In May of 2003, I asked colleagues to consider a concept that had been on my mind for many years.

"Somewhere inside the digits of pi is a representation for all of us -- the atomic coordinates of all our atoms, our genetic code, a coding of our motions and all our thoughts through time, all our memories.... Given this fact, all of us are alive, and hopefully happy, in pi. Pi makes us live forever. We all lead virtual lives in pi. We are immortal." - Cliff PickoverThis means that we exist in pi, as if in a Matrix. This means that romance is never dead. Somewhere you are running through fields of wheat, holding hands with someone you love, as the sun sets -- all in the digits of pi. You are happy. You will live forever.

Pi is a "transcendental number". It is a never-ending, patternless sequence of digits. Each digit appears with equal frequency. If pi is a "normal" number, then a representation of us is almost surely in it. Here are the first few digits:

1415926535 8979323846 2643383279 5028841971

6939937510 5820974944 5923078164 0628620899

8628034825 3421170679 8214808651 3282306647

0938446095 5058223172 5359408128 4811174502

This speculation caused a long discussion, some of which is reproduced here. (Note that researchers studying Quantum Immortality say you will live forever for other reasons.)

Note that I have written a book on this subject.
In this same book, I explain how our thoughts reside in the vibrational patterns of a cube of Jello.

Nick Lacasse responds: How is this conclusion reached, Cliff? To me it seems that it must be true because pi is an infinitely long, non-repeating number, so every possible combination of numbers must exist in every possible sequence in its digits. The trouble with this is, how do we know it doesn't repeat once it gets past the quadrillionth number? Can we really be CERTAIN that an infinite series doesn't repeat?

[Cliff responds, I believe it is well known that transcendental numbers don't repeat in the sense that you are worried about. Perhaps a math person in this Group can comment with certainty... That means that romance is never dead. Somewhere you are running through fields of wheat, holding hands with someone you love, as the sun sets -- all in the digits of pi.]

Ollyhardy responds: This is a fascinating speculation, on many levels.If the digits of pi include every possible finite sequence, than in principle a hypothetical super-and I mean really super- computer could "crank out" all of the permutations and thus produce a copyof everything that has ever lived. In theory, all of the information is there to produce a trilobyte, or Cliff at 15, or, interesting, Ciff at 80, or an infinite number of parallel reality Cliff's. Rogue physicist Frank Tipler wrote a book some years back called "The Physics of Immortality" in which he argued that under special circumstances, in a collapsing universe, right before the final collapse there would be enough power (watts) for a universal computer to produce simulations of everything that has ever lived in it. Tipler's thesis involves some rather elaborate and unlikely physics in order for his "Omega Point" computer to be able to somehow access and manipulate all of the infoirmation that has ever existed in our universe. However, if Tipler had been aware of what Cliff is saying about Pi, then in theory the information is already available, if only we knew what to look for in Pi. In principle, the information is already here, the main obstacle might be how to obtain the massive amount of power-sheer wattage- required to generate such simulations. Perhaps the power required to fuel such machines may one day be available to us, whet Now all we need is for Cliff to crank out a beta version of the hardware/software. Could you have somethiing on my desk by say, early next week?

From: "bobomutin": As for Cliff's statement on being encoded in the digits of pi , I am curious and skeptical. What does existing in pi mean, when pi is a neuro-biological creation of human creativity? Does PI exist when nobody knows of it? I would say, I don't know. Now neutrinos and radio waves existed ages before they as objects, were even conceived of. Is PI considered to be a relational object like a photon?

From: "tmredden": Are you suggesting that pi necessarily does contain an image of everything? I didn't realize that idea was concluded as factual. Pi may be a never ending and never repeating decimal, but if we could say that pi contained every image, then it would necessarily have to contain copies of itself, which would make it repeating. Because we know of something it cannot contain, it follows that there are other things it does not contain, as well. Are you suggesting that somewhere in pi is the ascii code for the 1993 Edition of the Encyclopedia Brittanica? Not just one but infinitely many copies of it? Are you sure? I don't want to be in pi!!! Let me out!

[Cliff says, yes, Pi almost surely contains the 1993 Edition of the Encyclopedia Brittanica. Moreover, it contains the Windows XP operating system. Moreover, it contains all your thoughts, coded in its digit string. You need not fear death or yearn for the woman you once loved but could never have. You have her in pi where you live forever.]

