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A Much Bigger Piece Of Pi
Punk_Rock_Johnny points to an AP story on Pi-obsessed Professor Yasumasa Kanada. A snippet from the story: "Kanada and a team of researchers set a new world record by calculating the value of pi to 1.24 trillion places, project team member Makoto Kudo said yesterday. The previous record, set by Kanada in 1999, was 206.158 billion places." Trillion!
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The number Six!
You can write Pi as 1 (base Pi).
Here's how it works. You'll need several boxes of toothpicks. Get a large piece of chart paper, and draw parallel lines on it, from one side to the other. The lines should be separated by a distance just slightly larger than the length of a toothpick.
From a height of about one metre, drop a measured number of toothpicks onto the chart paper, so that they all fall randomly somewhere on the paper. Count how many toothpicks are touching a line (or would be, if they weren't resting on another toothpick).
Repeat this process as many times as you can. Lots of people can do it at once. All that's important is that, each time you drop some toothpicks, you write down how many you dropped, and how many of those ended up touching a line. When you're done, find a total for each quantity.
You now have all the numbers you need to calculate Pi:
Now here's the formula you need to calculate Pi:
Fill them in the formula, and work out your own value of Pi!In the book version of Contact by Carl Sagan, but skipped in the Jodie Foster movie, was the notion that the aliens had discovered proof that the universe was created by a higher intelligence. A God or society of Gods far higher and more advanced than the aliens. The whole point of dragging Human-kind to that remote beach to talk with daddy was to tell Human-kind that it was time for them to look for God's signature on this universe.
As any artist, the creator signed the creation. Where? Deep into the insignificant but irrefutably valid digits of several of the fundamental mathematical constants such as pi and e.
The main character finds one of the signatures at the end of the book: if calculating digits of pi in base 11, after a few million or billion places, a 500x500 digit span is almost entirely zeros. If the span was rendered as a square of pixels, the non-zero digits drew a perfect circle inscribed in the square. A circle in a square. The key concept defining pi, in the digits of pi itself. The whole way the universe works is affected by that constant, so any such 'design' in it has, if you pardon the pun, a transcendental import.
Why base 11? It's left to the reader to decide, but I expect Sagan wrote it because it is considered one of the possible designs of the universe, one of the string theories is based on an 11-dimensional all-inclusive physics model. As the alien explains to the main character, it wouldn't be base 10, because what's the likelihood that the creator also happened to have ten fingers?
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Um, you have 1.24 trillion digits of pi. I think you can begin a statisticall analisys now.
If Mr. Edison had thought smarter he wouldn't sweat as much. --Nikola Tesla
42 really is the answer to life, the universe, and everything!!!
Pi is worse than irrational - it's trascendental. Merely irrational numbers can be expressed as simple expressions with finite numbers of terms, but transcendentals require an infinite number of terms.
Pi is represented usually by a fraction or relatively simple equation, it's just the division that makes the number go on for ever.
... but these are just approximations; 22/7 is a good enough approximation a lot of the time, but that's just an approximation too)
Nope. If pi was rational (a fraction), it wouldn't go on for ever without repeating. (reference)
In fact pi is irrational, i.e. there are no integers p, q such that pi = p / q. (proof)
You can approximate pi as a fraction, which is what projects like this do. (pi is approximately equal to 31/10, or 314/100, or 31416/1000, or
Here's the magic
You have a 1.24 trillion digit base ten number
10^1.24e12
Now we find out how many digits long it'll be in base 2, x
10^1.24e12 = 2^x
x = ln(10^1.24e12)/ln(2)
x = 1.24e12 * ln(10)/ln(2) = 4119190837660.6
Now divide by 8 to get bytes, and viola!
515e9
sorry, but in base pi, pi would be written as 10.
(fyi, i made the same mistake back in the day also)
make world, not war
Kanada and a team of researchers
MPAA forces have today invaded Canada, when asked their reasons they replied:
"While we were looking through through the binary version of Pi, and one of our special forces noticed that hidden in from digit 12,166,133,883 onwards was a c source to DeCSS. Obviously these terrorists must be stopped!"
When pointing out that it was Kanada, the researcher, and not Canada the country, the Canadian government sued for trademark violation.
The case is not expected to hold up, as it is doubtful canada will be able to proove it has the computing power to calculate Pi beyond 4 decimal places - and no confusion can occur.
3.14159265358979323846264338327950288...... ah, this is gonna take a while :-/
A google only has 100 zeros, thus 100 places.
:-P)
:-)
10^trillion is 1 followed by 1 trillions 0's... Assuming we are following the american system that would be equivalent to.
10^(10^12)
Okay... now.. let's get some interesting facts with this.
