The Straight Dope

Go Back   Straight Dope Message Board > Main > General Questions

Reply
 
Thread Tools Display Modes
  #1  
Old 06-19-2013, 02:11 AM
standingwave standingwave is offline
Guest
 
Join Date: Jul 2010
Math/Probability question from Sagan/Contact (Pi)

From the novel, not the movie.
Ellie, acting upon a suggestion by the senders of the message, works on a program which computes the digits of pi to record lengths in different bases. Very far from the decimal point (1020) and in base 11, it finds that a special pattern does exist when the numbers stop varying randomly and start producing 1s and 0s in a very long string. The string's length is the product of 11 prime numbers. The 1s and 0s when organized as a square of specific dimensions form a rasterized circle. The extraterrestrials suggest that this is a signature incorporated into the Universe itself. Yet the extraterrestrials are just as ignorant to its meaning as Ellie, as it could be still some sort of a statistical anomaly. They also make reference to older artifacts built from space time itself (namely the wormhole transit system) abandoned by a prior civilization. A line in the book suggests that the image is a foretaste of deeper marvels hidden even further within pi. This new pursuit becomes analogous to SETI; it is another search for meaningful signals in apparent noise.
Now granted, if you extend any transcendental (irrational? I'm a little fuzzy on the differences) you will eventually generate, well, pretty much everything. But my question is, can we pin any factor of improbability to the strings of ones and zeros (in base 11, of all things) described in the novel? Anyone want to take a stab at it? FWIW, we haven't yet reached 1020 but we seem to closing in.

Last edited by standingwave; 06-19-2013 at 02:12 AM.. Reason: forgot to include the wiki link
Reply With Quote
Advertisements  
  #2  
Old 06-19-2013, 02:45 AM
friedo friedo is online now
Charter Member
 
Join Date: May 2000
Location: Brooklyn
Posts: 20,610
An irrational number is a number which can't be written as a fraction of integers (a ratio.) Pi, e, and the square root of two are all irrational numbers.

Transcendental numbers are numbers which can not be the solution to a polynomial equation (whose coefficients are rational.) Pi and e are transcendental numbers but √2 is not because x2 = 2 has a pair of easy solutions.

As for the rest of it, I got no idea.
Reply With Quote
  #3  
Old 06-19-2013, 03:09 AM
Wendell Wagner Wendell Wagner is offline
Charter Member
 
Join Date: Jul 1999
Location: Greenbelt, Maryland
Posts: 11,837
Suppose it were true that the expansion of pi in base 11 contained nothing except 1's and 0's for some long stretch. You would expect a random number in base 11 to contain all 11 base-11 digits mixed together randomly. The chances that a random base-11 number of length n consists of just two of those digits is thus (2/11) ^ n. Suppose that n were 10,000. The chances would be thus (2/11) ^ 10,000 = .18181818... ^ 10,000, which is around 10 ^ -7404. It is thus excruciatingly unlikely that you would find a 10,000-long string of only 1's and 0's by looking through just 10^20 base-11 digits.

The plot then has Ellie put those 1's and 0's into a square (in the example I'm giving, a 100-by-100 square) and noticing this makes a picture showing something. The problem is that (according to the Wikipedia entry) the length of the stretch of uninterrupted 1's and 0's is "the product of 11 prime numbers." Of course, a product of 11 prime numbers can not be arranged in a square. To be arranged in a square, it would have to be a square number. Obviously, the product of 11 primes is not a square number.
Reply With Quote
  #4  
Old 06-19-2013, 04:18 AM
Ximenean Ximenean is offline
Guest
 
Join Date: Aug 2001
"su

I don't know about you, but whenever I see a coded message whose length is the product of 11 prime numbers, I tend to assume that it is supposed to be arranged as an 11-dimensional hyperrectangle. Silly aliens. If they wanted to send a two-dimensional image, just make the length the product of two primes.
Reply With Quote
  #5  
Old 06-19-2013, 06:06 AM
KarlGauss KarlGauss is offline
Out of the slimy mud of words
Charter Member
 
Join Date: Mar 2000
Location: Between pole and tropic
Posts: 6,937
Would it matter in what base the analysis is conducted? If there's a pattern in base 11, would there not also be a pattern in (every) other base?

BTW, if a MOD is reading this, you may want to add a SPOILER warning to the title of the OP. The highlight of the book for me when I read Contact was how the message was 'stored' and, even more, how it was created.
Reply With Quote
  #6  
Old 06-19-2013, 06:54 AM
ZenBeam ZenBeam is offline
Charter Member
 
Join Date: Oct 1999
Location: I'm right here!
Posts: 8,517
Quote:
Originally Posted by Wendell Wagner View Post
The plot then has Ellie put those 1's and 0's into a square (in the example I'm giving, a 100-by-100 square) and noticing this makes a picture showing something. The problem is that (according to the Wikipedia entry) the length of the stretch of uninterrupted 1's and 0's is "the product of 11 prime numbers." Of course, a product of 11 prime numbers can not be arranged in a square. To be arranged in a square, it would have to be a square number. Obviously, the product of 11 primes is not a square number.
So almost a square. Like 6480 is a product of nine prime numbers, and the product of 80 and 81. A figure 80 by 81 could be referred to as a square, even though it really isn't quite a perfect square.
Reply With Quote
  #7  
Old 06-19-2013, 07:09 AM
septimus septimus is online now
Guest
 
