Ok, so there ARE patterns, but to be analogous to what I was trying to convey, or at least my thought on it. ... The base numbering systems that we try to throw at pi are all flat or 2d representations (conversions from one base to another basically)
...When/if we ever hit on the right formula to 'unravel' pi, it will be like tuning in a crystal clear radio station and there will absolutely be no doubt to the solution. Until then ... well...
Beau, absolutely, you're right on it! There are 3D plots of Pi in various numbering systems and I do see things within. Take a look for yourself at the Function.Wolfram.Com site. Here's a small sample of images.
Pi is one of several trancendental numbers. Its real computational value is in allowing us to convert a circular quantity into a more standard reference of straight lines, square inches, or cubic inches and so on. The issue of such absurd accuracy breaks down if you take a fractal view of the world.
Nonetheless, I am deeply interested in trancendental numbers - especially the ones that are more ignored and less used -- like the Golden ratio, which also approximates the Fibinnocchi series. But I find my interest in the relationship of the ratio to useful application - not how many accurate decimal places can be extended.
In fact, I am not very interested in decimal maths on a a binary processor at all. It all seems like lemings chasing lemings rather than something original. The truth is the guy with the biggest grant and the fastest super computer has the advantage. I am looking for that little nugget of truth that the little guy can unsurface by independent thought and sole efforts. What's wrong with that?
But Pi is also transcendental, which means no finite sequence of polynomials can have its value. Given that the digits of Pi are know in excruciating detail, it makes a good burn in test of super computers.
I'm sure it's illegal to compute pi to too many decimal places. As with Borges' "The Library of Babel", the indefinite expansion of pi, converted to ASCII, must certainly include every copyrighted work ever published. For this reason, according to the Digital Millenium Copyright Act, possession of a program capable of such a feat of expansion would also be a crime. Moreover, I suspect that, by broaching the subject in a public forum, Humanoido would be deemed guilty of "contributory malfeasance by way of inducement," or some such legalese. But I won't tell!
Humanoido, I told you to beware of Phil finding out about this thread. Now you are really in trouble. I wonder if the abbreviation of Phi.Pi. means anything in Greek. Anybody from Greece on the forum?
I was rather hoping my comment might spur a discussion about the following: If a number is irrational, does that automatically imply than every n-digit sequence can be found somewhere in its expansion?
Phil, I say 'yes'. But I'll bet David Hilbert would have had a better answer than I do.
Humanoido should put his cogs to work calculating sqrt(2) to a trillion places. As the cogs proceed, one of them should attempt to match the output stream to an arbitrary sequences of digits selected in advance - perhaps "Sanya Richards runs fast and looks hot!" expressed in ASCII. Later we can worry about Shakespeare.
I was rather hoping my comment might spur a discussion about the following: If a number is irrational, does that automatically imply than every n-digit sequence can be found somewhere in its expansion?
-Phil
the answer is no. Pi may be irrational in base 10. but its digits do follow a pattern of sorts in other bases. So not all possible combination will end up happening. Interestingly it is in base 16(hex) that the most obvious pattern has been found. However I am not a mathematician and my interpretation may be wrong.
Pi may be irrational in base 10. but its digits do follow a pattern of sorts in other bases.
Irrationality is a property of the number itself, not of its representation in one base or another. A number that's "irrational in base x", so to speak, will be "irrational in all (rational) bases".
-Phil
Addendum: I added "(rational)" above, because pi, expressed in base pi, is just 10.
Beau, take a look at Stu's page for Pi in base 3. Definitely lots of patterns in this small clip.
Humanoido, those aren't "patterns" in the way any statistician would meaningfully use the word. The fact that you walked into the casino one day and saw the Roulette wheel history board lit up all in one color doesn't mean the wheel is biased; I've seen that four times myself. All it means is that you hang out in casinos way too much. If you look at a long run of truly random data you will see what look to human eyes like patterns; in fact, if you don't, it's a key clue that your data aren't random.
Pi is the same number whether you express it in base 2 or base 64. But in some bases there are interesting combinations or juxtapositions which grab the eye. Would it grab your attention if your random number generator put out the sequence 00000000? Well, if it can't spit out eight zeroes in a row, it isn't random because that particular sequence is just as likely to emerge from truly random data as any other particular sequence of eight digits. And this is true in any base, but we tend to notice it a lot more in hexadecimal than in binary. Knuth has an extensive discussion of this in Vol II of The Art of Computer Programming where he addresses the design of pseudo-RNG's.
