and later if you wanted more, you had to start over from
the beginning. By using continued fractions, and a novel coding
style, he was able to make his resumable. He could add more digits
any time he felt like it and had the spare time. This was a major
breakthrough at the time, because all previous efforts required
starting over from the beginning if you wanted more.
The third interesting thing about his calculations was the other
reason he chose to use infinite simple continued fractions. It is
still not known whether pi has any 'structure' or patterns to it.
It is known that it's irrational and transcedental, but it still
might have some pattern to it that would allow us more easily
calculate its digits. We just don't know. And patterns show up
more readily as a continued fraction rather than in some arbitrary
base that we humans call 'base 10'. As an example, 'e' and the
square root of two both have very simple, obvious patterns in their
continued fractions, even though they appear to be non-repeating in
base 10.
Chapter 7: The Chudnovskys
===========================
In the late 80's, the Chudnovsky's were using a symbolic math
package and derived a series that had some stunning benefits.
inf
___
< \ (6n)! (-1)^n (640320)^1.5
> {c1+n} * ------------------------ = --------------
<___/ (3n)! (n!)^3 (640320)^3n 6541681608*pi
n=0
With c1 = 13591409 / 545140134
It also includes a certain amount of self checking. If you truncate
the left half of the formula at 'N', then: For all primes P > 29,
the formula is evenly divisible by P (congruent to 0, modulo P) for
all N, with N < P <= 6N .
The advantages of this formula are:
First, it was resumable, like Gosper's was. You could add a few
more digits any time you had the spare computing time and power.
Although you did have to redo the calculations on the right side of
the formula.
Second, it was partionable. The calculations between each term were
independant enough that the work could be divided among many
computers and then combined into the final, whole, 'work' when they
all got done. So, instead of requiring a super computer, you could
just use some spare time on a LAN, which might have a few, or even
dozens of idle computers. You could even do it over the internet.
Assign a fixed range to compute among a thousand computers, and in a
few days (or whenever) when the results came in, just combine them,
and presto....
Third, the algorithm had a method of checking itself for errors.
This was the first method to have a way built into the calculations
to detect any errors made by the programming or the hardware.
Previously, the calculations had to be performed twice, with two
different programs and algorithms. That took considerable time.
This algorithm could be checked 'on the fly'. This self checking is
of critical importance, since both Gosper and Baily found errors in
their calculations that were caused by their computer hardware.
The disadvantages are: The intermediate numbers get fairly large.
The saved state takes up quite a bit of room. It takes about five
times as much intermediate working space than you get pi. After
you've "finished", you still have to do a division of a square root,
and a reciprocation, both of which have to be redone every time you
add more digits.
It's about the best you could want. About the only thing you might
want more would be: faster convergence (ie. more digits on each
iteration), a bit simpler math, smaller intermediate numbers, and
the ability to calculate the Nth decimal digit without having to
calculate all the preceeding ones first. And that latter one is not
likely to happen!
Chapter 8: Other methods
=========================
There are a vast number of formulas for pi. Some are an infinite
number of square roots. Some involve reciprocals of squares. Some
are based on various trigonometric identities. Others are based on
modular equations. Some are infinite series. Some are infinite
products. Some are infinite continued fractions.
There are even two based on the Fibonacci series! Yuri Matiyasevich
showed that:
6 log fcm(F[1]F[2]F[3]F[4]...F[n])
pi = lim û( ---------------------------------- )
n->ì log lcm(F[1]F[2]F[3]F[4]...F[n])
fcm is Formal Common Multiple, or simply the product of F[1], F[2],
etc.
lcm is Least Common Multiple.
F[1], F[2], F[3], etc. are the Fibonacci series. F[0]=0, F[1]=1,
F[2]=1, F[3]=2, F[4]=3, F[5]=5, F[6]=7, F[7]=12, etc.
