Another Pi question

[BioHazard]

Has anyone tried using any pattern recognition or decryption algorithms on pi (3.14...)?

[UncleBill]

Newsweek (http://www.newsday.com/news/nationworld/world/wire/sns-ap-japan-pi-calculation1207dec06,0,6244373.story?coll=sns-ap-world-headlines) saysAmong the most puzzling mysteries: Mathematicians are pretty sure, but still cannot prove conclusively, that the numbers following 3.141592... occur randomly.So I'd certainly guess they've tried as many algoriths as they could find. The prize for figuring that there IS a pattern woud be a coronation as King of Geeks.

[UncleBill]

or algorithms.

[William_Ashbless]

Well, depends what you mean by pattern. There are now algorithms for producing the k'th hexadecimal digit of pi without knowing all the intervening digits. Sadly, it doesn't work for base 10, and as far as I know a base 10 equivalent has not been found.

Nevertheless, this constitutes a 'pattern' in the sense that you can work out what the next digit is.

In fact, most means of working out pi to any number of decimal places constitutes 'a pattern' in the sense that it is mechanically deterministic that you can work out further and further digits, regardless of numeric base.

So what is meant by 'pattern'? Something 'pretty'? (What is 'pretty'?) Something that 'repeats'?

[BioHazard]

Originally posted by William_Ashbless
Well, depends what you mean by pattern. There are now algorithms for producing the k'th hexadecimal digit of pi without knowing all the intervening digits. Sadly, it doesn't work for base 10, and as far as I know a base 10 equivalent has not been found.

Nevertheless, this constitutes a 'pattern' in the sense that you can work out what the next digit is.

In fact, most means of working out pi to any number of decimal places constitutes 'a pattern' in the sense that it is mechanically deterministic that you can work out further and further digits, regardless of numeric base.

So what is meant by 'pattern'? Something 'pretty'? (What is 'pretty'?) Something that 'repeats'?

How? Link/brief description please.


I was just wondering what would happen if someone used a code breaking algorithm on it, and if anyone has ever tried it.

[DeepField]

The numbers "following 3.141592..." do NOT occur randomly. Randomly means that nobody can tell which is coming in any particular position, and that is definitely not the case.

What is true is that the succession of numbers does not seem to follow any particular order, or pattern, and in that respect Pi is not unique: there are plenty (infinite) of numbers with that characteristic.

Numbers that have a "pattern" belong to the RATIONAL set of numbers (those that can be expressed as the ratio of to integers), and those that not are IRRATIONAL numbers (they can't). Both sets comprise the REALS.

[erislover]

The media's use of random seems to be here: are the digits in pi's expansion normally distributed?

[Payne N. Diaz]

Pi good. Like pi. More pi, please.









What?

[ultrafilter]

Originally posted by DeepField
The numbers "following 3.141592..." do NOT occur randomly. Randomly means that nobody can tell which is coming in any particular position, and that is definitely not the case.

What is true is that the succession of numbers does not seem to follow any particular order, or pattern, and in that respect Pi is not unique: there are plenty (infinite) of numbers with that characteristic.

Numbers that have a "pattern" belong to the RATIONAL set of numbers (those that can be expressed as the ratio of to integers), and those that not are IRRATIONAL numbers (they can't). Both sets comprise the REALS.

Except, of course, that p is irrational, and it has a pattern (just not a simple one).

There are various notions of having a pattern. There are rationals, and constructibles, and algebraics, and computables, and then some.

The current claim is that p is absolutely normal. This means that, in any base b, the number of times you see a particular sequence of length k will approach 1/bk.

So in the base 10 representation of p, as you keep looking at more digits, you would expect to see, say, 314 about 1 in every 100 sequences of three digits--assuming that p is absolutely normal.

Here's something interesting: we know that most real numbers are absolutely normal, but we don't know of any specific ones that are.

