(PhysOrg.com) -- Shigeru Kondo is a seriously committed guy. Ever since discovering he had an interest in calculating pi (aka π) back in his college days, he’s been following the results achieved by others using massive supercomputers. Now, in his late 50's, with some help from Northwestern University grad school student Alexander Yee, he’s succeeded in calculating pi to ten trillion digits; on a home built PC yet.

Pi, the mathematical constant that describes the ratio of a circle’s circumference to its diameter, is generally rounded off to just two places, bringing it to 3.14. Believed to have been first described by Archimedes way back in the 3rd century BC, the ratio is used in all sorts of mathematical computations, not the least of which is in figuring out the area of a circle. But because pi is an irrational number, it’s value cannot be written as an fraction which means when written as a decimal approximation, it’s numbers go on infinitely, and perhaps more importantly, never repeat.

For hundreds of years, pi has held fascination for mathematicians, scientists, philosophers and even regular run of the mill people. Why this is so is hard to say, and so too is the seemingly endless progression of people that have set before themselves the task of calculating its digits. In spite of that, it’s possible that none has ever been so obsessed as Kondo. He’s spent the better part of a year with the singular task of finding the ten trillionth digit, and of course all those past the five trillionth and one digit leading up to the ten trillionth, since he found the five trillionth digit just last year.

Finding the value of pi to 10 trillion digits requires performing a lot of calculations (using software written by Yee), so many in fact, that Kondo had to add a lot more hard drive space than you’d find on your average PC. Forty eight terabytes to be exact. So intense was the computation that the computer alone caused the temperature in the room to hold steady at 104° F.

Also, it’s not as easy to keep a custom built super-sized PC going full steam ahead twenty four hours day for months on end, as it might seem. Hard drive failures and the threat of power disruption from the earthquake in Japan back in March threatened the project many times. And of course there was that power bill itself which ran to something close to $400 a month as the computer ground away.

But in the end, it was Kondo’s persistence that paid off. For his efforts he will be forever known (in the annals of science, and probably the Guinness Book of World Records) as the man who calculated the ten trillionth digit of pi. It’s 5.

**Explore further:**
Supercomputers crack sixty-trillionth binary digit of Pi-squared

**More information:**
www.numberworld.org/misc_runs/pi-10t/details.html

ja0hxv.calico.jp/pai/estart.html

*10/21/2011: The story has been updated.*

## NameIsNotNick

No it's not, it's 4. Better run that calculation again!

## NameIsNotNick

## Jaeherys

## NameIsNotNick

I suppose it's a hobby. It would be interesting if the sequence did eventually repeat... Why pi? because it's a very important irrational constant.

## Jaeherys

## Isaacsname

I wonder if he managed to find anything like Feynman's points ?

## VitalStatistic63

## antialias_physorg

Statistical analyses of the digits (and subsequences) in pi might be interesting. Is there a 'bias' to math (which might hint at an artificial universe)? If you want to go all strange one might even look at unusual clusters of digits (i.e. 'hidden messages' like in "Contact").

Ther might also be some value for crypto-algorithms based on transcendental numbers like pi or e.

## Jaeherys

Now *those* are some interesting points! Could it also be possible there are a finite number of digits of pi? I know it is supposed to be infinite but nonetheless is it even possible?

If it somehow did turn out to be finite what would that mean for math?

## LVT

## F111F

## CHollman82

It's an enigma, people doubt that it actually doesn't end or repeat...

## antialias_physorg

Pi has been proven to be transcendental by the Lindeman-Weierstrass theorem (no, don't ask me how. I just went to the wikipedia entry and didn't take the time to go through the math - and doubt I'd get very far if I tried).

All transcendental numbers are irrational - which means they cannot be expressed as a ratio of a/b (where a and b are integers). Any number with finite digits is rational (as well as some with infinite digits). Pi, being an irrational number, therefore must have an infinite number of digits.

One interesting aspect:

It is not entirely known whether the distribution of digits (in differing number systems) of pi is normal (i.e. whether all digits occur with the same frequency or not). For base 2 this does seem to be the case.

## El_Nose

-- this story is a few days old -- try slashdot.com sometime where this was addressed -- i am basically repeating their answer.

It helps set benchmarks for super computers. You calculate all of and including the nth digit of pi, then when the next generation of supercomputers comes out that can do this calculation in a few days they run the test to verify that everything seems to be in order.

## tjwied

## LVT

So 8slice*r = circumference

## SincerelyTwo

He just gave you the proof by exhaustion ... not exactly my favorite method, if I had a favorite method, but yeah. :)

## Parsec

These 2 concepts have nothing to do with each other.