The other formula is:
ì
pi=4 ä arctan(1/F[2n+1])
n=1
Again, this uses the Fibonacci numbers, and as n went from 1 to
infinity, you'd use the 2n+1'th Fibonacci number. Or, 2, 5, 12,
etc.
Of course, that formula is just a sophisticated way of saying:
arctan(1/F[2n+1])=arctan(1/F[2n])-arctan(1/F[2n+2])
and that of course is just another way of saying: pi/4=arctan(1/F2).
And since F[2] (the second Fibonacci number) is 1.... You end up
with pi/4=arctan(1).
And of course, since the 'Golden Mean/Ratio' is involved in the
Fibonacci numbers, it's also involved in pi! Each Fibonacci number
is integer part of the GM/R times the previous Fibonacci number.
You can also evaluate pi statistically, by randomlly dropping a
'needle' onto a ruled sheet of paper.
The formula:
inf
__
< \ 1 / 4 2 1 1 \
pi= > ---- | ---- - ---- - ---- - ---- |
<__/ 16^i \8i+1 8i+4 8i+5 8i+6 /
i=0
is actually capable of computing the i'th hexadecimal digit of pi
_without_ computing all of the previous digits. The total run time
for computing 0..i digits though is N^2, the same as the old Machin
/ Gregory series, so it isn't really practical.
You can use the distribution of bright stars across the sky to
approximate pi.
Chapter 9: Other related stuff.
===============================
If you graph y=1/x with the range of 1 to infinite, you get a fairly
typical curve. Its length is infinite, and the area under the curve
is infinite. If you rotate the curve through three dimensions,
getting a funnel, its surface area is still infinite, but its volume
is exactly pi cubic units! That means you can't paint the inside of
the funnel, because its area is infinite, but you can FILL IT!
Mathematics ought not be contridictory and I've often wondered
whether mathematicians should go back to the beginning of
mathematics and take a close look at everything.
There are many mnemonic phrases for remembering pi. Two common ones
are:
"How I want a drink, alcoholic of course, after the heavy leactures
involving quantum mechanics!"
"May I have a large container of coffee? Cream and sugar?"
There is also a poem modeled after Poe's The Raven that gives 740
digits.
Pi has been memorized to 42,000 digits.
Of course, the natural log, the trigonometric functions, and pi are
all intimatly linked together by:
e^(i*pi) = -1 and e^(ix) = cos x + i sin x
It is unknown whether pi is 'normal' in base 10. Normality means
that all the digits appear equally often.
Pi is an irrational number. This was proven by Lambert in 1771,
although even the somewhat dim witted Aristotle was vaguely aware of
the proof.
Although pi is not a rational number, it is not known whether it is
a near rational number. A rational number is a ratio of two
integers. With 'near rational' numbers, the numbers don't have to
be integers.
Although pi is a transcendental number, meaning it's not the root of
an algebraic equation with integer coefficients and postive
exponents, it's not known whether it is a root of an algebraic
equation with non integer coefficients and / or non positive
exponents.
We do not know whether pi+e or pi/e, or log pi are irrational or
transcendental.
We know little about its continued fraction, other the first 17.5
million terms computed by Bill Gosper.
It is not known whether pi eventually settles down into some
particular pattern of digits in any particular base. For example,
it's possible, although very unlikely, that it would fall into the
pattern: 3.1415926...010010001000010000010000001.... In this case,
it would not be a normal number, would still be irrational, and
still be a transcendental number, although it would be possible to
calculate any particular digit desired without computing any of the
previous digits.
It is not known whether there is any way to calculate the Nth digit
without calculating all the previous ones. If there is, it would
likely be limited to that particular base only. It would be
extremely unlikly that base would be our base 10.
It is not known whether the regular continued fraction of pi has any
particular pattern. If there is, that would indicate a pattern in
the number itself rather than in some particular base, since
continued fractions are base independant.