In a similar way, the pictures the stars seem to form in the sky do not indicate some pattern to their distribution; the naked eye stars are really pretty evenly and randomly distributed. The constellations are in our brains, not in the mechanism that arranged the stars. A similar starscape in which we don't see such patterns, such as Stephen Jay Gould once described from the arrangement of phosphorescent organisms on the roof of a cave, actually indicates order -- the organisms don't like to be too close together. The stars, by contrast, don't care.
Interesting example, mctrivia! So I guess the question, per Leon's comment is this: is pi normal?
And back to my original post (which pushed Humanoido's "overreact" button somehow), if pi is normal, and since pi has existed since the dawn of time, does it represent prior art for those copyrighted works embedded in its expansion?
A sequence can be normal in one base and not another:for example the following binary value
10011010
in base 2 it is normal having 4 0s and 4 1s. however in base 4 it is not having 3 2s and 1 1.
Also the digits of a sequece can be normal but not random. 1234123412341234 is normal but it is not random.
In a normal random sequence of large enough size you will be able to find anything you want in it. Pi even if the digits are normal is definitely not random.
People get mixed up thinking that irrational means there is no pattern to it. It does not. It means at no point can you find a set of digits that just keep repeating over and over. The example I gave has a very obvious pattern to it, but it is not a repeat of itself.
Pi also has a known pattern to it, it is just not very obvious. But in base 16 we can compute any digit of pi without knowing any of the digits before or after it.
"In mathematics, a normal number is a real number whose infinite sequence of digits in every base b is distributed uniformly in the sense that each of the b digit values has the same natural density 1∕b, also all possible b2 pairs of digits are equally likely with density b−2, all b3 triplets of digits equally likely with density b−3, etc." [emphases mine]
ok by that definiton and not the one in my head if pi is normal then it would contain the code for all programs ever written, the designs to every object ever created, the entire text of every book ever writen, even a vector drawing of every painting ever made.
However I think you are off a little. The people computing pi would not be the ones breaking copyright. It is the people writing the software that would be breaking God's copyright. After all he did write Windows 7 at the begining of the universe and encoded it into pi.
And would a patent run out for a God that is outside of the constraints of time(which any god would have to be in order to create the Universe and time itself.) It may have been a few billion years for us but for him he did write it, is writing it and will write it
Alas this opens up the possibility for a very interesting and pointless discussion on quantum mechanics, reality, cosmology, theology, and many other ologys.(and before anyone jumps down my throught about the seaming oximoronical meaning to that statement i am saying pointless because we can not know the answer to such a question until we ourselves can break the fourth dimension and transcend time. We can only see the whispers and brush strokes of a creation more beautiful and wonderful then we can even imagine*).
After all he did write Windows 7 at the begining of the universe and encoded it into pi
Wait a minute. We might say that God provided us with pi. However I'm going to argue that He never did write down all the digits of pi in any number system and did not even bother to document his creation with any written formulas or algorithms. Typical programmer.
The people computing pi would not be the ones breaking copyright. It is the people writing the software that would be breaking God's copyright. After all he did write Windows 7 at the begining of the universe and encoded it into pi.
Maybe we should start looking for skip codes in the digits of PI.
Pi also has a known pattern to it, it is just not very obvious. But in base 16 we can compute any digit of pi without knowing any of the digits before or after it.
This is absolutely not the case. The "patterns" which are apparent to human observers of the hex dump are not regular and are not predictively useful. The fact that there is an apparent density of them near the decimal point is a coincidence.
A famous mathematician, Kronecker, once said "God created the integers, all the rest is the work of man." That gets us into deep philosophical waters - are mathematical truths invented or have they always existed? They seem to have an existence independent of us, but that might be an illusion.
The ratio of a circle's circumference to its diameter is universal and invariant. So I would say that the value has always been there for us to discover, not to invent. But the various means we use to express and communicate that value are definitely inventions.