## sherriffwoody

## albenza

Hope he is happy with the result though. (Now use that power on something productive. :)

## VitalStatistic63

Not necessarily. I have had a passing interest in pi since high school. The formula given to us then was

pi/2 = (2n/2n-1 x 2n/2n 1) ... (for n=1:infinity)

so = (2/1 x 2/3) x (4/3 x 4/5) x (...)

If you write up a quick program or spreadsheet with this formula you can readily see that each extra digit of precision takes significantly longer than the previous one.

There are a few methods that I know of to calculate pi and probably many more I don't know, so theirs is probably way more efficient than mine. But I'll bet it still runs like molasses when you get up to the trillionth digit.

## antialias_physorg

...says the man whose CPU sits 99% idle most of the time.

## Silverhill

Now all we need is God's Own Computer to evaluate the digits, and to search the strings for accidental (or deliberate?!) meanings....

## YummyFur

nice.

## gwrede

## RayInLv

Big number..... But that other number is 10^-35th quite a bit smaller.

## sherriffwoody

I relise this, I've written a program to calculate pi to n myself and the difference on my laptop between 5000 decimal places and 10000 decimal places is larger double. But a few days to five trillion compared to months and months to get 10 trillion?

## Grizzled

PROVE you wrong??? Hmm, you may be interested to know that you HAVE been proved wrong ... just about the time of Archimedes at least. Probably even earlier. Time to catch up man :-)

## Grizzled

Those numbers are not only irrational (meaning they can't be represented as n/m fraction for any integer n and m values), they are also transcendental, meaning that they cannot be the root of any algebraic equation (polinomial).

The last statement is a LOT stronger than the first - all transcendental numbers are of course irrational. The reverse isn't true.

## Grizzled

Nah. Not after you added the last two words in your challenge. You see, you can't have a proof of something that isn't so.

## altino

When infinity is taken out of infinity,

only infinity is left over.

## Grizzled

Stop to think for a moment - suppose you've designed and built the latest and greatest ... how do you know it works? Not just works but works correctly? Even when pushed to the exremes of its capabilities?

Well, one way to try it is to desing (using it) an all-new thermonuclear reactor and see if it runs as expected...or not. A much cheaper alternative to examining a multi-kilometer crater might be checking how well it calculates the tenths (or hundredth) digit of pi.

Don't you agree?

## altino

Al = (14 ROOT 2) ÷ 4

= 3.1464466.....

:)

Comment on this, please.

## Humpty

## _etabeta_

## Grizzled

And what do the previous 1 trillion say? Just curious. You know, even if you limit yourself to 16 bits (how's that for antiquity?) you can still pack about 3 characters per two digits if you omit some less popular ones.

That makes it roughly 1.5 trillion characters you have just discovered. Wow. No, on second thoughts it's WOW!!! Do tell us - what does all that wisdom say?

## LongPurple

"If you want to go all strange one might even look at unusual clusters of digits (i.e. 'hidden messages' like in "Contact").

I'm wondering if there was any search for something like those pages upon pages of 1's and 0's in "Contact". I assume no pattern recognition was part of the project, just straight calculation.

## plasticpower

Thank you. That actually makes a lot of sense.

## spaceagesoup

The pi ratio is a powerful and prevalent natural institution. The better we understand it, the more we (probably) will find uses in the physical world around us.

And working on pi is a damn site better than all the non-contributors who whinge daily on this site, or the scores of others whose more intellectual pursuits amount to nuking their microwave dinners, flicking on the TV and screaming at the kids.

## antialias_physorg

Yes, but one can determine how likely it is that some sequence occurs.

Example: The sequence 9999 is to be expected to occur once within the first 10000 digits of pi (in base 10). but if it occurs 50 times in the frst 10000 digits instead of just once then the statistics are skewed to the point where there is only a very small likelyhood that this is a random fluke (though that likelyhood never reaches zero. We must always expect some false positives). If some 'complex message' (e.g. an inordinately long sequence of zeroes and ones) occurs way earlier than expected then we should take a look.

Though I really don't expect any such message to be in there.

## Ricochet

## antialias_physorg

Depends on what kind of patterns you're looking for. Simply looking for consecutive numbers that are 'unlikely' is pretty easy to do and would only take a couple of hours. Though depending on where you set your threshold for 'unlikely' and how many types of patterns you're looking for the amount of time needed will scale up.

E.g. a simple check for long swathes of unlikely ones and zeroes would take a couple of minutes.

## Skultch

Why? The perfectness of a circle can be described as its level of detail; its resolution, no? How perfect can a circle be could also be related to the question: how small can objects be? Right? Or does geometry lose its meaning at sub-atomic volumes?

## antialias_physorg

One is an abstract notion (pi) and the other is a physical approximation (real circles).

There are many uses of pi which have nothing to do with circles ( e.g. in number theory, analysis, ... )

The 'needed' accuracy is a matter of what you want to do with it. For certain number crunching tasks the more digits you have the better.