From: "Tufrmone" : The logical conclusion of what you are speculating is that there is not simply one of every person, there is an infinite number of variations of ourselves, each one one number different. There is also an infinite number of realities encompassing the genetic codes, the molecular structures and the past and future events of every living and non-living thing. A computer powerful and fast enough would theoretically be capable of using pi to decode, record and rerun every event from the birth of the universe to the present and into the future. One could for example watch Jesus give the sermon on the mount or see him raise the dead. The problem of course is that one could never know with certainty if it was our reality we were seeing or another reality similar but different from the events of our own past with respect to the one critical event we are at that moment concerned with. Tufr

Chuck G.: Our brain states may be encoded in the digtis of pi, but I hear the beer is cheaper in the square root of 2.

From: "Graham Cleverley": Nice thought, but based on a mathematical error. The digits of pi constitute a 'countable' infinity, like the set of integers - |N| or aleph0. Unless you can make a case for taking all the information you mention and listing it in sequential order, then it represents a higher order of infinity - e.g. |R| (the set of all real numbers) or alephN, N>0. |R| and aleph1 upwards contain elements that are not contained in |N|. [Cliff says, I think I'm correct in saying that the entire encylopedia is in pi. So it the Windows XP operating system and Shakespeare's Julius Cesar. Are you saying I am wrong?]

From: "Graham Cleverley"

[Cliff says, I'm correct in saying that the entire encylopedia is in pi. So it the Windows XP operating system and Shakespeare's Julius Cesar. Are you saying I am wrong?]Each of those things can be encoded as a series of bytes in sequence, and the three things themselves can be sequenced. As I said, if you can do that, it's a countable number (finite in this case) - it has lesser cardinality than the digits of pi. However, the (base 16 I assume) digits of pi are calculable in sequence via one of several algorithms. For your conclusion to be true, the sequence of bytes in those articles would also have to be calculable by that algorithm, which I doubt. The set of all possible infinite sequences is of cardinality greater than |N| (it equals |N|^|N|) so there are possible infinite sequences that are not contained in the digits of pi. That set might contain all the information you're talking about, but it's much (infinitely :- )) bigger than the number of digits in the expansion of pi. (Or any other transcendent number.) In any case, you were claiming rather more than those three things. I'm not sure all the information you're talking about can be sequenced (any more than the real numbers can be sequenced). [Cliff says, if you don't think I'm coded in pi, do you think that there is a number that codes me? Also, are you saying that I MIGHT not be coded in pi or that I AM NOT coded in pi? But if I don't have to be EXACTLY coded in pi, I am probably encoded in pi.]

Anton says: Pi does not contain the digits of the square root of 2, which one member suggested would code a reality where beer is cheaper! If does not contain square root of 2, then it is not infinite in regard to containing everything possible, and then all possible realities are not placed somewhere inside pi, and therefore we will drink expensive beer just because we made a mistake in which universe we would like to live. That will mean that there are possible universes divided by belonging to the particular transcendental number as a form of coding.

[Cliff says, Pi may not contain SQRT 2, because SQRT 2 is infinite, but doesn't PI almost surely code for all small finite sequences (like my DNA for example)? If it does not code ALL realities, it codes for realities that are CLOSE ENOUGH.]

From: ollyhardy I think Cliff's Pi idea is extremely interesting. If we put aside the goal of immortality for a second, we might consider other, more mundane uses for pi. A pi based algorithm might alllow for some unbelievably efficent data compression. Think about it. You want to copy the Encyclopedia Britannica, or a photo of your aunt, or the latest Scorcese flick? Just jump to the appropriate strings in pi. Of course, exactly how we would be able to recognize and any given string in pi as the blue print for something we know in th ereal world is easier said than done. Imagining sheerly speculative, sci fi type technology, I can envision a machine which scans objects, digitizes them and then searches for those strings in pi. It then stores the "addresses" or locations in pi where the information for any given object can be found. Once the machine knows "the address" or adresses in pi of any real world object, it can retrieve the information or build a copy as needed. Even more intersting, imagine getting creative with pi machine. Imagine a brain interfaced with a "Pi Machine" such that it could translate your thoughts into pi digits and then make your fantasies a reality? That'd be worth at least 50 bucks.