Join Date: Dec 2009
Quote:
Originally Posted by Ximenean View Post
I don't know about you, but whenever I see a coded message whose length is the product of 11 prime numbers, I tend to assume that it is supposed to be arranged as an 11-dimensional hyperrectangle. Silly aliens. If they wanted to send a two-dimensional image, just make the length the product of two primes.
Sagan wrote Contact about the same time as the 11-dimensional M-theory of strings emerged. Connection? ... And wouldn't it better if the image were on some sort of hyper-torus, and a treasure map?
Reply With Quote
  #8  
Old 06-19-2013, 07:33 AM
yellowjacketcoder yellowjacketcoder is offline
Member
 
Join Date: Jan 2013
Location: Atlanta
Posts: 1,454
Quote:
Originally Posted by KarlGauss View Post
Would it matter in what base the analysis is conducted? If there's a pattern in base 11, would there not also be a pattern in (every) other base?

BTW, if a MOD is reading this, you may want to add a SPOILER warning to the title of the OP. The highlight of the book for me when I read Contact was how the message was 'stored' and, even more, how it was created.
You want a spoiler warning for a 28 year old book? Would you also want a spoiler if I told you
SPOILER:
Romeo and Juliet die at the end
?

as far as the base, it would make a big difference. If you're someone that needs to think in different bases (ie, if you're a programmer that occaisionally needs to read binary or hex, like me), you realize that different bases make representing certain values very different.

For example, 0.25 in decimal is 0.01 in binary. That's not so bad. But when you try to do 0.1 decimal in binary, you get 0.00011001100110011001100110011001100110011...

The representation matters.
Reply With Quote
  #9  
Old 06-19-2013, 07:49 AM
KarlGauss KarlGauss is offline
Out of the slimy mud of words
Charter Member
 
Join Date: Mar 2000
Location: Between pole and tropic
Posts: 6,937
Quote:
Originally Posted by yellowjacketcoder View Post
as far as the base, it would make a big difference. If you're someone that needs to think in different bases (ie, if you're a programmer that occaisionally needs to read binary or hex, like me), you realize that different bases make representing certain values very different.

For example, 0.25 in decimal is 0.01 in binary. That's not so bad. But when you try to do 0.1 decimal in binary, you get 0.00011001100110011001100110011001100110011...

The representation matters.
I'm not saying the pattern would be easier or harder to detect, or prettier, but, that if there's a "pattern" i.e. a non random sequence of digits basex, it will be non random in every base, i.e. there will be a "pattern" in every base. Remember, part of the purpose of the encoded message was to signal that there was an intelligence 'out there', that the digits of pi were not random but had been 'designed'. The other part of the purpose was the content of the message itself - and that would be the same regardless of the base, just like the message would be the same whether it was written in French or English.

The representation doesn't matter.

Last edited by KarlGauss; 06-19-2013 at 07:49 AM..
Reply With Quote
  #10  
Old 06-19-2013, 08:31 AM
standingwave standingwave is offline
Guest
 
Join Date: Jul 2010
Quote:
Originally Posted by KarlGauss View Post
I'm not saying the pattern would be easier or harder to detect, or prettier, but, that if there's a "pattern" i.e. a non random sequence of digits basex, it will be non random in every base, i.e. there will be a "pattern" in every base. Remember, part of the purpose of the encoded message was to signal that there was an intelligence 'out there', that the digits of pi were not random but had been 'designed'. The other part of the purpose was the content of the message itself - and that would be the same regardless of the base, just like the message would be the same whether it was written in French or English.

The representation doesn't matter.
Interesting question. I'm tempted to agree but does it? I hear what you are saying, that a long string containing only two digits would still look like something in some other base, but this is already beyond my pay grade.

Dec: 3.14159265358979323846 ...
Hex: 3.243F6A8885A308D31319 ...
Reply With Quote
  #11  
Old 06-19-2013, 08:41 AM
standingwave standingwave is offline
Guest
 
Join Date: Jul 2010
Quote:
Originally Posted by standingwave View Post
Interesting question. I'm tempted to agree but does it? I hear what you are saying, that a long string containing only two digits would still look like something in some other base, but this is already beyond my pay grade.

Dec: 3.14159265358979323846 ...
Hex: 3.243F6A8885A308D31319 ...
Missed the edit window but wanted to add, from the novel:

Quote:
The anomaly showed up most starkly in Base 11 arithmetic, where it could be written out entirely as zeros and ones. Compared with what had been received from Vega, this could be at best a simple message, but its statistical significance was high. The program reassembled the digits into a square raster, an equal number across and down.
Most starkly in base 11. For whatever that's worth.
Reply With Quote
  #12  
Old 06-19-2013, 09:11 AM
Hari Seldon Hari Seldon is offline
Guest
 
Join Date: Mar 2002
I'm not sure the base doesn't matter. A good test would be to express the number \sum{i=1}^\infty 10^{-n!}. In base ten it looks like 0.110001000000000000000001... with ones in position 1,2,6,24,...,n!,... and zeroes elsewhere. Someone might want to see what it looks like in base 11. It is, BTW, transcendental. There is a theorem that says that an irrational number that is too closely approximated by rational numbers must be transcendental. (What does too closely mean? It means too closely with respect to the denominator of the rational number approximating it.)
Reply With Quote
  #13  
Old 06-19-2013, 09:30 AM
standingwave standingwave is offline
Guest
 
