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There’s a great scene in the end of the book “Contact” by Carl Sagan where the main character uses super powerful computers to find a raster image of a circle h

There’s a great scene in the end of the book “Contact” by Carl Sagan where the main character uses super powerful computers to find a raster image of a circle hidden deep in the digits of pi. This is supposed to be a moment of deep spiritual significance. But in fact, it’s basically an expected outcome given what we think we know about pi. At first glance these images were startling to me; that such recognizable images could be “hidden” in the primes... in the very structure of mathematics! But then I realized that with an infinite numbers of primes available to us, nothing is hidden. It’s more like the primes are simply building blocks for us to use in constructing images of whatever we want.

lonelappde 7y ago
Not quite true. Sagan's version was better than that analysis suggests: http://kasmana.people.cofc.edu/MATHFICT/mf55-spoiler.html http://kasmana.people.cofc.edu/MATHFICT/mf55-spoiler.html > as Sagan puts it in the story itself, the surprise is not that it appears that it appears "so early" in the sequence.
- jon_richards 7y ago
  I’ve always assumed that was also the inspiration for one of the earliest xkcd’s: https://xkcd.com/10/ https://xkcd.com/10/
- EvanAnderson 7y ago
  I'd never read that analysis of the ending of Contact. Thanks for sharing!
- yesenadam 7y ago
  It seems there was a large (unconscious?) religious dimension to SETI. Michael Shermer: [David Swift in his 1990 book SETI Pioneers] asked each of the SETI pioneers about their parents’ religiosity... most were raised in a religious household but that not one believes in anything like the traditional Judeo-Christian God... Astronomer Frank Drake, ostensively the SETI pioneer if there was one (and creator of the infamous “Drake equation” for computing the probability of ETIs), who was raised “Very strong Baptist. Sunday school every Sunday,” made this observation: “A strong influence on me, and I think on a lot of SETI people, was the extensive exposure to fundamentalist religion. You find when you talk to people who have been active in SETI that there seems to be that thread. They were either exposed or bombarded with fundamentalist religion. So to some extent it is a reaction to firm religious upbringing”. Similarly, John Kraus recalled: “We were very strong churchgoers, members of the Methodist church. I was brought up in a very religious atmosphere…there was never any thought of conflict between science and religion in my thinking or in my upbringing. Science and religion were simply both seeking ultimate truth but using different ways of going at it”. ... ... One might speculate that SETI, as a highbrow, elitist revolution, contains within it quasi-religious and spiritual overtones, in the sense that these scientists, while not believing in God, do believe in ETIs, uniformly portrayed as higher intelligences who, having survived what might be a tendency in species toward self-destruction once advanced technologies are created, must also be morally superior. To the extent that religion involves belief in and hope for transcendence or transcendent beings, SETI is a high-cultural form of religion, and UFOs a low-cultural form of religion. The Measure Of A Life: Carl Sagan and the Science of Biography http://www.theeway.com/skepticc/newsworthy06.html http://www.theeway.com/skepticc/newsworthy06.html P.S. I was very much a child of Sagan and SETI! I was 15 when Cosmos aired, and read every Sagan book I could get my hands on, got a large telescope, became obsessed with astronomy for years... Reading about this religious dimension years later surprised me, it had never occurred to me. (My family was atheist)
  - jacobush 7y ago
    Not saying you are from that background, but I think any upbringing that beats you over the head with something akin to "who are you going to believe, your deceiving eyes or me?" will prime you for SETI or similar "religion" later in life. That beating over the head could be an abusive parent, boarding school or whatever. (Not saying most religious practice is abusive or anything. But it can be.)
  - yesenadam 7y ago
    Also, if you loved (Sagan's) Cosmos - Dimitri Martin did a spoof version, Our Fascinating Planet, which is absolutely hilarious. https://www.youtube.com/watch?v=fYo-OdQ-5ic&list=PLRcB4n4CGcy8OG9eFOjCyZgooBirO-q8G https://www.youtube.com/watch?v=fYo-OdQ-5ic&list=PLRcB4n4CGc...
  - MrQuincle 7y ago
    I have been raised extremely religious and I completely understand obsession by the search for alien life. In some way it would "ground" at once my entire struggle to get out of this (human) religion. Sure, you can imagine an adapted religion cross the universe with multiple Jesuses etc., but it's much less convincing than the original. Actually, the reason I'm so interested in artificial general intelligence is, I think, driven by the same reasons. If we can create fellow human beings, it somehow is like creating your own aliens on earth. That, and the idea that we need them to solve the tough problem we nowadays face as mankind.
unlinked_dll 7y ago
There's a corollary here about the very human nature of seeking to find order in chaos. I'm not sure how to voice it exactly but it cuts to the philosophy of mathematics, especially probability and statistics. We are beings that seek to attach order and patterns to randomness and noise even when there might be none. And when it is a pattern or recognizable slice of information within something than contains very few, if any of those slices we are astounded and ponder the greater nature of whatever we observe.
Dylan16807 7y ago