[sailor]

We have had this discussion several times and it boils down to semantics. Tell me what you mean by random and I'll tell you if pi is random.

Imagine this: I have a device, a random number generator, and it generates digits from 0 to 9 randomly, all with equal probability. We start the machine and start recording the digits: 2458623571245896321458795632153325687441256368421691587736919725483. . . . . We can say the series was random as it came out as we could not predict the next digit. But now that we have it, is it "random" any more? After all, we do know what digit comes after ech previous digit. Now it is not random any more. In the same way, pi is random in the measure that it is unknown.

[ultrafilter]

btw, when we say that some quantity is distributed "at random", that's generally taken to mean that it's uniformly distributed over some set.

[DrMatrix]

Originally posted by DeepField

Numbers that have a "pattern" belong to the RATIONAL set of numbers (those that can be expressed as the ratio of to integers), and those that not are IRRATIONAL numbers (they can't). Both sets comprise the REALS..12345678910111213141516. . . has a pattern, but it's not a rational number.

[Mangetout]

The idea is mentioned in Carl Sagan's novel Contact; the unknown, long-gone, all-powerful aliens who created the transit system also manipulated the universe in such a way as to leave a message encoded in the decimal digits of pi.

[Tuco]

Originally posted by erislover
The media's use of random seems to be here: are the digits in pi's expansion normally distributed?

I would hope they were uniformly distributed, not Normally.

[Jabba]

Ultrafilter declared:
Here's something interesting: we know that most real numbers are absolutely normal, but we don't know of any specific ones that are.
Indeed, we know that almost all real numbers are normal. However, we know that
0.12345678910111213.....
is normal, though the proof is harder than one might suspect.

[Hari Seldon]

One reason the Chudnovsky brothers gave for building a computer to find the first billion or so digits of pi was to see if any pattern emerged. So far, no one has seen any. And yes it seems to be normally distributed so the sequence 314 will occur once every 1000 (not 100) times you look at three consecutive digits. And a sequence of a 1000 consecutives 0s will occur about once in every 10^1000 consecutive digits. That said, it is not random in the formal sense since you can transmit as many digits as you like in a couple handsful of words ("Compute the first 1,000,000 digits of pi.")

Just for the record, rationals are characterized by having indefinitely repeating sequence of digits. Irrationals are the other kind. Quadratic irrationals are characterized by having indefinitely repeating continued fractions, which have the form
n + 1
--------------------------------------
m + 1
----------------------------
p + 1
----------------------
q + .
.
.
AFAIK, there is no such way of characterizing higher degree algebraic numbers.

[Hari Seldon]

AAgh. The editor completely screwed up my nice continued fraction. I give up, but what was posted made no sense.

[erislover]

Originally posted by Tuco
I would hope they were uniformly distributed, not Normally. Um... whoops!

[Napier]

I thought it odd that pi has many occurrences of aba where a and b are digits - in other words, a digit repeats with something else thrown in between.
3.14159265358979323846
1 then 5 then 9 then 3
or at least it looks that way in the beginning.
I asked Mathematica for Pi to maybe 100,000 places, then tested for this - and tested for other quantities of intervening digits besides 1. But no pattern - it's just that way in the beginning.

[BioHazard]

Mangetout: Thats what I was thinking when I posted this.


Originally posted by William_Ashbless

Well, depends what you mean by pattern. There are now algorithms for producing the k'th hexadecimal digit of pi without knowing all the intervening digits. Sadly, it doesn't work for base 10, and as far as I know a base 10 equivalent has not been found.


Once again, how? Link/brief description please.


BTW in my 12th grade math class I discovered a repeating pattern in pi. My teacher had pi to the 500th digit on the top of the walls, going all around the room. I forgot what I did, but it was a repeating pattern. I showed my teacher, she looked it over and said that I had found a pattern. Unfortunatly I discovered that the pattern was there because the posters were the mass produced kind and the teacher had put them up wrong with no overlapping, but they were supposed to be overlapped by one number on each side.