Cliff wrote: [Cliff says, Pi may not contain SQRT 2, because SQRT 2 is infinite, but doesn't PI almost surely code for all small finite sequences (like my DNA for example)? If it does not code ALL realities, it codes for realities that are CLOSE ENOUGH.] Yes,but it is interesting that even pi is infinitive it does not contain every combination of numbers which will imply that trancendental numbers are infinitives which are finitely diverse or that all trancendental numbers are one number but seen started on different "point" of beginning. Anton

From: "Chuck Gaydos" A number can fail to ever repeat and still not contain every finite sequence. How about .1001110000111110000001111111...? It never repeats yet doesn't encode the number 1½. Pi could fail to repeat but still not contain every possible finite sequence. -Chuck

[Cliff says, it doesn't matter. For all intents and purposes, Pi codes for you. It doesn't matter if it actually codes for a "you" that has several atoms misplaced.]> Todd wrote: Pi may be a never ending and never repeating decimal, but if we could say that pi contained every image, then it would necessarily have to contain copies of itself, which would make it repeating. Interesting remark.Pi is containing self not as copy but it is containing self as being pi BUT it does not contain square root of 2 ,as one member suggested to be place where beer is cheaper. If does not contain square root of 2 then it is not infinitive in regard of containing everything possible and then all possible realities are not placed somewhere inside pi and therefore we will drink expensive beer just because we made a mistake in which universe we would like to live. That will mean that they are possible universes divided by belonging to the particular transcendental number as form of coding. Anton

From: "Tufrmone" to Chuck: The Pi sequence Chuck has cited will eventually encode for 1.5. Actually it will encode for 1.5 a whole lot sooner than it will encode for the events surrounding Jesus, which will also be encoded in one form or another. The sequence isn't simply Pi itself but any encoded sequence in the sequence itself as in every third digit followed by the 42nd and divided by the first 5. Eventually you will find a sequence to accurately represent every possible event and thing imaginable including fractions. Tufr

[Cliff says, my mathematician friend from IBM says: "If the binary representation of pi is interpreted as a program in some computer language, perhaps it encodes a simulation of the universe that includes itself, you and me... or maybe it is a movie of your live encoded in some yet-to-be-discovered version of MPEG..."

Chuck Gaydos wrote:A number can fail to ever repeat and still not contain every finite sequence. How about the sequence...... It never repeats yet doesn't encode the number 1½. Pi could fail to repeat but still not contain every possible finite sequence. Chuck [Cliff says, it doesn't matter. For all intents and purposes, Pi codes for you. It doesn't matter if it actually codes for a "you" that has several atoms misplaced.]

From: "Tufrmone" : Dear Cliff: I have not seen this aspect of Pi discussed before and it raises a number of immediate questions in my mind. The first one is that assuming Pi does encode for every event and thing which ever occurred, may have occurred, may occur in the future and will occur in the future --- does this provide a solution to the apparent fact that the delivery of material into a black hole violates the laws of thermodynamics because would theoretically be destroying information in a manner which cannot be recovered. If Pi is a record, such information could theoretically be recovered couldn't it? Tufr

From: ollyhardy: Cliff: A couple of questions about Pi.... Just because all of the information describing us is in Pi, why does that mean that we are necessarily immortal per se? Doesn't it really just mean that all of the critical information necessary to "compute us" is available, requiring still someone or some "thing" else to take the trouble to compute us? After all, all of the information to create the movie "Goodfellas", or you yourself and me for that matter was in Pi when the dinosaurs ruled the earth, but the data only became such things as we call Goodfellas and Cliff respectively when they were naturally selected for computation. We can take our left hand and our right hand and do "karate chops" anywhere along the infinite string of digits that is Pi and "compute" what lies between our hands, but what does it really mean? Also, what number is "pi" itself in? I have your book "Keys To Infinity" and my friend tells me that this idea that "we all live forever in PI" is in there, but I can't find it. Is he right? If so, do you know off hand what section of the book I might find it in? I have 6000 questions about this fascinating subject you brought up, but unfortunately,or fortunately as the case may be, I have many more questions than time, so I have to leave it here for now.

[Cliff says, you MUST read the SF novel Permutation City by Greg Egan. It gets close to this sort of thinking. As you know, Chapter 7 in Keys to Infinity has a lot of fascinating pi information. I don't recall if I made this exact comment here or in another book. ]

From: "mwganson" : Regarding a brain interfaced with a "Pi Machine": This is an interesting idea. You just keep the address of the start of the encoded string. The problem is that the address is a number so long that it takes more space to store that value than the space needed to store the data that is encoded. As an example, in order to find the first occurrence of a particular string of 4 digits, say, 1234, we don't find it until we get to the 13,386th digit in pi. If the address is longer than the data, not only do we need more space to store the address, we also have the overhead of producing the sequence each time. --Mark

From: "Graham Cleverley" [Cliff says, if you don't think I'm coded in pi, do you think that there is a number that codes me?] In theory, your physical body structure at a particular time could be encoded as a long but finite string of bytes, and therefore as a number. I suppose also that in theory that number could be used to reconstitute you, given the technology to do it. Which is what makes 'matter transmitters' theoretically possible, though they would be better called 'matter duplicators'. Whether you have a part that is non-physical (one or more 'souls') I really have no idea. If you do, then I also have no idea whether that could be similarly encoded.