Join Date: Jul 2010
Quote:
Originally Posted by Wendell Wagner View Post
Suppose it were true that the expansion of pi in base 11 contained nothing except 1's and 0's for some long stretch. You would expect a random number in base 11 to contain all 11 base-11 digits mixed together randomly. The chances that a random base-11 number of length n consists of just two of those digits is thus (2/11) ^ n. Suppose that n were 10,000. The chances would be thus (2/11) ^ 10,000 = .18181818... ^ 10,000, which is around 10 ^ -7404. It is thus excruciatingly unlikely that you would find a 10,000-long string of only 1's and 0's by looking through just 10^20 base-11 digits.

The plot then has Ellie put those 1's and 0's into a square (in the example I'm giving, a 100-by-100 square) and noticing this makes a picture showing something. The problem is that (according to the Wikipedia entry) the length of the stretch of uninterrupted 1's and 0's is "the product of 11 prime numbers." Of course, a product of 11 prime numbers can not be arranged in a square. To be arranged in a square, it would have to be a square number. Obviously, the product of 11 primes is not a square number.
Thanks. This is pretty much what I was looking for. Excruciatingly unlikely. I like that.
Reply With Quote
  #14  
Old 06-19-2013, 09:31 AM
Great Antibob Great Antibob is offline
Guest
 
Join Date: Feb 2003
Quote:
Originally Posted by standingwave View Post
Now granted, if you extend any transcendental (irrational? I'm a little fuzzy on the differences) you will eventually generate, well, pretty much everything.
Just as a side note, this is probabilistically true for any real number, but it's not been proven true for most, if any, of the interesting numbers.

Any number that satisfies this property is called a normal number (with some nomenclature on whether they are normal in one number base or all of them). We know for certain that a few constants are normal, but sqrt(2), pi, and e are not among them. We do suspect they are normal, though.

Last edited by Great Antibob; 06-19-2013 at 09:32 AM..
Reply With Quote
  #15  
Old 06-19-2013, 10:09 AM
naita naita is online now
Guest
 
Join Date: Jun 2002
Quote:
Originally Posted by KarlGauss View Post
I'm not saying the pattern would be easier or harder to detect, or prettier, but, that if there's a "pattern" i.e. a non random sequence of digits basex, it will be non random in every base, i.e. there will be a "pattern" in every base. Remember, part of the purpose of the encoded message was to signal that there was an intelligence 'out there', that the digits of pi were not random but had been 'designed'. The other part of the purpose was the content of the message itself - and that would be the same regardless of the base, just like the message would be the same whether it was written in French or English.

The representation doesn't matter.
Yes it does.
base eleven 10101010 = base ten 19649564
base eleven 11111111 = base ten 21435888

And I just tried to make a universe where pi in base 11 is 3.0110100110010110 -> random sequence of base 11 digits, which has a rasterised "circle" in the first 16 digits, and although the project failed due to excel not being able to handle more than 15 significant digits, I now know that in base ten mypi starts with 3.00902199191354245
Reply With Quote
  #16  
Old 06-19-2013, 10:23 AM
standingwave standingwave is offline
Guest
 
Join Date: Jul 2010
Quote:
Originally Posted by Great Antibob View Post
Just as a side note, this is probabilistically true for any real number, but it's not been proven true for most, if any, of the interesting numbers.

Any number that satisfies this property is called a normal number (with some nomenclature on whether they are normal in one number base or all of them). We know for certain that a few constants are normal, but sqrt(2), pi, and e are not among them. We do suspect they are normal, though.
A good point. Random doesn't necessarily mean a normal distribution, though I am often guilty of using the terms interchangeably.

Back to the novel, what would it mean really if we discovered some sort of artifact inside Pi? How could such a thing exist? As Arroway says in the novel: "You're telling me there's a message in eleven dimensions hidden deep inside the number pi? Someone in the universe communicates by... mathematics? But... help me, I'm really having trouble understanding you. Mathematics isn't arbitrary. I mean pi has to have the same value everywhere. How can you hide a message inside pi? It's built into the fabric of the universe."

Pi isn't arbitrary unlike some physical constants which may or may not be. For one example, it's also the result of several infinite series. How does a designer embed a message in that?
Reply With Quote
  #17  
Old 06-19-2013, 10:40 AM
Keeve Keeve is offline
Guest
 
Join Date: Aug 2000
Quote:
Originally Posted by standingwave View Post
Missed the edit window but wanted to add, from the novel:
Quote:
The anomaly showed up most starkly in Base 11 arithmetic, where it could be written out entirely as zeros and ones. Compared with what had been received from Vega, this could be at best a simple message, but its statistical significance was high. The program reassembled the digits into a square raster, an equal number across and down.
Most starkly in base 11. For whatever that's worth.
Yes, that is quite familiar to me, and sure sounds like an exact quote from the novel. I don't remember anything about the product of eleven prime numbers. Gonna check it when I get home.
Reply With Quote
  #18  
Old 06-19-2013, 10:52 AM
Frylock Frylock is online now
Guest
 
Join Date: Jun 2001
Quote:
Originally Posted by standingwave View Post
A good point. Random doesn't necessarily mean a normal distribution, though I am often guilty of using the terms interchangeably.