A famous mathematician, Kronecker, once said "God created the integers, all the rest is the work of man." That gets us into deep philosophical waters - are mathematical truths invented or have they always existed? They seem to have an existence independent of us, but that might be an illusion.
I definitely disagree with this statement. PI and (sqrt(5)+1)/2 are both directly found in nature.
The digits of Pi probably do contain encoded versions of arbitrary texts. But the trick would be in finding them in the time before the universe dies. For example, if you want to find a 10 MB long binary sequence of digits, and the digits of Pi are randomly distributed. I would think the odds any particular 10 MB string having that sequence would be low.
Total number of bits = 10*1024*1024*8
Odds per bit = .5
Probability = .5 ^ 10*1024*1024*8
Now my calculator doesn't go that high, but a Java program using big numbers could compute that probability. To get that probability close to one you would need to process a whole lot of 10 MB sequences.
Now Windows 7 is on the order of GB, so the probability gets much worse, and you're likely to find many near miss copies that are buggy before you find the golden sequence.
Now Windows 7 is on the order of GB, so the probability gets much worse, and you're likely to find many near miss copies that are buggy before you find the golden sequence.
You are probably more likely to find a version with all the bugs fixed LOL.
@everyone, talks too loudly about the reality of numbers and we'll be up to our armpits in philosophers accusing us of neo-Platonist delusions. Better to say, it appears as if numbers have an objective reality, but that's only a working assumption.
The digits of Pi probably do contain encoded versions of arbitrary texts. But the trick would be in finding them in the time before the universe dies. For example, if you want to find a 10 MB long binary sequence of digits, and the digits of Pi are randomly distributed. I would think the odds any particular 10 MB string having that sequence would be low. Total number of bits = 10*1024*1024*8. Odds per bit = .5. Probability = .5 ^ 10*1024*1024*8. Now my calculator doesn't go that high, but a Java program using big numbers could compute that probability. To get that probability close to one you would need to process a whole lot of 10 MB sequences.
Martin, this is a very good analogy and a key point. While we may not have the resources today to accomplish greater digits to show the reality of numerical patterns of A, as well with B, etc., on the probability curve they can exit. Simply stated, and as Hawking has shown, there is chronological progression, for example, to states of the Universe, say from point A to B. In a later state, every possible combination of the Universe will exist, simply because as time changes, there is enough time to run through every version statistically. As a simile, therefore, if we had enough time and resources too, the value of Pi would statistically run through these possibilities. For now, however, it may be in the best interest of discovery to run these computations extending the digits of Pi and look for variation and permeability.
Comments
Nonetheless, I am deeply interested in trancendental numbers - especially the ones that are more ignored and less used -- like the Golden ratio, which also approximates the Fibinnocchi series. But I find my interest in the relationship of the ratio to useful application - not how many accurate decimal places can be extended.
In fact, I am not very interested in decimal maths on a a binary processor at all. It all seems like lemings chasing lemings rather than something original. The truth is the guy with the biggest grant and the fastest super computer has the advantage. I am looking for that little nugget of truth that the little guy can unsurface by independent thought and sole efforts. What's wrong with that?
http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational
But Pi is also transcendental, which means no finite sequence of polynomials can have its value. Given that the digits of Pi are know in excruciating detail, it makes a good burn in test of super computers.
One learns something new everyday.......
__________________
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-Phil
Humanoido should put his cogs to work calculating sqrt(2) to a trillion places. As the cogs proceed, one of them should attempt to match the output stream to an arbitrary sequences of digits selected in advance - perhaps "Sanya Richards runs fast and looks hot!" expressed in ASCII. Later we can worry about Shakespeare.
the answer is no. Pi may be irrational in base 10. but its digits do follow a pattern of sorts in other bases. So not all possible combination will end up happening. Interestingly it is in base 16(hex) that the most obvious pattern has been found. However I am not a mathematician and my interpretation may be wrong.
http://mathworld.wolfram.com/NormalNumber.html
Greets Daniel!
I'm in Horsham PA! Small world!
KK
-Phil
Addendum: I added "(rational)" above, because pi, expressed in base pi, is just 10.