Also, are you saying that I MIGHT not be coded in pi or that I AM NOT coded in pi?

That you might be, but that you might not, because there are sequences of numbers that are not included in pi. Whether you are or not is theoretically a testable hypothesis: how the test would turn out is unknown. And if you are encoded in pi, it doesn't mean I would be. I might be included in 2*pi though - or e. As I hinted earlier, I could more happily live with the assertion that all the information you mentioned is included somewhere in the set of all transcendent numbers.

[But if I don't have to be EXACTLY coded in pi, I am probably encoded in pi.]

I guess the lower the exactness the higher the probability.

From: "Graham Cleverley"

[Cliff says, my mathematician friend from IBM says: "If the binary representation of pi is interpreted as a program in some computer language, perhaps it encodes a simulation of the universe that includes itself, you and me... or maybe it is a movie of your live encoded in some yet-to-be-discovered version of MPEG..."]The important things there are the words 'perhaps' and 'maybe'. I go along with that.

From: "Chuck Gaydos": Whether or not I'm encoded in Pi depends on the coding method. If we choose a coding method such that you're encoded in Pi there's no guarantee that I am as well. Since you don't insist that every atom be exactly accounted for there would be a vast number of strings of digits that could be said to encode me, but do any of them exist in Pi? A number can fail to repeat and still not contain all possible finite patterns. Are there a trillion consecutive zeros in Pi? It seems likely to the best of our knowledge, but maybe not. It might be that none of the strings that encode me exist. Of course, you could choose a coding method that includes both of us, but that's kind of arbitrary. I'd say I was encoded in the encoding method and not in Pi itself. If you choose a sufficiently complex coding method then the entire universe can be encoded by a single digit. It's unremarkable that we're all encoded in Pi if the coding method forces it. -Chuck

From: "mwganson" : Are there a trillion consecutive zeros in Pi? Of course there are. The possibility that there are not is so small that it is impractical to even consider it. We can determine the probable number of times p that a consecutive string of z zeros exists within a given subset of the first d digits of pi as: p == 1/10^z * d For example, suppose we are wondering whether there are any strings of 5 consecutive zeros within the first 100,000 digits of pi. p == 1/10^5 * 100000 p == 1 This means we can expect to find 1 such string, which is exactly how many there are. The string of 5 zeros has the same mathematical probability of occurring as any other 5 values, such as 12345 or 99999. Each of these strings will probably occur once each. I will test this by searching for, and counting, the occurrences of these 10 strings: 00000, 11111, 22222, ..., 99999. Stay tuned... Okay, I'm back.

String: occurrences 00000: 1

11111: 1

22222: 1

33333: 1

44444: 0

55555: 2

66666: 1

77777: 0

88888: 0

99999: 4

Total: 4+0+0+1+2+1+1+1+1 == 11

We would have expected to have a total of 10 occurrences of all of the searched for strings. Instead, I found 11, which is close enough for me. (Arguably, however, the string 99999 only occurs 3 times since there is a single occurrence of 6 consecutive 9's -- 999999, which I counted as 2 separate occurrences of 5 consecutive 9's.) Half of the strings (5 of them) had exactly 1 occurrence each. There were 3 absent strings, but this was made up for by multiple occurrences of 2 of the strings (2 times for 55555 and 4 times for 99999).

I can post a text file of about 100K in size, which contains all of these 100,000 digits to the Files area if anyone is interested in seeing it. You can probably find something similar through Google, though... Be right back... Okay, I'm back. Here is a link to just such a file:

http://www.geocities.com/thestarman3/math/pi/df/PI100KDP.ZIP and, for other sizes:

http://www.geocities.com/thestarman3/math/pi/picalcs.htm Now, back to the question of 1 trillion consecutive zeros. How many digits of pi must we extract before we can reasonably expect to find a single occurrence of 1 trillion (10^12) consecutive zeros?.p == 1/10^z * d 1 == 1/10^(10^12) * d