[ultrafilter]

Originally posted by Jabba
Indeed, we know that almost all real numbers are normal. However, we know that
0.12345678910111213.....
is normal, though the proof is harder than one might suspect.

I believe that we don't know that to be normal in every base, just base 10. But I could be wrong.

[BioHazard]

Whats the chance of the sequence: 020915080126011804 showing up in pi?

[ultrafilter]

Originally posted by Jabba
Indeed, we know that almost all real numbers are normal. However, we know that
0.12345678910111213.....
is normal, though the proof is harder than one might suspect.

I believe that we don't know that to be normal in every base, just base 10. But I could be wrong.

[Desmostylus]

Keep looking, BioHazard, your name's in there somewhere.

[Achernar]

Originally posted by BioHazard
Whats the chance of the sequence: 020915080126011804 showing up in pi?As I understand it, every sequence shows up in pi. Some people have asked if this can be used to compress messages - just tell someone where in the expression of pi your message is. However, it would be no good, because in general, the first time a sequence N shows up is later than the Nth digit. So the position indicator would be bigger than the original message itself.

[DarrenS]

Originally posted by Mangetout
The idea is mentioned in Carl Sagan's novel Contact; the unknown, long-gone, all-powerful aliens who created the transit system also manipulated the universe in such a way as to leave a message encoded in the decimal digits of pi.

That was nice of them - they left a message specially for us creatures with 10 fingers. While pi may be universal, and natural, the base 10 number system is arbitrary. Or did they encode a message for each possible number base? I know, it's only a book :)

Hehe, I just noticed that smilies still show even in a spoiler tag - and when you select the happy smiley it turns into a frown.

[jcgmoi]

I hear you BioHazard.

Here's a 1998 popular article on the Borwein-Plouffe Algorithm:
http://www.maa.org/mathland/mathtrek_3_2_98.html

A more technical discussion:
http://www-2.cs.cmu.edu/~adamchik/articles/pi/pi.htm

And another:
http://numbers.computation.free.fr/Constants/Algorithms/nthdigit.html

It is possible to compute the nth decimal digit of pi but not with the same efficiency as Bailey-Borwin-Plouffe.

Off topic, but pi has now been calculated to more than 1 trillion decimal digits; actually, 1,241,100,000 digits.
http://www.maa.org/mathland/mathtrek_3_2_98.html

[Shalmanese]

I think pi is random in the sense that it passes all known randomness tests.

[ftg]

Originally posted by Shalmanese
I think pi is random in the sense that it passes all known randomness tests.

One of the most widely used definitions of "random" is based on Kolomogorov complexity: The size of the program to generate the string. Since you can generate huge number of digits of Pi using a tiny program, Pi is most definitely not random using this test.

Virtually any Computer Scientist will describe Pi as very nonrandom.

[ultrafilter]

Originally posted by ftg
One of the most widely used definitions of "random" is based on Kolomogorov complexity: The size of the program to generate the string. Since you can generate huge number of digits of Pi using a tiny program, Pi is most definitely not random using this test.

Virtually any Computer Scientist will describe Pi as very nonrandom.

But p probably would pass statistical randomness tests, which is probably what Shalmanese had in mind.

[Mangetout]

Originally posted by DarrenS
That was nice of them - they left a message specially for us creatures with 10 fingers. While pi may be universal, and natural, the base 10 number system is arbitrary. Or did they encode a message for each possible number base? I know, it's only a book :) Good point; As far as I recall it's not specifically mentioned, but if they were advanced enough to actually bend the universe in such a way as to modify a constant, I think we could assume that they would have engineered a certain degree of universality in the message.

[Wikkit]

Originally posted by Hari Seldon

n + 1
-----------------------------
m + 1
------------------------
p + 1
-------------------
q + .
.
.