Back to the novel, what would it mean really if we discovered some sort of artifact inside Pi? How could such a thing exist? As Arroway says in the novel: "You're telling me there's a message in eleven dimensions hidden deep inside the number pi? Someone in the universe communicates by... mathematics? But... help me, I'm really having trouble understanding you. Mathematics isn't arbitrary. I mean pi has to have the same value everywhere. How can you hide a message inside pi? It's built into the fabric of the universe."

Pi isn't arbitrary unlike some physical constants which may or may not be. For one example, it's also the result of several infinite series. How does a designer embed a message in that?
I don't think it's possible. But what one could do is create a universe in which pi is very likely to come to be seen as an important number to its inhabitants.

It is possible that my ignorance is showing, but I do not think it's physically necessary for pi to fill an important physical role in just any possible physical universe. ("possible" here has to be constructed very broadly, though. A physical universe in which spheres aren't likely to strike inhabitants as important? Hard to imagine! But I don't think impossible.)
Reply With Quote
  #19  
Old 06-19-2013, 12:00 PM
septimus septimus is online now
Guest
 
Join Date: Dec 2009
Quote:
Originally Posted by standingwave View Post
Pi isn't arbitrary ... How does a designer embed a message in that?
That's what made Sagan's concept so beautiful to me! Any old Creator God can hide a message in Planck's constant, or something like that. But to hide a message in an arithmetic fact is spookily awesome.

BTW, I have mixed feelings about pi's normalcy, assuming it is. It seems nice that pi rendered in Ascii contains the Complete Works of Shakespeare and the Complete History of the U.S.A. published in the 23rd century. But ... before you ever get to them in pi's expansion, you'll have to wade through gazillions of Hamlet versions where the Prince and Ophelia elope in Act II. And before you get to the correct History, there will be incorrect versions which end suddenly with the Cuban Missile Crisis ... or in which GWB is elected to a third term in 2008.
Reply With Quote
  #20  
Old 06-19-2013, 12:23 PM
Frylock Frylock is online now
Guest
 
Join Date: Jun 2001
Quote:
Originally Posted by septimus View Post
That's what made Sagan's concept so beautiful to me! Any old Creator God can hide a message in Planck's constant, or something like that. But to hide a message in an arithmetic fact is spookily awesome.

BTW, I have mixed feelings about pi's normalcy, assuming it is. It seems nice that pi rendered in Ascii contains the Complete Works of Shakespeare and the Complete History of the U.S.A. published in the 23rd century. But ... before you ever get to them in pi's expansion, you'll have to wade through gazillions of Hamlet versions where the Prince and Ophelia elope in Act II. And before you get to the correct History, there will be incorrect versions which end suddenly with the Cuban Missile Crisis ... or in which GWB is elected to a third term in 2008.
See Borges' The Library of Babel, which I occasionally teach in my intro to philosophy course.
Reply With Quote
  #21  
Old 06-19-2013, 01:59 PM
standingwave standingwave is offline
Guest
 
Join Date: Jul 2010
Quote:
Originally Posted by septimus View Post
That's what made Sagan's concept so beautiful to me! Any old Creator God can hide a message in Planck's constant, or something like that. But to hide a message in an arithmetic fact is spookily awesome.

BTW, I have mixed feelings about pi's normalcy, assuming it is. It seems nice that pi rendered in Ascii contains the Complete Works of Shakespeare and the Complete History of the U.S.A. published in the 23rd century. But ... before you ever get to them in pi's expansion, you'll have to wade through gazillions of Hamlet versions where the Prince and Ophelia elope in Act II. And before you get to the correct History, there will be incorrect versions which end suddenly with the Cuban Missile Crisis ... or in which GWB is elected to a third term in 2008.
Yes, it is indeed an awesome concept. There's a little something for everyone in that ending. Theists can say it proves the existence of a god but the scientific approach would be to say that's it's evidence of design but proceeding to the next obvious question: where did the designer(s) come from? Is it turtles all the way down?

Sturgeon's Law applies. Of course, in pi's case, it's more like 99.999999... percent of everything is crap.

Then there's the whole pi vs. tau debate. I would have been impressed had Ellie's "father" had said, "Silly humans. Basing a constant in terms of diameter while measuring angles in terms of radii. No wonder you've made so little progress."
Reply With Quote
  #22  
Old 06-19-2013, 02:05 PM
pancakes3 pancakes3 is offline
Guest
 
Join Date: Nov 2008
Quote:
Originally Posted by KarlGauss View Post
I'm not saying the pattern would be easier or harder to detect, or prettier, but, that if there's a "pattern" i.e. a non random sequence of digits basex, it will be non random in every base, i.e. there will be a "pattern" in every base. Remember, part of the purpose of the encoded message was to signal that there was an intelligence 'out there', that the digits of pi were not random but had been 'designed'. The other part of the purpose was the content of the message itself - and that would be the same regardless of the base, just like the message would be the same whether it was written in French or English.