1/d == 1/10^(10^12)

d == 10^(10^12) (after cross multiplication) So, after (just?) an expansion of 10^(10^12) digits, we can reasonably expect to find an occurrence of a string of 1 trillion consecutive digits. I say (just?) because, even though 10^(10^12) is unimaginably huge, it is not even a drop in the ocean compared to infinity. After an expansion of 10 * 10^(10^12), we can reasonably expect to find 10 such strings of a trillion zeros. After an expansion of, say, a billion times the amount needed to find a trillion consecutive zeros (10^9 * 10^(10^12)), which is still not even a drop in the bucket as compared to infinity, we could expect to find a billion such strings of a trillion consecutive zeros. Does this mean there is *definitely* at least one string of a trillion consecutive zeros within the first 10^9 * 10^(10^12) digits of pi? No. But, the probility is extremely high (I'd say: (10^9-1)/10^9? -- Someone please correct me if I'm wrong.) that there is at least one such string within that finite expansion. --Mark

From: "Chuck Gaydos": All of this assumes that the digits of Pi are normally distributed. They appear to be so far but we haven't looked at truly large quantities of them. They're not randomly generated after all. -Chuck

From: "bobomutin": Chuck's observation seems sound regarding coding. I am still not sure why Cliff made the statement below. What observation, research, paper, mathematician(s) are his compelling statements based on? Pythagoras? Kant, Reimann, someone more contemporary? Puzzled and waiting, on Yahoo.

From: "Graham Cleverley": The fact that they're not random is one of the things that makes me doubt I'm encoded in it.

[Cliff says, can you explain why? If you assume pi is "normal", why wouldn't you think you are coded in it? When I use the word "you" I allow for slight discrepencies. For example, I still call it "you" even if a few atoms are out of place. After all... As we age, the molecules in our bodies are constantly being exchanged with our environment. With every breath, we inhale the world lines of hundreds of millions of atoms of air exhaled yesterday by someone on the other side of the planet. In some sense, our brains and organs are vanishing into thin air, the cells being replaced as quickly as they are destroyed. The entire skin replaces itself every month. Our stomach linings replace themselves every five days. We are always in flux. A year from now, 98 percent of the atoms in our bodies will have been replaced with new ones. We are nothing more than a seething mass of never-ending world lines, continuous threads in the fabric of spacetime. What does it mean that your body has nothing in common with the body you had a few years ago? If you are something other than the collection of atoms making up your body, what are you? You are not so much your atoms as you are the pattern in which your atoms are arranged. For example, some of the atomic patterns in your brain code memories. We are persistent spacetime tangles. In my book Time: A Traveler's Guide, in a diaram, a person is represented by a set of four atom threads that have come close together. (An "atom thread" is the spacetime trail of an individual atom.) Note that an atom can leave one person's array and become part of another person. Very likely you have an atom of Jesus of Nazareth coursing through your body.The nth digit doesn't just depend on the previous one, it depends on all the previous ones, because it must take the whole number closer to the value of pi. The numbers in my encoding would not have to follow that rule (but one of their own, I assume, which might well conflict).

From: "Graham Cleverley": Something else I should have added. The set of the digits of pi is sequenced and therefore has cardinality |N|, just like the set of integers. However the set [2,4,6,8,10,....] also has cardinality |N| but it does not include the odd numbers. Incidentally, the set of all infinite sequences in pi also has cardinality |N|, whereas the set of all infinite sequences has cardinality |N|^|N| which is greater.

From: "Cliff Pickover": Hi, I'm not sure why I am getting push back on this. If we assume that the digits of "pi" are normal (and indications are that they probably are), then it seems clear to me that we are in pi. And, never mind the fact that pi may code us a few atoms out of place. As we age, the molecules in our bodies are constantly being exchanged with our environment. With every breath, we inhale the world lines of hundreds of millions of atoms of air exhaled yesterday by someone on the other side of the planet. In some sense, our brains and organs are vanishing into thin air, the cells being replaced as quickly as they are destroyed. The entire skin replaces itself every month. Our stomach linings replace themselves every five days. We are always in flux. A year from now, 98 percent of the atoms in our bodies will have been replaced with new ones. We are nothing more than a seething mass of never-ending world lines, continuous threads in the fabric of spacetime. What does it mean that your body has nothing in common with the body you had a few years ago? If you are something other than the collection of atoms making up your body, what are you? You are not so much your atoms as you are the pattern in which your atoms are arranged. For example, some of the atomic patterns in your brain code memories. We are persistent spacetime tangles. In my book Time: A Traveler's Guide, in a diagram, a person is represented by a set of four atom threads that have come close together. (An "atom thread" is the spacetime trail of an individual atom.) Note that an atom can leave one person's array and become part of another person. Very likely you have an atom of Jesus of Nazareth coursing through your body.