[knock knock]

Some people already made the point that "patterned" does not mean "rational," with the example 0.12345678910111213..... Even a transcendental number can be patterned. For instance,
Sk=1infinity 10-(k!) =0.1100010000000000000000010.... is transcendental (http://hades.ph.tn.tudelft.nl/Internal/PHServices/Documentation/MathWorld/math/math/l/l328.htm) but has a visible pattern.

Pi, as far as anybody knows, is random based on any statistical test or visible pattern in the digits, though it can be computed by fairly simple programs.

[BioHazard]

Originally posted by DarrenS
That was nice of them - they left a message specially for us creatures with 10 fingers. While pi may be universal, and natural, the base 10 number system is arbitrary. Or did they encode a message for each possible number base? I know, it's only a book :)

Hehe, I just noticed that smilies still show even in a spoiler tag - and when you select the happy smiley it turns into a frown.

This spoiler thing is really cool. Actually Im pretty sure that it was not base 10 that she was using as she searched through pi. I think it was base 12. Im not sure. So the proof mentioned in Contact was irrelevent because any sequence imaginable is in there? Terry pratchet wrote a book based around what Sagan was hinting at, a long time before Sagan did. The book called Strata.




A happy smiley :) when I select it turns into an evil smiley when I select it.

[Shalmanese]

Base 11 actually. Although I hardly see why somebody knowing this particular fact would ruin their reading of the story.

[BrandonR]

Let's say someone DID find a pattern to pi or some other universal "mystery" if you will... Most people say that person would be rich, but how? Who would pay him for that information? And how WOULD you get payed? Would you just walk up to Universities or companies and tell them you discovered ______ and they need to pay you to tell them?

[ozziemaland]

>Let's say someone DID find a pattern to pi

The New Yorker profiled the work of the Chudnovsky brothers, back in the early '90s IIRC, and the article stressed the wave-like patterns that were found when the repetitions of the digits were plotted graphically; much like waves of the oceans.

[Scott Dickerson]

1) The motive behind the OP may be: what if some sort of analysis allowed one to actually "decipher" the decimal expansion of Pi, in such a way that it was found to contain an endless number of messages, not only in all languages known but in all POSSIBLE languages, with a content that was, in crucial respects, mutually consistent? (Maybe this is the idea in the Sagan novel; I'm also reminded of a Star Trek TNG episode in which ancient aliens encoded a video greeting in the DNA structure of all galactic life-forms). If someone insists on an evidentiary "proof of the existence of God," it's hard to imagine a better one.

But that's just a speculation as to the OP's motive.

2) This distribution of prime numbers along the number line is also said to be "random" by our high school math teachers. But it turns out to have what I think is termed "order of the second degree." Imagine creating a string of beads: black beads for nonprimes, white beads for primes, each representing an integer and ordered from one to N (N has to be fairly large). Make the first bead the centerpoint, and wrap the string compactly about that point in an outward-going spiral--creating a disk of beads, so to speak. If you have good eyesight, you will now see that the white beads form a very obvious pattern, rather like a poppy. If I'm interpreting this correctly: you cannot predict where the next prime will be, but you can assign varying degrees of probability to its appearance.

My point being--I wonder if some version of the "bead test" would reveal a VISUAL form in the expansion of Pi. Would any of you folks care to perform this test and report the results here?

Now that my juices are flowing...start with the first digit of the expansion of Pi ("1") and pretend that as you go to the right, you are actually going left, in terms of decimal values. Ie, an unending string that begins 14159265358979323...etc. Now perform the bead test on that number (the rule is that you never have a lesser number after a greater, and you never jump to, for instance, three-digit numbers until you have exhausted the two-digit ones per the preceding clause). So the first few are:
1
4
15
92
653
5897
What will the bead test show regarding the distribution of primes in this series? Anything visually interesting?

[BioHazard]

I really like that idea, Scott. Someone try it? IANAProgrammer but I do think it would be relativly easy to make a program to do that. It would need to use the pi decimal finding algorithm as a base.