The representation doesn't matter.
I don't know if there was a pattern. It was just that when represented in base-11, it only used 1's and 0's and none of the other numbers which could be a huge coincidence or it could mean something more. It'd be something all together if the code wasn't just in 1's and 0's but also in 1-0-1-0-1-0-1-0... etc. alternating pattern.

Someone else earlier said it had to be non-random based on the probability of only getting 1's and 0's in a set length. Well... perhaps it was non-random but that doesn't mean it has to fit a pattern. That's the entire point behind irrational numbers, no? Pi doesn't fit a pattern yet it's certainly not a pattern - the very base from which this code is derived?
Reply With Quote
  #23  
Old 06-19-2013, 02:14 PM
Frylock Frylock is online now
Guest
 
Join Date: Jun 2001
A number can be irrational yet still follow a pattern in its decimal (or any other base) expansion. For example, this is an irrational number:

0.12345678910111213141516171819202122232425262728293031............
Reply With Quote
  #24  
Old 06-19-2013, 03:12 PM
standingwave standingwave is offline
Guest
 
Join Date: Jul 2010
Quote:
Originally Posted by Frylock View Post
A number can be irrational yet still follow a pattern in its decimal (or any other base) expansion. For example, this is an irrational number:

0.12345678910111213141516171819202122232425262728293031............
Going all the way back to undergraduate math (engineering) would it be fair to say that that series is normally distributed but (obviously) not random?

I struggled with integral calculus (second semester) but, strangely enough, I warmed to differential equations and infinite series.
Reply With Quote
  #25  
Old 06-19-2013, 03:12 PM
pancakes3 pancakes3 is offline
Guest
 
Join Date: Nov 2008
Right. I understand that irrationality is being formed as a fraction, but I was speaking more to the idea that the code being non-random necessitates a pattern.
Reply With Quote
  #26  
Old 06-19-2013, 03:24 PM
Great Antibob Great Antibob is offline
Guest
 
Join Date: Feb 2003
Quote:
Originally Posted by standingwave View Post
Going all the way back to undergraduate math (engineering) would it be fair to say that that series is normally distributed but (obviously) not random?
I'm not entirely sure what you mean by "random" and "normally distributed" (but I have a good idea).

If you mean the distribution of the digits, it's going to be uniformly distributed. Each digit is going to appear roughly the same number of times.

It's also a normal number. That particular number - 0.1234567891011121314... is known as Champerowne's constant. It's known to be normal in base ten. That is, you'll find every possible finite sequence of digits in it somewhere (which is not quite the formal definition but close enough for us).

Normally distributed (but not a 'normal number') in terms of statistics means something looks like it's distributed like a Gaussian function. In this case, the obvious candidate are the number of times each digit appears. But, as noted above, the distribution of digits is uniform in this case.

The number is also not random, if by 'random', you can't easily predict the next digit based on some pattern found in the digits of the decimal expansion.

In this sense, the digits of pi are 'random'. But of course, pi is a constant, so the decimal expansion is entirely deterministic in the sense that they are entirely calculable and predictable if you know which digit you want.

Last edited by Great Antibob; 06-19-2013 at 03:26 PM..
Reply With Quote
  #27  
Old 06-19-2013, 03:48 PM
standingwave standingwave is offline
Guest
 
Join Date: Jul 2010
Quote:
Originally Posted by Great Antibob View Post
I'm not entirely sure what you mean by "random" and "normally distributed" (but I have a good idea).

If you mean the distribution of the digits, it's going to be uniformly distributed. Each digit is going to appear roughly the same number of times.

It's also a normal number. That particular number - 0.1234567891011121314... is known as Champerowne's constant. It's known to be normal in base ten. That is, you'll find every possible finite sequence of digits in it somewhere (which is not quite the formal definition but close enough for us).

Normally distributed (but not a 'normal number') in terms of statistics means something looks like it's distributed like a Gaussian function. In this case, the obvious candidate are the number of times each digit appears. But, as noted above, the distribution of digits is uniform in this case.

The number is also not random, if by 'random', you can't easily predict the next digit based on some pattern found in the digits of the decimal expansion.

In this sense, the digits of pi are 'random'. But of course, pi is a constant, so the decimal expansion is entirely deterministic in the sense that they are entirely calculable and predictable if you know which digit you want.
Well, I'm not entirely sure what I mean by "random" which is perhaps part of the problem. It is a difficult concept. In one sense, the successive digits of pi are random but strictly speaking it's completely deterministic so it can't be considered truly random.
Reply With Quote
  #28  
Old 06-19-2013, 09:23 PM
Keeve Keeve is offline
Guest
 
Join Date: Aug 2000
Google Books has a copy of Contact, with much of it readable on-screen. Chapter 24 (page 430) has the piece that I remember:
Quote:
The anomaly showed up most starkly in Base 11 arithmetic, where it could be written out entirely as zeros and ones. Compared with what had been received from Vega, this could be at best a simple message, but its statistical significance was high. The program reassembled the digits into a square raster, an equal number across and down. The first line was an uninterrupted file of zeros, left to right. The second line showed a single numeral one, exactly in the middle, with zeros to the borders, left and right. After a few more lines, and unmistakable arc had formed, composed of ones. The simple geometrical figure had been quickly constructed, line by line, self-reflexive, rich with promise. The last line of the figure emerged, all zeros except for a single centered one. The subsequent line would be zeros only, part of the frame.

