This pi reference / history is placed into the public domain by  its
author, Carey Bloodworth.


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 finish 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)  (125) } is the 'squared' symbol.
(252)  (124) | is the 'nth' power symbol.
(251)  (123) { is the square root symbol
(247)  (119) w is the 'approximatly equal' symbol.
(246)  (118) v is the 'divide' symbol
(243)  (115) s is the 'less than or equal' symbol: <=
(242)  (114) r is the 'greater than or equal' symbol: >=
(241)  (113) q is the 'positive negative' symbol: +-
(240)  (112) p is the 'congruent' symbol.
(236)  (108) l is the 'infinity' symbol.
(228)  (100) d is the 'sumation' symbol.
(172)  ( 44) , is the 'one fourth' symbol.
(171)  ( 43) + is the 'one half' symbol.
(138)  ( 10)   is an 'e' with an accent.  Converts into a line feed.

Unfortunately,  charcter  (227),  the  'pi'  character  can _NOT_ be
entered into this text document, because that is used as the 'end of
line' character in the .QWK  message  packet  format that is used by
many BBSs.  If somebody tried to post it on a BBS, it'd screw up.


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.


My own personal pi computation  experience started in the early 80's
when I computed pi on  an  8  bit  micro  using  various  arctangent
formulas  in  assembly.   After  many years away from the subject, I
started again in 1996 on  the  PC.   I  wrote a nice portable arctan
program in C, and then a fairly simple  AGM  formula.   In  1998,  I
started  up  again,  and  eventually  reached 32 million digits on a
Cx486/66 in 36.5 hours, of  which  28  hours was computation and the
rest was disk I/O. (It is,  of  course,  considerably  faster  on  a
faster  computer.)  Also, my v2.3 program has reached a massive 512m
digits of pi on somebody else's computer.  My programs usually  tend
to  be  considerably better than the famous Japanese program SuperPi
(written by Kanada and friends), so  although I may not be the best,
I at least know enough about the subject to be able to write a  text
like this.

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  Vite  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.  Vite's was still based on the  Archimedian  polygon,
but it did point a new direction.

                       2
pi  =  -------------------------------------
        * ( + ) * ( + (+))...


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 + 1
           ---
            2  + 3
                 ---
                  2 + 5
                      ---
                       2 + 7
                           ---
                            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*3        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 (1/3).

pi       1         1      1       1
--  =   ---*( 1 - --- + ----- - ----- ... )
 6      3        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-tan  B    12

         2tan2B     120
tan 4B=---------- = ---
       1-tan 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:

1/1 + 1/2 + 1/3 + ... = pi/6
1/1 + 1/3 + 1/5 + ... = pi/8
1/1 - 1/2 + 1/3 - ... = pi/12

He also liked the idea that Machin had and developed:

arctan 1/p = arctan 1/(p+q) + arctan q/(p + 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 = (x)/(1+x)

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.

He also used and popularized Jones' abreviation symbol for pi.   The
same symbol we all now know and love.

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.  It's just a  quick  estimate  of  how
much work you'd have to do to use that formula.

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
unconsciously 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.  I myself
found cache problems  with  my  old  486  system.   And somebody who
computed 256m digits with my program discovered a problem  with  his
brand new Pentium-II mainboard, even though it appeared to work fine
with everything else.  Errors _do_ happen.


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]=1/2

Then iterate from 1 to 'n'.

A[n]=(A[n-1] + B[n-1])/2
B[n]=(A[n-1]*B[n-1])
C[n]=A[n] - B[n]

                      n
PI[n]=4A[n+1] / (1-(  (2^(j+1))*C[j]))
                     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!  (Of
course, the AGM's full precision operations are more complex, but it
still has a major, major advantage.)

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(n) 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.

Although  there  are  newer iterations, such as the Borwein formulas
below, the AGM is still the fastest, the easiest to  implement,  and
all around best iteration.



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]=6-42 and y[0]=2-1.

         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])

Then: 0  <  a[n]  -  1/pi  <  16*4*e^(-2*4*pi),  which  means a[n]
converges to 1/pi quartically (4 times.)

B)
Let s[0]=5*(5-2) and a[0]=1/2.