## Skultch

## Ricochet

## YummyFur

## Deesky

I think you're too hung up on physical measurement of physically drawn circles. All circles are 'perfect', as is any other geometric shape as defined mathematically. There is no need to equate a mathematical concept with the physical world constrained by the limits of measurement.

## antialias_physorg

Yes we would (if the numbers are truly random - something not yet proven!)...though the likelyhood that these types of lists/works of art occur at digit numbers far, far, FAR exceeding the number of atoms in the universe is extremely high.

Or conversely: Finding any one of these in a list of digits - with each atom in the universe used to store one digit - is incredibly unlikley.

## Skultch

(weird; Pi is in the formula for Planck time, but not length hmmm)

## Jeddy_Mctedder

## Lazernugget

Really? How? You must have a lot of computer space. lol.

## antialias_physorg

Circles are an abstract mathematical concept. It is the set of all points equidistant from a geometrical point. What does this have to do with particles?

## Grizzled

How's that for predictive power?

## S_Bilderback

## S_Bilderback

Remember, infinity isn't empirical or a number - it's a human perception of the absents of a limit.

## Skultch

Circles have something to do with particles when we put particles in the shape of a circle. Get your head out of the textbook. It's called a thought experiment, and you aren't even trying.

## Skultch

## Grizzled

P.S. And yes, you are coming across as exceedingly aggressive. That's not a good sign in a suposedly scientific discussion. Not even as a thought experiment :-)

## Grizzled

Not all bases are created equal. If you delve into the fundamentals of math.analysis, you will discover that it depends on the definition of numerical axis whic in turn depends on the properties of the... rational numbers! If you try to redefine everything starting from scratch and use irrational base... Ummm, the very first question becomes - irrational in what sense? You haven't even defined what "irrational" means yet. Never mind any of its critically important properties. Like what kind of a set do those numbers form - is it a ring for instance? Speaking of irrational universe in this context is a little bit immature.... unless of course you meant it as a hint on what you think of the Universe at large :-)

## Skultch

I understand what an abstract math concept is. (Calc-1 was stupidly easy for me, albeit long ago, which is why I'm asking you guys this) What you are really saying, without actually saying it, is that perfection is unattainable. My question goes right to the heart of *why* a circle can only be abstract. The only reason that a material circle cannot be achieved is practical. I'm trying to transcend math here; to attempt to link it to the material.

There. It's exceedingly frustrating when people just echo a definition of a concept without supporting it with an explanation of "why," like what you guys are doing with your circle definition. Just repeating it does no one any good.

## Skultch

why? please.

Supposedly being the key word there. We aren't actually having a scientific discussion, or any kind of discussion. Yet.

## Deesky

Einstein would say no. Quantum Mechanics would beg to differ.

I doubt that is possible.

But seriously, you've both asked and answered your own question but were dissatisfied by the answer so you ask the question again.

You confuse the issue by bringing up the mathematical concept of a perfect circle with observed reality. Your basic question appears to be as in the first quote of this post. And the answer is, as I've alluded to, that no one knows for sure.

There are arguments for both views, but it seems that the QM view is likely correct (ie, spacetime is discrete/granular at the finest level).

## Blakut

Yes you can. You'll get a slightly bigger circle, keeping the distance between the particles the same. You can have circles of all sizes, provided you have enough particles.

## 137

## Skultch

Blakut, I should have said that we couldn't increase the diameter for that scenario, which I realize is practically ridiculous. There's no way they could be that close to each other and not interact.

I guess I should be asking: 1.) if Pi is (or could be) a fundamental constant needed to describe reality? and 2.) then how accurate could we theoretically get before any further accuracy is pointless?

## thefurlong

That isn't necessarily true. Nobody has actually proven PI to have this property. It is a popular misconception.

A variable can be random without having a normal distribution. In fact, the distribution can be quite irregular, and some sequences can completely preclude specific sequences. As a trivial example, consider a rational number whose digits only include 1 and 0, but the sequence is otherwise unpredictable (random). Then, obviously, the sequence 222 will never appear.

## Deesky

While pi is a mathematical constant, it is not a physical constant and therefore is no more or less important than any other mathematical constant in describing reality.

I think we've reached that point long ago. No one uses these ridiculously long sequences of computed digits in any practical way to solve everyday or even non-everyday problems. I would say that the interest in increasing accuracy is of some esoteric interest in certain fields of mathematics (and with computer geeks!), but that's about it.

## albenza

Nope. It's working for the LHC when not used for compiling or encoding. ;)

## hush1

So long the unprovable properties of the arbitrarily defined remain arbitrary, so long pi will remain transcendental.

## hush1

In the broadest numerical sense pi comes from all indistinguishable points. Points with labels are lines or distinguishable.