Hiding in the alternating patterns of digits, deep inside the transcendental number, was a perfect circle, its form traced out by unities in a field of noughts.
The next page is missing from the Google Books availability, but unless someone can show me wrong, I suspect that this business about "the product of eleven prime numbers" is bogus. And in fact, the very first comment on that Wikipedia article's talk page says the same thing.

On the other hand, I do have a problem with Sagan's use of the word "perfect" in the final sentence of the section I quoted. Call it a circle, but no rasterized circle can be called "perfect".
Reply With Quote
  #29  
Old 06-20-2013, 12:31 AM
standingwave standingwave is offline
Guest
 
Join Date: Jul 2010
Quote:
Originally Posted by Keeve View Post
Google Books has a copy of Contact, with much of it readable on-screen. Chapter 24 (page 430) has the piece that I remember:The next page is missing from the Google Books availability, but unless someone can show me wrong, I suspect that this business about "the product of eleven prime numbers" is bogus. And in fact, the very first comment on that Wikipedia article's talk page says the same thing.

On the other hand, I do have a problem with Sagan's use of the word "perfect" in the final sentence of the section I quoted. Call it a circle, but no rasterized circle can be called "perfect".
Yeah, conflation. Not eleven primes. Not eleven dimensions. Base eleven. (Meaning "God" has ten fingers and some other appendage he's particularly proud of?) Two primes for two dimensions, indicating "I am an array. Print me." Didn't Sagan et al use this on the Voyager and Pioneer messages?

And yes, I agree. In practice, no physical circle can be considered perfect. Only in the abstract is any circle perfect. I interpreted the passage to mean as perfect as any rasterized circle of that dimension could be but I could be fan-wanking. Here's the rest of the passage and the end of the novel, if people are concerned about spoilers... (emphasis mine which I discuss below.)
Hiding in the alternating patterns of digits, deep inside the transcendental number, was a perfect circle, its form traced out by unities in a field of noughts.

The universe was made on purpose, the circle said. In whatever galaxy you happen to find yourself, you take the circumference of a circle, divide it by its diameter, measure closely enough, and uncover a miracle--another circle, drawn kilometers downstream of the decimal point. There would be richer messages farther in. It doesn't matter what you look like, or what you're made of, or where you come from. As long as you live in this universe, and have a modest talent for mathematics, sooner or later you'll find it. It's already here. It's inside everything. You don't have to leave your planet to find it. In the fabric of space and in the nature of matter, as in a great work of art, there is, written small, the artist's signature. Standing over humans, gods, and demons, subsuming Caretakers and Tunnel builders, there is an intelligence that antedates the universe. The circle had closed.

She found what she had been searching for.

The End.
Kilometers?! Just kilometers? Am I missing something? Please to be checking my math. I've had a couple of beers. But 10E20 characters (the distance of the presumed artifact) written at 10 characters per inch is 1E20 inches. By comparison, a light year is just 3.7E17 inches. So isn't 10E20 digits half a dozen round trips to the novel's (and movie's) aforementioned Vega?

Carl just making the ultimate understatement? I think more accurately illustrating the physical distance of printing out the digits really drives home the sheer scale we're looking at here. Something Carl usually excelled at.

Aside: One of the first c programs I ever wrote was the Leibniz expansion for pi because it was a) simple and b) so horribly inefficient that even with a 4.77MHz computer the digits would appear in reasonable time-frames. It would happily run for hours before a digit stabilized. Quite fun to watch. Digits on the right, flickering. Digits on the left, nice and fixed.
Reply With Quote
  #30  
Old 06-20-2013, 02:20 AM
moriah moriah is offline
Guest
 
Join Date: Jun 1999
In base pi, pi is 1.0000........
Reply With Quote
  #31  
Old 06-20-2013, 02:44 AM
drewtwo99 drewtwo99 is offline
Member
 
Join Date: Aug 2011
Location: Houston, Texas
Posts: 8,577
Quote:
Originally Posted by moriah View Post
In base pi, pi is 1.0000........
mind BLOWN.
Reply With Quote
  #32  
Old 06-20-2013, 03:11 AM
Wendell Wagner Wendell Wagner is offline
Charter Member
 
Join Date: Jul 1999
Location: Greenbelt, Maryland
Posts: 11,837
It's possible to have non-positive-integer number bases:

http://en.wikipedia.org/wiki/Non-sta...umeral_systems

I remember in high school reading about negative integer bases (not in anything used in our high school, of course) and finding it interesting.
Reply With Quote
  #33  
Old 06-20-2013, 05:35 AM
septimus septimus is online now
Guest
 
Join Date: Dec 2009
Quote:
Originally Posted by moriah View Post
In base pi, pi is 1.0000........
Nitpick: Did you mean 10.000 ?
Reply With Quote
  #34  
Old 06-20-2013, 06:48 AM
Colophon Colophon is offline
Guest
 
Join Date: Sep 2002
Quote:
Originally Posted by moriah View Post
In base pi, pi is 1.0000........
This is true; however it becomes impossible to accurately count how many circles you have.
Reply With Quote
  #35  
Old 06-20-2013, 06:51 AM
Keeve Keeve is offline
Guest
 