> it’s basically an expected outcome given what we think we know about pi. There is a massive gulf between searching deep into random digits to find something, down to completely unimaginable depths, entire universe-fulls of digits discarded... and having it be right there, near the surface, in one of the most common numbers that exists. The book talks about the latter. What this page is about is a third option. It's not searching through primes or other math-derived numbers. It's constructing a number that already has a pattern, which is trivial, and then it's searching nearby numbers until it hits a prime.
baddox 7y ago
The way I would describe it is that the primality of a number is almost completely unrelated to these images. The image is chosen beforehand, then a set of integers is chosen that each looks recognizably close to that image, then that set is searched for one that is prime (which is not unusual). Primality is the chosen attribute simply because it’s widely known and considered to be interesting in popular mathematics. People would be less impressed by images constructed with the binary representation of numbers divisible by 1,000,000, even though that may be a rarer property than primality for the magnitude of numbers considered.
- hwillis 7y ago
  truly it is an amazing congruence of mathematics and linguistics that one of my favorite words appears in the real series: 8008135.
  - infradig 7y ago
    It looks better on a calculator.
- throwaway744678 7y ago
  Your argument is true; though constructing images with numbers divisible by 1000000 is far easier. Take any image you want, and replace the last figures by zeroes. You'll just get a small artefact in the bottom right corner!
  - baddox 7y ago
    Not in binary, right?
jedimastert 7y ago
This is why I love the idea of the "Library of Babel". The basic idea is that it's a vast "library" of every permutation of letters and punctuation in a book. Almost every book would be complete gibberish, but there much also be every book ever written, already created and waiting to be discovered. Is any idea truly created, or simply discovered first? Or is the simple fact of having too many possibilities man that there's no possible way you can "discover" anything at all, and anything much be invented first to be looked for?
- jameshart 7y ago
  well, among other things that library contains an infinite number of proofs that P=NP. Also an infinite number of proofs of the opposite. Also an infinite number of base64-encoded FLACs of the original cast recording of a rock opera based on each of those proofs, performed by David Bowie and Justin Bieber. Obviously, an infinite number of them contain logical flaws. An infinite number of them are surprisingly listenable. You’re welcome to devote time to trying to discover them...
  - SamBam 7y ago
    > well, among other things that library contains an infinite number of proofs that P=NP. Also an infinite number of proofs of the opposite. Of course, given our current understanding of math and logic, although there would be an infinite number of proofs both for and against P=NP, only ones on one side of the argument could ever be correct, and all the others must contain logical errors, however subtle. Assuming that the proof of P=NP isn't somehow outside of human comprehension, it means that, over an infinite period of time, a human could go over all those infinite proofs and still pick out the correct answer.
criddell 7y ago
> with an infinite numbers of primes available to us, nothing is hidden Just because there's an infinite number of primes, it doesn't necessarily mean that everything is in there, does it? For example, would you expect to find Romeo and Juliet in pi? It might be there, but I don't think it has to be there, does it?
- andreareina 7y ago
  > A sequence is normal if and only if every block of equal length appears with equal frequency. (A block of length k is a substring of length k appearing at a position in the sequence that is a multiple of k: e.g. the first length-k block in S is S[1..k], the second length-k block is S[k+1..2k], etc.) This was implicit in the work of Ziv and Lempel (1978) and made explicit in the work of Bourke, Hitchcock, and Vinodchandran (2005). [1] If pi is normal (as it is conjectured to be), then yeah it has to. [1] https://en.wikipedia.org/wiki/Normal_number#Properties https://en.wikipedia.org/wiki/Normal_number#Properties
  - criddell 7y ago
    This is amazing! So somewhere pi is a computer program that when run would set up a server with an entry for every person that has ever lived or ever will live with complete information on that person. You could theoretically see a real-time movie of anybody from life until death. Crazy! Do you know if k has to be finite?
    - SamBam 7y ago
      K would have to be finite, yes, or there could never be another string after it. But k can be arbitrarily large. Yes, Normal Numbers are awesome to think about when you first learn them. Ever tiny detail of everything that has ever happened in the history of the world, written in incredibly beautiful prose that would bring the entire planet to its knees on reading it, is somewhere in the digits of pi. Of course, so are a lot of lies -- every possible variant of lie. And lots and lots and lots of junk. Indeed, you can make a program to search pi for any string of letters, and if your program is powerful enough you may find a 10 or 11 digit sequence of your choosing, but no more than that. And of course, searching for the string you pre-chose seems a lot more underwhelming than stumbling across the story of your life. https://www.angio.net/pi/ https://www.angio.net/pi/
homonculus1 7y ago
Note also that the line breaks are chosen by a human, and if you look at the bottom right corner of some of the images you can see random-looking digits that don't contribute to the image, but do contribute to the "primeness". There's the actual size of the search space, the rest of it is just picked to look that way. It's still a huge space and doesn't discount the effort of finding candidates and checking them, but it does demystify the image a bit.