As for my OP, I was thinking more along the lines of Sagan's book, Contact, with the "evidence" being proof of something greater than us.

[ultrafilter]

Originally posted by Scott Dickerson
1) The motive behind the OP may be: what if some sort of analysis allowed one to actually "decipher" the decimal expansion of Pi, in such a way that it was found to contain an endless number of messages, not only in all languages known but in all POSSIBLE languages, with a content that was, in crucial respects, mutually consistent? (Maybe this is the idea in the Sagan novel; I'm also reminded of a Star Trek TNG episode in which ancient aliens encoded a video greeting in the DNA structure of all galactic life-forms). If someone insists on an evidentiary "proof of the existence of God," it's hard to imagine a better one.

But that's just a speculation as to the OP's motive.

If p is absolutely normal, then it all is in there at some point. If not, 1415 can be made to represent anything you like by means of a suitable decoder. ;)

2) This distribution of prime numbers along the number line is also said to be "random" by our high school math teachers. But it turns out to have what I think is termed "order of the second degree." Imagine creating a string of beads: black beads for nonprimes, white beads for primes, each representing an integer and ordered from one to N (N has to be fairly large). Make the first bead the centerpoint, and wrap the string compactly about that point in an outward-going spiral--creating a disk of beads, so to speak. If you have good eyesight, you will now see that the white beads form a very obvious pattern, rather like a poppy. If I'm interpreting this correctly: you cannot predict where the next prime will be, but you can assign varying degrees of probability to its appearance.

My point being--I wonder if some version of the "bead test" would reveal a VISUAL form in the expansion of Pi. Would any of you folks care to perform this test and report the results here?

Now that my juices are flowing...start with the first digit of the expansion of Pi ("1") and pretend that as you go to the right, you are actually going left, in terms of decimal values. Ie, an unending string that begins 14159265358979323...etc. Now perform the bead test on that number (the rule is that you never have a lesser number after a greater, and you never jump to, for instance, three-digit numbers until you have exhausted the two-digit ones per the preceding clause). So the first few are:
1
4
15
92
653
5897
What will the bead test show regarding the distribution of primes in this series? Anything visually interesting?

Again, if p is absolutely normal, no large-scale pattern will emerge. If not, who knows?

[murphydog]

Imagine the excitement, then disappointment that early mathamaticians had when they calcuated pi longhand and found
that decimal places 762 thru 767 were all "9". They were probably tearing thier hair out when the discovered that the place 768 was "8".

[bryanmcc]

Originally posted by murphydog
Imagine the excitement, then disappointment that early mathamaticians had when they calcuated pi longhand and found
that decimal places 762 thru 767 were all "9". They were probably tearing thier hair out when the discovered that the place 768 was "8".

Murphydog, according to this (http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pi_chronology.html) site, that many digits of pi weren't calculated until 1947, and by then it was done with a computer. The most digits calculated by hand was 620 in 1946 (apparently by the same guy).

-b

[Scott Dickerson]

Bryanmcc, betcha ol' Murphydog was making a joke!

[bryanmcc]

Whoosh.

[murphydog]

No joke. I have a QB program that will calculate pi to any number of decimal places. It's free but it's a dos program. You need
QBasic to run it. Request it (ronfrancis@att.net) if anybody wants it.

[Shalmanese]

Originally posted by bryanmcc
Murphydog, according to this (http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pi_chronology.html) site, that many digits of pi weren't calculated until 1947, and by then it was done with a computer. The most digits calculated by hand was 620 in 1946 (apparently by the same guy).

-b

Yes, he calculated it by hand 3 times, got different answers each time and it turned out his first answer was the closest anyway.

[bryanmcc]

Originally posted by Shalmanese
Yes, he calculated it by hand 3 times, got different answers each time and it turned out his first answer was the closest anyway.

Talk about someone with too much time on their hands.

-b