Join Date: Aug 2000
Quote:
Originally Posted by standingwave View Post
Kilometers?! Just kilometers? Am I missing something? Please to be checking my math. I've had a couple of beers. But 10E20 characters (the distance of the presumed artifact) written at 10 characters per inch is 1E20 inches. By comparison, a light year is just 3.7E17 inches. So isn't 10E20 digits half a dozen round trips to the novel's (and movie's) aforementioned Vega?
The "product of 11 primes" isn't the only figment of the imagination of the author of that bogus article. 10E20 doesn't appear anywhere in Contact either. The fault is not with Sagan.
Reply With Quote
  #36  
Old 06-20-2013, 07:01 AM
Bryan Ekers Bryan Ekers is online now
Guest
 
Join Date: Nov 2000
Quote:
Originally Posted by septimus View Post
Nitpick: Did you mean 10.000 ?
You are technically correct. The best kind of correct.

In base-pi, 1.0000... equals, not too surprisingly, 1.0000....(in the more familiar base 10).

In fact, 1 = 1 in any base equal or greater than 1.

In base ½, though, where the value of ½ can be represented by the character ½, the value of 1 is represented by...

½




Yes, I think that's correct.
Reply With Quote
  #37  
Old 06-20-2013, 07:45 AM
Keeve Keeve is offline
Guest
 
Join Date: Aug 2000
Quote:
Originally Posted by Bryan Ekers View Post
In base ½, though, where the value of ½ can be represented by the character ½, the value of 1 is represented by...

½

Yes, I think that's correct.
I'm not so sure.

I'm not saying you're wrong. All I'm saying is that I get a headache trying to figure out what numerals would be used for bases lower than 1, let alone how the numbers would look.

In other words, for a base lower than 1, would "2" be a legitimate numeral? "9" isn't legitimate for base 8. In fact, even "8" isn't legitimate for base 8, just as base 10 doesn't have a numeral which represents ten. I just can't wrap my head around this whole thing at all.
Reply With Quote
  #38  
Old 06-20-2013, 08:12 AM
Bryan Ekers Bryan Ekers is online now
Guest
 
Join Date: Nov 2000
Quote:
Originally Posted by Keeve View Post
I'm not so sure.

I'm not saying you're wrong. All I'm saying is that I get a headache trying to figure out what numerals would be used for bases lower than 1, let alone how the numbers would look.

In other words, for a base lower than 1, would "2" be a legitimate numeral? "9" isn't legitimate for base 8. In fact, even "8" isn't legitimate for base 8, just as base 10 doesn't have a numeral which represents ten. I just can't wrap my head around this whole thing at all.
Well, we can arbitrarily assign whatever symbols we want, as was the case for values of 10-15 in the hexadecminal system, represented by a,b,c,d,e and f.

In a base lower than 1 (or in base 1 itself, for that matter, or indeed any base lower than 2) the symbol "2" has no meaning, just as the symbol @ isn't useful or meaningful when counting in base 10. This was satirized somewhat in an episode of Futurama where Bender, a robot, had a nightmare consising of an endless cloud of ones and zeros and was telling his human friend Fry about it:

Bender: And I think I saw a 2 in there!
Fry: Take it easy, Bender. There's no such thing as 2.


And I was actually wrong, before. In base ½, the value of ½ is represented by 10, of course. Silly mistake on my part. Chew on this, though: in that base 11 represents the value of 1½

Last edited by Bryan Ekers; 06-20-2013 at 08:15 AM..
Reply With Quote
  #39  
Old 06-20-2013, 08:15 AM
drewtwo99 drewtwo99 is offline
Member
 
Join Date: Aug 2011
Location: Houston, Texas
Posts: 8,577
What would base pi even mean, really? How would you count from 0 to pi (evidently 10.0000....) in base pi? Evidently we have 0 and 1, what number comes next? Don't tell me it's 2!

Last edited by drewtwo99; 06-20-2013 at 08:16 AM..
Reply With Quote
  #40  
Old 06-20-2013, 08:25 AM
Colophon Colophon is offline
Guest
 
Join Date: Sep 2002
Quote:
Originally Posted by drewtwo99 View Post
What would base pi even mean, really? How would you count from 0 to pi (evidently 10.0000....) in base pi? Evidently we have 0 and 1, what number comes next? Don't tell me it's 2!
0

1

...

π - Profit!!!
Reply With Quote
  #41  
Old 06-20-2013, 08:36 AM
Bryan Ekers Bryan Ekers is online now
Guest
 
Join Date: Nov 2000
Well, for base-pi, let's assume we have a symbol set of [0, 1, 2, 3], so....

0 = 0
1 = 1
2 = 2
3 = 3
10 = pi
11 = pi+1 = 4.14159...
12 = pi+2 = 5.14159...
13 = pi+3 = 6.14159..
20 = 2 x pi

It feels wrong, I admit, because the steps between the values are not even, but this will happen in any base that's not a whole number, if we want "1" to mean 1. We could declare some arbitrary iteration to make the steps equal, like pi/4...

0 = 0
1 = pi/4
2 = pi/2
3 = 3pi/4
10 = pi

I suppose in this case, the "base" is technically the denonimator. We could as easily use pi/5.

Now, try coming back with a counting scheme where the denominator is not a whole number, as in pi/e .....