lonelappde 7y ago

Not quite true. Sagan's version was better than that analysis suggests: http://kasmana.people.cofc.edu/MATHFICT/mf55-spoiler.html http://kasmana.people.cofc.edu/MATHFICT/mf55-spoiler.html > as Sagan puts it in the story itself, the surprise is not that it appears that it appears "so early" in the sequence.

jon_richards 7y ago
I’ve always assumed that was also the inspiration for one of the earliest xkcd’s: https://xkcd.com/10/ https://xkcd.com/10/
EvanAnderson 7y ago
I'd never read that analysis of the ending of Contact. Thanks for sharing!
yesenadam 7y ago
It seems there was a large (unconscious?) religious dimension to SETI. Michael Shermer: [David Swift in his 1990 book SETI Pioneers] asked each of the SETI pioneers about their parents’ religiosity... most were raised in a religious household but that not one believes in anything like the traditional Judeo-Christian God... Astronomer Frank Drake, ostensively the SETI pioneer if there was one (and creator of the infamous “Drake equation” for computing the probability of ETIs), who was raised “Very strong Baptist. Sunday school every Sunday,” made this observation: “A strong influence on me, and I think on a lot of SETI people, was the extensive exposure to fundamentalist religion. You find when you talk to people who have been active in SETI that there seems to be that thread. They were either exposed or bombarded with fundamentalist religion. So to some extent it is a reaction to firm religious upbringing”. Similarly, John Kraus recalled: “We were very strong churchgoers, members of the Methodist church. I was brought up in a very religious atmosphere…there was never any thought of conflict between science and religion in my thinking or in my upbringing. Science and religion were simply both seeking ultimate truth but using different ways of going at it”. ... ... One might speculate that SETI, as a highbrow, elitist revolution, contains within it quasi-religious and spiritual overtones, in the sense that these scientists, while not believing in God, do believe in ETIs, uniformly portrayed as higher intelligences who, having survived what might be a tendency in species toward self-destruction once advanced technologies are created, must also be morally superior. To the extent that religion involves belief in and hope for transcendence or transcendent beings, SETI is a high-cultural form of religion, and UFOs a low-cultural form of religion. The Measure Of A Life: Carl Sagan and the Science of Biography http://www.theeway.com/skepticc/newsworthy06.html http://www.theeway.com/skepticc/newsworthy06.html P.S. I was very much a child of Sagan and SETI! I was 15 when Cosmos aired, and read every Sagan book I could get my hands on, got a large telescope, became obsessed with astronomy for years... Reading about this religious dimension years later surprised me, it had never occurred to me. (My family was atheist)
- jacobush 7y ago
  Not saying you are from that background, but I think any upbringing that beats you over the head with something akin to "who are you going to believe, your deceiving eyes or me?" will prime you for SETI or similar "religion" later in life. That beating over the head could be an abusive parent, boarding school or whatever. (Not saying most religious practice is abusive or anything. But it can be.)
- yesenadam 7y ago
  Also, if you loved (Sagan's) Cosmos - Dimitri Martin did a spoof version, Our Fascinating Planet, which is absolutely hilarious. https://www.youtube.com/watch?v=fYo-OdQ-5ic&list=PLRcB4n4CGcy8OG9eFOjCyZgooBirO-q8G https://www.youtube.com/watch?v=fYo-OdQ-5ic&list=PLRcB4n4CGc...
- MrQuincle 7y ago
  I have been raised extremely religious and I completely understand obsession by the search for alien life. In some way it would "ground" at once my entire struggle to get out of this (human) religion. Sure, you can imagine an adapted religion cross the universe with multiple Jesuses etc., but it's much less convincing than the original. Actually, the reason I'm so interested in artificial general intelligence is, I think, driven by the same reasons. If we can create fellow human beings, it somehow is like creating your own aliens on earth. That, and the idea that we need them to solve the tough problem we nowadays face as mankind.

unlinked_dll 7y ago

There's a corollary here about the very human nature of seeking to find order in chaos. I'm not sure how to voice it exactly but it cuts to the philosophy of mathematics, especially probability and statistics. We are beings that seek to attach order and patterns to randomness and noise even when there might be none. And when it is a pattern or recognizable slice of information within something than contains very few, if any of those slices we are astounded and ponder the greater nature of whatever we observe.