                 25
s[n+1] = --------------------
         (z + x/z +1) * s[n]

x=5/s[n]

y=(x-1)+7

z=((y+(y-4*x^3)))^(1/5)

                          s[n]-5
a[n+1]=s[n]*a[n] - 5 * (------- + (s[n]*(s[n]-2*s[n]+5)))
                             2

then 0 < a[n] - 1/pi < 16*5*e^(-5*pi), which means a[n]  converges
to 1/pi quintically (5 times.)


They also  mention,  and  independantly  derive,  an  old  Ramanujan
formula:

C)

 1      2n 3   42n+5
 - =   (  )  ---------
pi  n=0   n   2^(12n+4)


The also found a series that adds 25 digits for each iteration:

D)

        
        __   (-1)(6n)![212,175,710,91261+1,657145,277,365+
 1     <  \             n(13,773,980,892,67261+107578,229,802,750)]
-- = 12 >   ---------------------------------------------------------
pi     <__/     (n)!^3(3n)![5,280(236,674+30,30361)]^(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:

                
                __
  1      8    <  \  (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.  Hackers weren't interested in bothering
anybody, where as a  cracker  lives  for that sole reason.  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.  (Although that last one is
possible with  another  formula,  its  speed  is  so  slow  as to be
impractical.)


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    ( ----------------------------------  )
      n->        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:

     
pi=4  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.   (A  slightly  more
complex formula can compute other constants too.)

There is also a formula to compute the n'th digit of pi in any base,
including base 10, and not  just hexadecimal like the formula above.
This was published in a paper by Mr. Bailey, Mr. P. Borwein, and Mr.
Plouffe in the april  1997  issue  of  Mathematics  of  Computation.
Fabrice Bellard has an improved  version  on  his web page, but it's
still a bit too complex to show.  At best, it's still O(N^2) with  a
fairly    large    O().     (http://www-stud.enst.fr/~bellard/   and
http://www.lacim.uqam.ca/plouffe/Simon/articlepi.html)         Still
interesting though.

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  trivial  to
calculate  any particular digit desired without computing any of the
previous digits.

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.

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 (within a rotation or so)
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,  and  even  after  several
kilometers, two different size  wheel  measurements will often agree
within half a meter or so.

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.

(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.)


Apendix A:  Pi time line
=========================


~2000 BC
Babylonians use pi=3 1/8
Egyptians use pi=(16/9)=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 (ie: square) 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=10=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=10=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)=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 and calculates Ludolph Van
Ceulen's digits with far less effort.

1654
Huygens proves the validity of Snellius' refinement.

1655
Wallis finds the first infinite rational product for pi.

Brouncker converts Wallis' product into a continued 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 pi.

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.

(And since ENIAC didn't do a verification run, that  actually  makes
the 1,120 digits the verified record.)

1954
Smith and Wrench continue to  1160  digits, of which 1157 agree with
ENIAC.  And according to modern 'rules', this becomes  the  official
record.

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.

1973
Guilloud 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

Jean Guilloud reaches 2,000,050 digits.

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 reach 525,229,270 digits.

Kanada and Tamura reach 536,870,898 digits.

Chudnovskys use an IBM-3090 and reach 1,011,196,691 verified  digits
of pi.

Kanada and Tamura reach 1,073,741,799 digits.

1991
Chudnovskys reach 2,260,000,000 digits

1994
Chudnovskys reach 4,044,000,000 digits

1995
Kanada and Takahashi compute 3,221,225,466 digits

Kanada and Takahashi compute 4,294,967,286 digits

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.

Bailey,  Borwein,  and  Plouffe  comput the 40,000,000,000'th binary
digit of pi

1996
Fabrice Bellard computes the 200,000,000,000'th binary digit of pi.

Fabrice Bellard computes the 400,000,000,000'th binary digit of pi.

1997
Kanada and Takahashi compute 51,539,607,552 digits of pi.

Fabrice Bellard computes the 1,000,000,000,000'th binary digit of pi.

1999
Kanada  and  Takahashi   computes   68,719,470,000  decimal  digits.
Computer used was a (parallel) HITACHI SR8000.  32.9 hours.  296g of
main memory used.

Dominique Delande computes 1,073,741,824 digits (1,073,741,686  were
correct)  on a desktop computer.  A world record for a desktop.  The
program was  v2.3  of Carey Bloodworth's pi-agm program.  ~520m main
memory used, with 8g of disk.  Pentium-II @450mhz.



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,   Volume   2:
Semi-numerical algorithms.
  A   decent  reference  on  various  fast  multiplication  methods,
  although FFTs are highly system tunable.

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 or 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.