The first six digits of pi itself show up six times in the first 10
million digits. The first six digits of 'e' show up 9 times. The
first eight digits of the square root of two shows up at the
52,638th decimal. Is any of this relevant? Nobody knows. It is
statistically likely that things like this happen, but nobody can
prove that it is only pure chance.
There are four known 'pi forward' primes: 3, 31, 314159, and
31415926535897932384626433832795028841.
There are 7 known 'pi back' primes: 3, 13 51413, 951413, 2951413,
53562951413, and 97853562951413. There are no more through the
432nd decimal.
It is not known where there are any PiFor square numbers.
355/113 is the 'best' ratio for pi, because of the small size of the
numbers, and it is accurate to 6 decimals. (That rational is from
pi's regular continued fraction, and isn't something that was simply
'created'.) If you write it backwards and add one to the denominator
so you get 553/312, you get the square root of pi correct to four
decimals.
The square root of 10 is accurate to one decimal. Square root of
two plus the square root of three is accurate to two decimals. The
cube root of 31 is correct to 3 decimals. The square root of 9.87
is correct to a rounded four decimals. The square root of 146 times
13/50 is correct to a rounded six decimals.
Divide 2,143 (the first four counting numbers) by 22, then take the
square root twice. You get pi correct to eight decimals.
The first 144 decimals of pi add up to 666. And 144=(6+6)*(6+6).
And the three digits starting at the 666th position is 343, which
happens to be 7*7*7.
(I know that some of the above is more than a little odd, and well
into mysticism, but I figured I'd throw in a little of everything.)
The Great Pyramid of Egypt has a circumference to height ratio of
2pi. Of course, this is to be expected since it's likely they used
a wheel to make their horizontal measurements. Why a wheel? Well,
you've got to make measurements some how. A rope will stretch quite
a bit over that great of distance. Even a modern rope would
probably stretch half a meter. A steel tape measure can stretch
several centimeters. The papyrus rope that was used back then might
have stretched as much as 5-10 meters! A wheel on the other hand
will give you very similar measurements regardless of how many times
it's measured, or what the weather is, or how hot it is at that time
of day. Even today a calibrated wheel is used to measure tracks and
courses.
The height is 146.599m. One Egyptian cubit is 0.5235m. That means
the height is 146.599/0.5235m=280.03 cubits high. A very convenient
number (ie: a whole number). The width is 230.364m, which is 440.04
cubits wide. Not a very convenient number. But if you make your
width measurments with a wheel that is one cubit in diameter, it
will have a circumference of 1.6446m. And a width of 140.07
revolutions of a cubit diameter wheel. And the number 140 just
happens to be half the height of the pyrmaid is in cubits.
In spite of some people prefering mysticism, super science on the
Egyptians part, Atlantis, aliens from space, etc., the numbers fit
and the process is possible, and simple. And since it solves the
problem, and is practical, and implementable, and is simpler than
other explanations, Science requires it to be accepted until further
notice.
There is some variation in the measurements. There are many
measurements for the four widths, and the height has to be estimated
because it's missing the apex, and all four sides are different
sizes anyway. But, if you use an Egyptian cubit, the numbers fit so
incredibly close that it simply can't be dismissed.
Everybody has heard about 'squaring the circle'. As many can tell
you, it's not too hard to square the circle, but under the 'rules of
the game', it had to be done under Euclidean geometry, which only
allowed a straight edge, a compass, and a finite number of steps.
Basically, what it all boiled down to, was they were trying to prove
whether pi was transcendental! In other words, whether it could be
a root of an equation with integer coefficients and positive
exponents. And of course, it was proved in 1882 that pi is
transcendental, and therefore that you can't square the circle under
Euclidean geometry.
Apendix A: Pi time line
=========================
~2000 BC
Babylonians use pi=3 1/8
Egyptians use pi=(16/9)ý=3.16
~1200 BC
Chinese use pi=3.
~440 BC
Hippocrates of Chios squares the lune.