From: "nick_hobson": Is it possible to make money betting on the digits of pi, without actually calculating pi? Nick

From Mark: You are correct in that I assume the digits are randomly distributed. I believe this is a good working assumption. The fact that the digits of pi bring pi to a specific, non-random, value is not relevant. As an analogy, consider the estimated number of atoms in the observable universe: 10^78 (some estimates have it at 10^81). Now, the *exact* number, if we could count them, is probably not be exactly 10^78. The actual number would probably look a lot like a random string of about 78 digits, with each of the 10 possible digits (0 through 9) being represented rougly an equal number of times each. In other words, we'd expect to see 7 or 8 zeros, 7 or 8 ones, etc., despite the fact that we are looking at a number that is not itself a randomly generated value. We can reasonably expect that after n digits of expansion we will have roughly equal representations of all 10 digits in pi. The larger the value for n, the closer the count totals for each digit will approach n/10. As I have shown, within the first 10^9 * 10^ (10^12) digits, we will have a billion different strings of a trillion consecutive zeros. This also holds true for any particular sequence of a trillion digits, including the first trillion digits of pi, a trillion consecutive nines, etc. The encoding scheme is quite irrelevant, whether it be mpeg, dna, machine language, ascii text, or whatever, just so long as the encoding sequence is capable of encoding the data, any finite string of digits is sure to exist not just once, but an infinite number of times. There are an infinite number of strings of a trillion consecutive zeros within the infinite expansion of pi. I have done some more calculating. Given any randomly generated stream of an infinite number of (base 10) digits, the probability that there exists a number n, such that the first n digits in the stream is exactly duplicated by the second n digits in the stream is 1/9. This is based on the sum of the infinite series defined as 1/10^n where n ranges from 1 to infinity.

Mathematica code:

Sum[1/10^n,{n,1,Infinity}]

output: 1/9

I calculated this because I was curious to know if there was such a number n for pi where the first n digits of pi were exactly duplicated by the next n digits. I wondered what the probability for this was, and whether it was closer to 0 or to 1. The fact that if such a number n exists for pi, then such a number must be quite large, means that the probability of this ever happening for pi is quite low. The bulk of the value of 1/9 comes when n is very low. With each successive value for n, the probability gets lower and lower. Given this calculation, I conclude that the probability that the first n digits of pi are duplicated by the next n digits anywhere along the stream is extremely unlikely and very close to 0. But, pi is not the only transcendental number out there. Given 9 different transcendental numbers, the chances are good that at least one of them will have a cycle of repeated digits, most probably occurring within the first 10 digits. --Mark

From: ollyhardy: This is essentially a statement of the pattern theory of identity, an idea that is at least as old as the ancient Greeks who spoke of the ship of Theseus. The Greeks imagined a ship- the ship of Theseus- which over the years, owing to wear and tear, has to have a piece here replaced and a piece there replaced until finally, after a hundred years or so, not one component of the ship of Theseus was a part of the original construction. Is it the same ship? If the ancient Greeks were aware of the fact that all of the atoms making up the ship of Thesus were being replaced so frequently that there was probably not a single original atom in it's structure by the time the first 4 x 4 had to be replaced, it would have rendered the answer to this ancient riddle acadamic. Of course it's the same ship; it's the pattern that determines identity, not the physical continuity of the pattern's components. When people express doubts about the pattern theory of identity, it is usually in the context of discussions about simulated beings. Some people find it hard to accept that a machine which faithfully dupilcated the patterns that make us up would really be us. The problem is not with the pattern theory of identity, which is really the only viable identity theory, but with their understanding of it. The idea that there is some static "stuff' that we are made up of from moment to moment is an illusion. We are all like ships of Theseus.

From Mark: Cliff, It doesn't matter that we are in a constant state of change. We can pinpoint our atomic makeup to an instant in time and use that. In fact, given the infinite nature of pi, we can expect that there are continuous encodings strung sequentially representing our exact atomic makeup for every second of our existence from birth to death. By this, I mean that you could create an encoding scheme that would allow you to map every atom in a human being. This encoding scheme would create a series of base 10 digits, which could then be decoded and used to reproduce a 3D map of the human being all the way down to the atomic level. Now, use this encoding scheme to take a "snapshot" of any person every second of that person's life. (Or, every microsecond, if you prefer.) If this person lives to be n seconds of age, then we have n sets of snapshots encoded sequentially in the scheme. We can delimit the snapshots any way you like, such as by separating them by a stream of a trillion consecutive zeros. For every person that has ever lived there exists an infinite number of these life strings within the infinite expansion of pi. If we view the string of digits in pi as the stream of digits in an mpeg movie, there exists an infinite number of strings of such digits.that exactly duplicate the new Matrix movie. There are also versions with different endings, in different languages, even with different actors and actresses. There are even versions where you and I get to be the stars in it! --Mark

From: "ENAM" : I think the problem simplified down to whether the representation of a system as complex as me in a string of binary digits is same as me ,and I think the answer is yes ,my reasoning is simple ,whatever atoms or elimentary perticle creates me ,they are nothing but a representation of some complex mathematical formulation ,and within it it holds no extraordinary principle ,except how it will change according to global time or coordinate or what ever, so if I am really nothing but a representation then in whatever format I exist ,they don't differ by a long way ,however some represtentation may act differently then other ,such as I am now writing the letter ,but my binary self may not be doing that .... :) Enam.