Dylan16807 7y ago

> it’s basically an expected outcome given what we think we know about pi. There is a massive gulf between searching deep into random digits to find something, down to completely unimaginable depths, entire universe-fulls of digits discarded... and having it be right there, near the surface, in one of the most common numbers that exists. The book talks about the latter. What this page is about is a third option. It's not searching through primes or other math-derived numbers. It's constructing a number that already has a pattern, which is trivial, and then it's searching nearby numbers until it hits a prime.

baddox 7y ago

The way I would describe it is that the primality of a number is almost completely unrelated to these images. The image is chosen beforehand, then a set of integers is chosen that each looks recognizably close to that image, then that set is searched for one that is prime (which is not unusual). Primality is the chosen attribute simply because it’s widely known and considered to be interesting in popular mathematics. People would be less impressed by images constructed with the binary representation of numbers divisible by 1,000,000, even though that may be a rarer property than primality for the magnitude of numbers considered.

hwillis 7y ago
truly it is an amazing congruence of mathematics and linguistics that one of my favorite words appears in the real series: 8008135.
- infradig 7y ago
  It looks better on a calculator.
throwaway744678 7y ago
Your argument is true; though constructing images with numbers divisible by 1000000 is far easier. Take any image you want, and replace the last figures by zeroes. You'll just get a small artefact in the bottom right corner!
- baddox 7y ago
  Not in binary, right?

jedimastert 7y ago

This is why I love the idea of the "Library of Babel". The basic idea is that it's a vast "library" of every permutation of letters and punctuation in a book. Almost every book would be complete gibberish, but there much also be every book ever written, already created and waiting to be discovered. Is any idea truly created, or simply discovered first? Or is the simple fact of having too many possibilities man that there's no possible way you can "discover" anything at all, and anything much be invented first to be looked for?

jameshart 7y ago
well, among other things that library contains an infinite number of proofs that P=NP. Also an infinite number of proofs of the opposite. Also an infinite number of base64-encoded FLACs of the original cast recording of a rock opera based on each of those proofs, performed by David Bowie and Justin Bieber. Obviously, an infinite number of them contain logical flaws. An infinite number of them are surprisingly listenable. You’re welcome to devote time to trying to discover them...
- SamBam 7y ago
  > well, among other things that library contains an infinite number of proofs that P=NP. Also an infinite number of proofs of the opposite. Of course, given our current understanding of math and logic, although there would be an infinite number of proofs both for and against P=NP, only ones on one side of the argument could ever be correct, and all the others must contain logical errors, however subtle. Assuming that the proof of P=NP isn't somehow outside of human comprehension, it means that, over an infinite period of time, a human could go over all those infinite proofs and still pick out the correct answer.

criddell 7y ago

> with an infinite numbers of primes available to us, nothing is hidden Just because there's an infinite number of primes, it doesn't necessarily mean that everything is in there, does it? For example, would you expect to find Romeo and Juliet in pi? It might be there, but I don't think it has to be there, does it?

andreareina 7y ago
> A sequence is normal if and only if every block of equal length appears with equal frequency. (A block of length k is a substring of length k appearing at a position in the sequence that is a multiple of k: e.g. the first length-k block in S is S[1..k], the second length-k block is S[k+1..2k], etc.) This was implicit in the work of Ziv and Lempel (1978) and made explicit in the work of Bourke, Hitchcock, and Vinodchandran (2005). [1] If pi is normal (as it is conjectured to be), then yeah it has to. [1] https://en.wikipedia.org/wiki/Normal_number#Properties https://en.wikipedia.org/wiki/Normal_number#Properties
- criddell 7y ago
  This is amazing! So somewhere pi is a computer program that when run would set up a server with an entry for every person that has ever lived or ever will live with complete information on that person. You could theoretically see a real-time movie of anybody from life until death. Crazy! Do you know if k has to be finite?
  - SamBam 7y ago
    K would have to be finite, yes, or there could never be another string after it. But k can be arbitrarily large. Yes, Normal Numbers are awesome to think about when you first learn them. Ever tiny detail of everything that has ever happened in the history of the world, written in incredibly beautiful prose that would bring the entire planet to its knees on reading it, is somewhere in the digits of pi. Of course, so are a lot of lies -- every possible variant of lie. And lots and lots and lots of junk. Indeed, you can make a program to search pi for any string of letters, and if your program is powerful enough you may find a 10 or 11 digit sequence of your choosing, but no more than that. And of course, searching for the string you pre-chose seems a lot more underwhelming than stumbling across the story of your life. https://www.angio.net/pi/ https://www.angio.net/pi/

homonculus1 7y ago

Note also that the line breaks are chosen by a human, and if you look at the bottom right corner of some of the images you can see random-looking digits that don't contribute to the image, but do contribute to the "primeness". There's the actual size of the search space, the rest of it is just picked to look that way. It's still a huge space and doesn't discount the effort of finding candidates and checking them, but it does demystify the image a bit.