~434 BC
Anaxagoras attempts to square the circle.
~420 BC
Hippias discovers the quadratix.
~335 BC
Dinostratos uses the quadratix to square the circle.
~300 BC
Archimedes established 3 10/71 < pi < 3 1/7 and
pi=211875:67441= 3.1416. He also uses the archimedean spiral
to rectify the circle.
~225 BC
Appolonius improves the Archimedean value. Unknown to what
extent.
~200 AD
Ptolemy uses pi=377/120=3.14166..
~250 AD
Chung Hing uses pi=û10=3.16..
Wang Fau uses pi=142/45=3.1555.
263
Liu Hui uses pi=157/50=3.14.
450
Tsu Chung-Chi establishes 3.1415926 < pi < 3.1415927.
~500
Aryabhatta uses pi=62832/2000=3.1416.
~550
Brahmagupta uses pi=û10=3.16..
1220
Leonardo of Pisa (Fibonacci) finds pi=3.141818.
<1436
Al-Kashi of Samarkand calculates pi to 14 places.
1573
Valentinus Otho finds pi=355/133=3.1415929.
1583
Simon Duchesne finds pi=(39/22)ý=3.14256.
1593
Francois Viete finds pi as an infinite irrational product composed
of an infinite series of square roots.
Adriaen van Roomen finds pi to 15 decimal places.
1596
Ludolph van Ceulen calculates pi to 32 places, later to
35 places.
1621
Snellius refines the Archimdedean method.
1654
Huygens proves the validity of Snellius' refinement.
1655
Wallis finds the first infinite rational product for pi.
Brouncker converts Wallis' product into a continuedd fraction.
1665-1666
Newton discovers calculus and calculates pi to at least 16 decimal
places. Wasn't published until 1737 (posthumously).
1671
Gregory discovers his now classic arctangent series.
1674
Leibniz 'discovers' that arctangent(1) is pi/4 and is calculable
with the then new Gregory sieres.
1705
Sharp calculates pi to 72 decimal places
1706
Machin calculates pi to 100 places using:
pi/4=4arctan(1/5)-arctan(1/239)
His relationship is the first ever of this kind. It greatly
accelerates the convergence of the arctangent series, and it was the
slow convergence that made calculating arctan(1) impractical.
William Jones uses the now common pi symbol, as an abreviation for
'periphery' for the first time.
1719
De Lagny calculates pi to 127 places.
1747
Euler uses Jones' 'pi' symbol for the first time. His prestige and
'weight' in the mathematics world make it the standard symbol fo pi.
1748
Euler publishes the "Introductio in analysis infinitorum",
containing Euler's Theorem and many series for pi.
1755
Euler derives a very rapidly converging arctangent series.
Uses pi/4=5arctan(1/7)+2arctan(3/79) and his own arctangent series
and calculates 20 digits in less than an hour.
Develops the arctangent relationships, allowing an many arctangent
relationships to be developed. What Machin did that was so special
and revolutionary had now become mundane and trivial. (For the
reader's sake, I will omit many, many minor calculations and
formulas that resulted from Euler's discovery.)
1766 (1761?)
Lambert proves the irrationality of pi. This means that pi can not
be the ratio of two whole numbers (ie. a rational number with
integers.)
1775
Euler suggests that pi is transcendental. A transcendental number
is one that can not be the solution of an algebraic equation where
the coefficients are integers, and the exponents are positive
integers.
1794
Legendre proves the irrationality of pi and piý.
Vega calculates pi to 140 decimal places.
1840
Liouville proves the existence of transcendental numbers.
1844
Strassnitzky and Dase calculate pi to 200 places. Dase was a
calculating prodigy and did the calculations _in_his_head_! At
Gauss' suggestion, Dase was actually paid for his time and effort,
making this the first time that money was paid for 'computer' time.
1853
Shanks publishes his calculation to 530 digits. Verifies it to at
least 441 digits.