From: "Graham Cleverley": Cliff you are getting push baack for two reasons. One is that you don't take into consideration that there are (infinitely) many different orders of infinity. A smaller one cannot contain elements that map to all the elements of a larger one. The digits of pi are a set of cardinality aleph0, also written as |N|, the same as the set of all integers. We know that because each nth digit of pi can be mapped to the integer n. A little harder to see is that the set of infinite sequences in pi is of the same cardinality. This is because the only infinite sequences in it are, spelling it out, the one that starts with the first digit, the one that starts with the second digit, and so on. They can therefore be mapped to the set of integers, which means they are another set of order |N|. You can therefore take the union of the two sets, but that does not increase the cardinality, since it is provable that |N| + |N| = |N|. The set of finite sequences in pi is harder. But take the first digit: it starts one finite sequence for each following digit (3, 3- 1,3-1-4,3-1-4-1 etc. There are therefore |N| of them. There are also |N| starting with the second digit, |N| with the third and so on. There are therefore |N|*|N| finite sequences - but |N|*|N| also equals |N|. Add all the three together and you still have only |N| elements. That is not enough for instance to map all the real numbers, which have cardinality |R| > |N|. So, before you can validly make your assertion you have to demonstrate that the set of things you want to map is of cardinality <= |N|. And that is only a necessary condition, not a sufficient one. Then there is the second problem, the distribution of the digits of pi. They do not seem to be random in any strictly definable sense. If they were, then the n+1th digit could be any one of the ten. In fact it is constrained to be one and one only of them. But even if they were random, they might not be normally distributed. They might instead be like the faces of a die - all equally likely, whereas if they were normally distributed the numbers most common would be 5 and 6. You are therefore making two assumptions without adequate foundation. Primarily that |N| has sufficient cardinality and secondarily that the digits are random. It's not as though all infinities were equal.

Tufr says: Continuing this thread for a moment, there is something as equally intriguing in the Pi sequence as everyone's human genome. All things being equal, sequenced in Pi is a book or a very very large set of books entitled somethlng like - the The Nature and Secrets of Universal Reality or some such thing.

Once decoded and printed the reader will find that the book is indeed the legitimate answer to man's every question about the universe, his place in it, and each and every question the reader ever had or will ever have, including what stock will jump 6 points on Tuesday and what horse will win the 5 at Belmont and the numbers for every state lottery from now to Kingdom come.

Some versions will contain a genuine moment to moment biography of every person who ever lived or will ever live. Others will contain detailed accurate instructions to create just about anything and everything that can be created, made or manufactured.

In fact, based upon the theory of Pi as expressed in this group, there are nearly an infinite number of such books. Moreover, there is at least one subset containing a personalized dedication to the specific reader from the great unknown.

For example there is at least one version specifically dedicated to Clifford Pickover, thinker extraordinare. Of course, the whole trick is simply to find it.

I suppose a very very large computer programmed with the right sequences could make a stab at it. The chances of obtaining meaningful and accurate results would probably be no greater than winning the state lottery on a date certain.

Jim says: even if one accepts Cliff's proposal that Pi contains all possible information sets, there is a key difference between the existence of *information* (whether it is PC source code, genetic code, or a coded representation of the current state of every atom in my body) and the retrieval/execution of that information. Every one of my cells contains my genetic code -- yet I don't feel bad when I get my hair cut and watch a few billion cells get swept into the trash, and I don't take any comfort in the (unlikely) possibility that one of these cells might one day be cloned.

"A little rhetorically overwrought, I think. But in any case, it depends on an open conjecture." (professional mathematician, noted skeptic and author)

"People will believe in anything if it promises immortality." (retired physicist)

"Even if there is ultimately a discreteness to everything, it still doesn't follow - the above is only true if the above are finite, and that pi is normal, which I seem to remember is still to be proved.."