The absolutely smallest length measurable by quantum theory is the planck length which is approx 10^-34 m. Needless to say, if we have a diameter of an incredibly small perfect circle, we'll know it's circumference beyond what is possible by quantum theory (but since there are no perfect circles, and quantum theory adds probability, this doesn't mean anything really useful.
Now, since we know the smallest measurable... lets look at what the estimates for the size of the universe are. Recent estimates put it as 10 billion light years in radius source
Which works out to about... (assuming american notation on billion)
10^9 * 300,000,000 m/s* 365*24*3600 ~= 10^25 m
Okay... now if we were to measure the circumference to as accurate as allowed by quantum theory we'd have.
pi*2.10^25 ~= 6.28*10^26 10^27 with an accuracy of about 34 decimals...
So... to get perfect accuracy as allowed by quantum theory we would have at most 35 decimal places afterwards... therefore, we'd need pi with an accuracy of
~10^63...
We have pi with an accuracy of 10^(10^12) which is
63 : 10^12 ~= 1: 1.59x10^11
Way more accuracy then we really need.
That's absolutely insane, but it is fun math.
Just some food for thought.
~ kjrose
I can't believe you report this and don't even include the value of Pi he calculated in the article!
I guess I'll have to wait for one of the page widening trolls to post it.
Why?
Well, if you read the article, you would know why. Mapping out a very large number like that is useful for testing the accuracy of supercomputers. Also, the research process spins off lots of discoveries. Someone who mapped out pi to 1.24 trillion decimal places probably learned a couple neat tricks along the way.
Comment removed based on user account deletion
Pi, like everything else, compresses down to one bit, given the correct decompression algorithm. (It is generally nonsense to talk about how well something compresses without specifying something about the algorithm you mean to use.)
Usually, "X compresses down to one bit for a correct algorithm" is a snarky answer, but in this case, it actually makes sense. Generally one has to define those algorithms as a table, where "X" is what the decompression function returns for "1", which definately feels like cheating. In this case, though, one can provide a finite algorithm to compute as many digits of pi as you please, so it makes sense.
In fact, we compress pi down to one or two bytes, with a mathematically defined decompression sequence you can use if you want, all the time. In fact, I've done it three times in this post already, where two different two byte sequences stood in for the infinite series that is that number. Can you find them?
You, Sir, despite your low member number, would get an F- for information theory at the university I was tought and now teach.
There is nothing that compresses to one bit. There is such thing as a most efficient way of encoding any message. Counted in bits. and no, not just one bit. One bit would just contain enough information to say "Pi" or "Not Pi". "Not Pi" would according to my intuition not be an acceptable answer, you also have to say "What kind of 'Not Pi'". And that takes bits. You forgot that your algorithm is supposed to possibly generate all possible messages, or else it's "not fair".
Pi would not compress at all, given it's an infinitely long number. (To be precise, it's length would be reduced from inf to inf/(alphabet entropy) which is still inf, although a "smaller" inf). If you are content with a finite number of digits, its length would be reduced by about a little more than three bits per decimal (because log2(10)=3.???) with any decent entropy encoder. You could try to reduce this further by taking two decimal digits at once, but unfortunately it would not work, as not only are Pi's digits uniformly distributed from 0 to 9, pairs of digits are also distributed uniformly from 0-99, so you would remain with 6.???? bits (log2(100)) per decimal digits pair.
Another approach you might take, if you want infinite precision (silly on a finite machine), or more generally random precision, is to write a code in a predetermined programming language, in this case a series developement, or whatever the number thorists use nowadays to calculate pi, and decide that the "decompression algorithm" is a compiler (that is perfectly legal, as any finite message can be passed that way, eg "#include <iostream> int main(){cout << "The message";}").
My idea is that the c compression algorithm would be beat by a perl compression. Maybe try in BrainFuck, it might beat perl, but BF sucks at multiplications.
Anyway, the most optimal compression for pi is probably saying "Pi" by itself. Any decent geek knows at least one way to calculate that/ find it on project gutenberg/whatever. But don't ever think that you could compress it to two bytes or less : you gotta be sure that I will not understand "the string of decimal digits a.k.a. Pi, do write it in numbers when decompressing", not just "mu turned over", "Pi the string" or "Private investigator". This certainty takes bytes.
Another example is : "you cannot encode '3 4 8 15 3.141592653 78 54' as '3 4 8 15 pi 78 54', because that would increase the number of symbols in the alphabet, and all the other symbols would have to contain more bits as a result, so the compressed message length would suffer- hope there are a lot of 'pi' in the compresed message".
I must leave now, gotta go bowling with friends. Start your flames, I can see blatant holes in my reasonments. Hope you get the point. Mailing a link to the message to my signal theory professor (formally one of my bosses), so I will suffer if I told bullshit.