Shanks publishes his calculation to 607 digits, and giving the
details of all the calculations to 530 places. Due to an error,
only 527 digits were correct.
1855
Richter calculates pi to 500 decimal places.
1873
Hermite proves the transcendence of e (natrual log).
1873
Shanks publishes a paper containing 707 digits. He also attempts to
correct his errors from his 1853 book but blunders again, with
several digits in error starting at the 460th place.
His second paper that same year gives his final calculation, and
corrects all the previous errors, but a typographical error makes
the 326th digit incorrect.
1882
Lindemann proves the transcendence of pi. This also proves that you
can not square the circle using standard Euclidean geometry (ie. a
straight edge and compass and a finite number of steps.)
1897
Indiana (U.S.) state legislature almost passes a law declaring that
pi is equal to 16/5 (the value varies depending on the reference.)
They do it because the author of the 'discovery' offers to let them
use it for free, while other states would have had to pay royalties
on it. This may also be the source of a rumor about the government
doing it for religious reasons.
1945
Ferguson finds Shanks' calculation wrong starting at the 527th
place.
1946
Ferguson publishes 620 decimal places.
1947
Ferguson calculates 808 places using a desk calculator.
Wrench and Smith calculate 818 places. 710 places were officially
verified between Wrench / Smith and Ferguson.
1948
Wrench and Ferguson resolve descrepancy starting with the 723rd
digit, and finally produce a published value of 808 digits of pi
with guaranteed accruacty.
1949
Smith and Wrench resume their calculations and obtain 1,120 digits.
Before checking could be completed,....
ENIAC is programmed to compute 2,037 decimals. Takes 70 hours.
1954
Smith and Wrench continue to 1160 digits, of which 1157 agree with
ENIAC.
1954-1955
NORC is programmed to compute 3,089 decimals and does it in 13
minutes.
1957
A Pegasus computer computed 10,021 digits in 33 hours. Due to an
error, only 7480 were correct.
1958
An IBM 704 in Paris calculated 10,000 in 1 hour and 40 minutes.
(Only 40 seconds were required to reach Shanks' 707 digits.)
1959
Pegasus in England computes 10,000 digits.
IBM 704 in Paris computes 16,167 decimal places in 4.3 hours.
1961
IBM 7090 calculates 20,000 digits in 39 minutes.
Shanks and Wrench use an IBM 7090 calculates 100,265 digits in 8
hours, 16 (or 43) minutes
1966
Gilloud and Fillatoire, using an IBM 7030 (STRETCH) in Paris
compute, and checks 250,000 decimal places in a total of 45 hours
and 20 minutes.
1967
Gilloud and Dichampt use a CDC 6600 and compute 500,000 decimal
places in 44 hours and 45 minutes.
1968
Strassen discovers that multiplication of very large numbers can be
done using a Fast Fourier Transform.
1970
Strassen discovers how to speed up his method, avoiding 'complex'
numbers. More complicated than regular FFT though.
1972
Eugene Salamin discovers a faster, more efficient way to calculate
pi, using an Arithmetic-Geometric Mean. Actually rediscovers work
done by Gauss. Paper not formally published until 1976. This
method, and all later ones, depend on the multiplication of very
large numbers, and normally uses Strassen's FFT method. It was
mentioned in MIT's famous 'hakmem' memo of 1972.
1976
Gilloud and Bouyer use a CDC-7600 and compute 1,000,000 digits in 23
hours and 18 minutes.
1981
Miyoshi and Nakayama use a FACOM M-200 to reach 2,000,038 digits.
1982
Tamura and Kanada use a HITAC M-280H to reach 4,194,293 digits in 2
hours and 53 minutes. This is the first real use of Salamin's AGM
formula.
1983
Kanada used a Hitachi S-810 and calculated 16 million digits, with
10 million verified, in less than 30 hours.