"I suspect the puckish Pickover of engaging in parody here. I'm not sure of the target: John Barrow, Frank Tipler, or perhaps Max Tegmark. In any event, I thinking he's kidding, because it must be obvious that to represent is not to replicate. It must also be obvious that if a segment of pi perfectly represents our lives, it must also represent our deaths. The 'immortal' part is a nonsequitor...

Grog knows, I'm no mathematician, but I have yet to encounter a single one of these infinite replication speculations that doesn't strike me as problematic. Here's one general stumbling block:

We don't know whether organic experience can be entirely digitized. A
CD recording of Pavarotti may be a damn good representation of the
singer, but it is a sampling, not a replication. Unless you can make
a one-for-one digital representation of organic experience, you
cannot claim to replicate it. This, in a quantum world, strikes me as
unlikely. It would likely require infinite probability ranges within
a finite set of numbers representing you."

Jim Cox writes:

Perhaps I can shed some light on this matter. The property of the decimal expansion of a number containing every possible FINITE (my emphasis) sequence of digits is called (appropriately enough) being a lexicon. Many numbers are lexicons.

All Borel normal are lexicons. This includes all numbers which are truly random in a certain precise mathematical sense. It is not known if PI is a lexicon. However there are simply constructed lexicons for example 0.123456789 10 11 12 13 14 15 16 17 18 19 ... (I put the spaces for pedagogical reasons). constructed by just counting! This number will contain the library of congress infinitely often as ASCII code. 0, 1, 01, 11, 100, etc will all occur infinitely often.

Now this number is not random but contains every possible pattern of digits (finite patterns) infinitely often. It doesn't contain the square root of 2, since this is infinite. But it certainly contains Greg Bear's Blood Music infinitely often as well as every episode of Monty Python's flying circus. Basically a random real number (many equivalent definitions of random, see for example Greg Chaitin) between 0 and 1 will have this remarkable property. But I am not sure if PI is a lexicon. I believe that it is but I don't believe that it has been proven. -- Jim Cox

Martin responds: Another question about pi: Is pi normal ? It looks like e is not normal. See:

http://www.mathpages.com/home/kmath519/kmath519.htm

Chuck writes:

The square root of two is encoded in pi. Somewhere in those digits
there's a computer program that can generate the digits of the square
root of two. That's an encoding method. An encoding method doesn't
mean that the actual digits are there.
-Chuck

Pete writes:

No computer program anywhere could ever generate the digits of the square root of two or of Pi or of any other irrational or transcendental number, Chuck. So no, the square root of two is not "encoded" in Pi.

Even the vast assortment of known mathematical functions we have to "generate" Pi are not really Pi generators, in reality they can never generate more than finite approximations of the actual infinite value of Pi. Any and all of those are so trivial in content compared to the true infinite entity as to be totally negligible as true absolute and complete representations of Pi itself. They all can only physically be used to generate exactly zero percent of the true infinite string of digits comprising the exact value of Pi, and all of the known calculated results therefore actually differ from the true infinite string of digits of Pi by an infinite amount of digits.

Nothing is "encoded" in Pi except Pi itself. Finally, the term "encoding" as you use it is the same as saying that the symbol "2" is an "encoding" of the abstract mathematical value 2, which is true bit is totally trivial.

Pete B

My real point is that it is only the human mind that actually encodes anything, not the contents of patterns in abstract entities. Without humans to recognize such patterns, Pi does not encode anything, not even its own value; the **mathematical-described relationships** that involve Pi would still exist whether humans were around or not, but there would be no encoding of anything in that (or in any other such infinite string), it would simply be a case of chaos and absence of chaos (or order if you will) existing in the same universe.

A computer programmer encodes a set of instructions into a formal program. But the actual physical form of the program is either symbols of some arbitrary shape inscribed by the programmer in ink on paper or perhaps various kinds of arbitrary patterns of pixels on a computer screen stored as other arbitrary electronic states in a memory chip. The encoding is not in the symbols, not even in the pattern of those sets of symbols; it is in the human mind that runs the whole show and actively employs those otherwiuse inert meaningless patterns to accomplish some task. The only encoding is the human invention of transforming certain kinds of symbols or pixel patterns into other kinds of electronic states in order to ultimately accomplish some useful task.

Encoding is strictly a human endeavour. The symbols are merely human inventions used as tools to accomplish that function.

Pete B

Ned says: Oh God, Pi contains a scene in which I am making love to Britney Spears. Fabulous! I am now a happy man.

The discussion of our lives being encoded in pi continues on the
next page. Here, we'll also delve deeper into the mathematics
of finding ourselves in pi.
Go to Page 2 of the discussion, now! |