Humanoido, those aren't "patterns" in the way any statistician would meaningfully use the word. The fact that you walked into the casino one day and saw the Roulette wheel history board lit up all in one color doesn't mean the wheel is biased; I've seen that four times myself. All it means is that you hang out in casinos way too much. If you look at a long run of truly random data you will see what look to human eyes like patterns; in fact, if you don't, it's a key clue that your data aren't random.
Pi is the same number whether you express it in base 2 or base 64. But in some bases there are interesting combinations or juxtapositions which grab the eye. Would it grab your attention if your random number generator put out the sequence 00000000? Well, if it can't spit out eight zeroes in a row, it isn't random because that particular sequence is just as likely to emerge from truly random data as any other particular sequence of eight digits. And this is true in any base, but we tend to notice it a lot more in hexadecimal than in binary. Knuth has an extensive discussion of this in Vol II of The Art of Computer Programming where he addresses the design of pseudo-RNG's.
In a similar way, the pictures the stars seem to form in the sky do not indicate some pattern to their distribution; the naked eye stars are really pretty evenly and randomly distributed. The constellations are in our brains, not in the mechanism that arranged the stars. A similar starscape in which we don't see such patterns, such as Stephen Jay Gould once described from the arrangement of phosphorescent organisms on the roof of a cave, actually indicates order -- the organisms don't like to be too close together. The stars, by contrast, don't care.
is irrational. But not all combinations will result
And back to my original post (which pushed Humanoido's "overreact" button somehow), if pi is normal, and since pi has existed since the dawn of time, does it represent prior art for those copyrighted works embedded in its expansion?
-Phil
10011010
in base 2 it is normal having 4 0s and 4 1s. however in base 4 it is not having 3 2s and 1 1.
Also the digits of a sequece can be normal but not random. 1234123412341234 is normal but it is not random.
In a normal random sequence of large enough size you will be able to find anything you want in it. Pi even if the digits are normal is definitely not random.
People get mixed up thinking that irrational means there is no pattern to it. It does not. It means at no point can you find a set of digits that just keep repeating over and over. The example I gave has a very obvious pattern to it, but it is not a repeat of itself.
Pi also has a known pattern to it, it is just not very obvious. But in base 16 we can compute any digit of pi without knowing any of the digits before or after it.
Only irrational numbers are normal. 10011010 is simply normal, not normal!
Phil:
PI appears to be normal, but no one has managed to prove it.
-Phil
However I think you are off a little. The people computing pi would not be the ones breaking copyright. It is the people writing the software that would be breaking God's copyright. After all he did write Windows 7 at the begining of the universe and encoded it into pi.
And would a patent run out for a God that is outside of the constraints of time(which any god would have to be in order to create the Universe and time itself.) It may have been a few billion years for us but for him he did write it, is writing it and will write it
Alas this opens up the possibility for a very interesting and pointless discussion on quantum mechanics, reality, cosmology, theology, and many other ologys.(and before anyone jumps down my throught about the seaming oximoronical meaning to that statement i am saying pointless because we can not know the answer to such a question until we ourselves can break the fourth dimension and transcend time. We can only see the whispers and brush strokes of a creation more beautiful and wonderful then we can even imagine*).
Wait a minute. We might say that God provided us with pi. However I'm going to argue that He never did write down all the digits of pi in any number system and did not even bother to document his creation with any written formulas or algorithms. Typical programmer.
So God does not get the copyright on any of it:)
This is absolutely not the case. The "patterns" which are apparent to human observers of the hex dump are not regular and are not predictively useful. The fact that there is an apparent density of them near the decimal point is a coincidence.
-Phil
I definitely disagree with this statement. PI and (sqrt(5)+1)/2 are both directly found in nature.
Total number of bits = 10*1024*1024*8
Odds per bit = .5
Probability = .5 ^ 10*1024*1024*8
Now my calculator doesn't go that high, but a Java program using big numbers could compute that probability. To get that probability close to one you would need to process a whole lot of 10 MB sequences.
Now Windows 7 is on the order of GB, so the probability gets much worse, and you're likely to find many near miss copies that are buggy before you find the golden sequence.
You are probably more likely to find a version with all the bugs fixed
@everyone, talks too loudly about the reality of numbers and we'll be up to our armpits in philosophers accusing us of neo-Platonist delusions. Better to say, it appears as if numbers have an objective reality, but that's only a working assumption.