1984
Chudnovsky's develop a new formula for pi. It has slower
convergence than other methods, and overall takes more effort for
the same number of digits, but their method is resumable,
partionable, and self checking.
1985
Yuri Matiyasevich discovers a connection between pi and the
Fibonacci numbers.
Bill Gosper uses a small workstation to calculate 17.5 million
digits using a continued fraction. He verifies 10 million digits of
it against Kanada's value. He strongly points out the need for some
way to automatically verify the calculations, since the history of
pi calculations is littered with errors. He also points out the
need for resumability in the calculations, since starting over from
the beginning so you can calculate more digits is somewhat self
defeating.
1986
David Bailey calculates and verifies 29 million digits, using the
first delivered Cray-2. He uses two formulas discovered by the
Borweins.
1987
Kanada calculated 134,217,000 on a NEC SX-2.
1989
Chudnovskys use a Cray 2 and an IBM 3090 and reach 480 million
digits.
Chudnovskys use an IBM-3090 and reach 1,011,196,691 verified digits
of pi.
1994(?)
Kanada computes 4,294,960,000 digits.
1995
Kanada computes 6,442,450,944 (3*2^31) digits. Upon checking, only
the last 6 digits are different, meaning an official record of
6,442,450,938 digits.
Bibliography
============
Baily, David: private communications, 1987
Brief discussion of his prime modulus transform, and obtained two
copies of his paper.
Baily, David: The Computation of pi to 29,360,000 decimal digits
using Borwein's Quartically convergent algorith. 1987. Obtained my
copy from author, it probably ended up in something such as
Mathamatics of Computation.
Gives a general overview of how he used a Cray-2 to calculate pi,
and a brief overview of statistical analysis. Might be worth
hunting for.
Ballantine, J. P.: The Best (?) Formula for computing pi to a
thousand places. American Mathematical Monthly, October, 1939,
pages 499-501
Builds a bit on Lehmer's previous paper, and points out a few ways
to improve the time of the hand calculations. If you intend to do
any arctangent calculations (by hand or computer), might be worth
getting.
Beckmann, Petr: A History of pi. 1974. St. Martin's press / Golem
Press.
An excellent general overview of the history of pi and mathematics
in general. The definative, most complete record of pi. It is a
little opinionated in places, and shows its age with the limited
computer references, mentions of the Soviet Union, and lack of
modern formula techniques, but well worth reading for the history
and entertainment. FIND IT! Gives quite a few references, but
you aren't likely to find them anymore.
Borwein, J. M. and P. B.: The Arithmetic-Geometric Mean and fast
computation of elementary functions. SIAM Review, Vol 26, No. 3,
July 1984. Pages 351-366
Talks a bit about the AGM and pi, etc. Gives 21 references. Not
really worth hunting down.
Borwein, J. M. and P. B.: An explicit Cubic Iteration for pi. BIT
26, 1986.
Gives a algorithm for pi that triples the number of correct digits
at each iteration. No real big deal. Copy it if you stumble on
it, but not worth worrying about.
Borwein, J. M. and P. B.: Pi and the AGM: A study in Analytic Number
theory and Computational Complexity. John Wiley, 1986.
A very mathematical presentation of pi, the arithmetic geometric
mean, Ramanujan, and a whole bunch of other stuff. Very deep math
in places! Obviously, it has a large number of references. Get
your library to get it via the interlibrary loan!
Borwein, J. M. and P. B.: Ramanujan And Pi. Scientific American.
Date unknown. Probably mid to late 80s.
Talks about Ramanujan, his methods for pi, gives a few formulas,
both his and theirs. If your library has back issues of SciAm, go
ahead and photocopy it!
Borwein, J. M. and P. B., and Bailey, David: Ramanujan, Modular
equations, and approximations to pi, OR, How to compute one billion
digits of pi. Date unknown. Probably 1986 or 1987. Source:
unknown. I obtained my copy directly from the Borwein's and don't
know where their paper finally appeared.
A very interesting paper! Basically a very simplified part of
the Borwein's book.. Gives several usefull formulas, has a
discussion about the math behind them. Talks about the
implementation issues. Gives 39 references, many of which are for
their mathematical work, rather than pi itself.
Chudnovsky, David and Gregory: Largest Supercomputers Battle Over
pi. Date and origin are unknown. I think mine is a partial
photocopy of a preliminary paper.
Talks about their new formula that is resumable, partionable, and
has built in self-checking ability. Also briefly mentions Gosper
and Bailey. If you can find anything about their method, it
almost certainly will be worth copying.
Gosper, Bill: private communications
Very brief discussion of his continued fraction method.
Knuth, Donald: Art of Computer Programming, Volumne 2:
Semi-numerical algorithms.
The definative work on things such as multiplying two large
numbers together, which is used in modern pi algorithms.
Kurosaka, Robert T: pi, e, and all that. Byte Magazine, September
1985. Pages 409-414.
Talks about pi, e, and related stuff. Shows how to make an
infinite funnel that has 'pi' cubit units of volume, but infinite
surface area. Talks about the 'Monte Carlo' method of pi
calculation. Points out the relationship between the Golden Mean,
and the Fibonacci series.
Lehmer, D. H.: On Arccotangent relations for pi. American
Mathematical Monthly, December 1938, pages 657-664
Gives an interesting overview of how to select the 'perfect'
arccotangent formula for hand calculation of pi. (Arccotan(x) is
the same as arctan(1/x).) Also introduces the 'measure' of the
formula's 'goodness'. Includes numerous arccotangent
relationships. If you plan on doing any arctangent calculations
(by hand or computer), it might be worth getting.
Matiyasevich, Yuri: A new formula for pi. 1986. my photocopy
doesn't say where it came from.
Mentions the two pi formulas involving the Fibonacci numbers.
Miel, George: An algorithm for the calculation of pi. American
Mathematical Montly, October, 1979, pages 694-697
Shows how to use the arctangent relationships to come up with your
own, custom, arctangent formula. If you plan on doing any
arctangent calculations (by hand or computer), it might be worth
getting.
Salamin, Eugene: Computation of pi using Arithmetic-Geometric Mean.
Mathematics of Computation, Vol 30, Num 135. July 1976, pages
565-570
Gives a very mathematical (naturally) presentation of his AGM
method, which was later found to have been discovered 150 years
previously by Gauss. It's been referenced so many times it's not
really 'new' anymore and worth hunting for, but if your library
has MathComp, you might as well photocopy it, and everything else
you can find.
Science News: Occasionally, this weekly science newspaper will have
bits about the latest pi happenings.
Wagon, Stan: Is pi Normal. The Mathematical Intellingencer, vol 7,
Num 3. Pages 65-67.
Briefly talks about pi's 'normality', transcendentalism, etc.
Gives Salamin's AGM formula. Worth copying if you stumble on it,
but not worth hunting for.
Wozniak, Stephen: The Impossible Dream: Computing 'e' to 116,000
places with a personal computer. Byte magainze, June 1981.
Discusses his effort to calculate 116,000 digits of 'e' (the
natural logarithm) on a 48k byte Apple II. Not really relevant to
pi, but interesting for his continued fraction method, and general
technique.
Wrench Jr., J. W.: The evolution of extended decimal approximations
to pi. The Mathematics Teacher, December 1960, pages 644-650.
Gives an interesting overview of the history of the arctangent
calculations. Gives numerous arctangent formulas, and talks about
his own personal experience in the calculation of 808, then 1120
places with a desk calculator before ENIAC. Also gives 55
references to the history of pi hunting. Unfortunately, most of
those are so old that you'll never find them, or they'll be in
some other language that you can't read. Definetly worth
photocopying if you can find it, and have an interest in the
history of pi calculations. Wrench later had a paper in
Mathematics of Computation.