Me like counting scheme.
Reply With Quote
  #42  
Old 06-20-2013, 08:48 AM
Colophon Colophon is offline
Guest
 
Join Date: Sep 2002
Quote:
Originally Posted by drewtwo99 View Post
What would base pi even mean, really? How would you count from 0 to pi (evidently 10.0000....) in base pi? Evidently we have 0 and 1, what number comes next? Don't tell me it's 2!
Seriously, though, you wouldn't "count to pi" in a base-pi numbering system any more than you would try to count to pi in base 10. If you're counting in integers, you count in integers - it's just that any integer above 3 in base pi is going to be written as a "decimal" (pimal?) expansion:

1

2

3

10.2201... (which means, of course, "(1 x pi) + (0 x 1) + (2 / pi) + (2 / pi^2) + (0 / pi^3) + (1 / pi^4) + ...")

11.2201...

12.2201...

20.2021...

21.2021...

22.2021...

30.1212... (or this could be written as 100.1212..., given that 1010 is greater than pi^3 (or 100π)


See more discussion on the XKCD forums here: http://forums.xkcd.com/viewtopic.php?f=17&t=36351

Last edited by Colophon; 06-20-2013 at 08:53 AM..
Reply With Quote
  #43  
Old 06-20-2013, 08:54 AM
Colophon Colophon is offline
Guest
 
Join Date: Sep 2002
Too late to edit, but I meant of course that 10 in base 10 is greater than pi SQUARED, not cubed.
Reply With Quote
  #44  
Old 06-20-2013, 09:35 AM
drewtwo99 drewtwo99 is offline
Member
 
Join Date: Aug 2011
Location: Houston, Texas
Posts: 8,577
Doh! Colophan, you are absolutely right. I am not sure why I was thinking "count to pi" in base pi would be particularly difficult. Counting to 10 in base 10 is nice because 10 is an integer and as you said, when we talk about "counting to" something we mean integers.

And so, counting in base pi upwards isn't nearly as bad as I thought it was. Thank you for making it so simple!

Last edited by drewtwo99; 06-20-2013 at 09:35 AM..
Reply With Quote
  #45  
Old 06-20-2013, 10:19 AM
Clothahump Clothahump is offline
Charter Member
 
Join Date: May 2000
Location: Houston, TX
Posts: 11,414
Egads. When come back, bring pi.
Reply With Quote
  #46  
Old 06-20-2013, 10:21 AM
standingwave standingwave is offline
Guest
 
Join Date: Jul 2010
Quote:
Originally Posted by Keeve View Post
The "product of 11 primes" isn't the only figment of the imagination of the author of that bogus article. 10E20 doesn't appear anywhere in Contact either. The fault is not with Sagan.
Actually, it's not entirely inaccurate. I think part of the confusion was Ellie's "father's" enigmatic way of encouraging her to take a look at pi. But speaking vaguely and metaphorically. What he told her she might find (eleven dimensions, base ten, etc.) was in fact not what she found... Quoting from the novel: (emphasis mine)
"Your mathematicians have made an effort to calculate it out to..."

Again she felt the tingle.

"... none of you seem to know.... Let's say the ten-billionth place. You won't be surprised to bear that other mathematicians have gone further. Well, eventually--let's say it's in the ten- to-the-twentieth-power place--something happens. The randomly varying digits disappear, and for an unbelievably long time there's nothing but ones and zeros."

Idly, he was tracing a circle out on the sand with his toe. She paused a heartbeat before replying.

"And the zeros and ones finally stop? You get back to a random sequence of digits?" Seeing a faint sign of encouragement from him, she raced on.

"And the number of zeros and ones? Is it a product of prime numbers?"

"Yes, eleven of them."

"You're telling me there's a message in eleven dimensions hidden deep inside the number pi? Someone in the universe communicates by... mathematics? But... help me, I'm really having trouble understanding you. Mathematics isn't arbitrary. I mean pi has to have the same value everywhere. How can you hide a message inside pi? It's built into the fabric of the universe."

"Exactly." She stared at him.

"It's even better than that," he continued. "Let's assume that only in base-ten arithmetic does the sequence of zeros and ones show up, although you'd recognize that something funny's going on in any other arithmetic. Let's also assume that the beings who first made this discovery had ten fingers. You see how it looks? It's as if pi has been waiting for billions of years for ten-fingered mathematicians with fast computers to come along. You see, the Message was kind of addressed to us."

"But this is just a metaphor, right? It's not really pi and the ten to the twentieth place? You don't actually have ten fingers."

"Not really." He smiled at her again. "Well, for heaven's sake, what does the Message say?" He paused for a moment, raised an index finger, and then pointed to the door.
For those who haven't read the novel, the tingling Ellie feels in her brain is where the alien is raping her mind.

Last edited by standingwave; 06-20-2013 at 10:24 AM.. Reason: typo
Reply With Quote
  #47  
Old 06-20-2013, 11:19 AM
Robot Arm Robot Arm is offline
Guest
 
Join Date: Jun 2000
Quote:
Originally Posted by septimus View Post
That's what made Sagan's concept so beautiful to me! Any old Creator God can hide a message in Planck's constant, or something like that. But to hide a message in an arithmetic fact is spookily awesome.

BTW, I have mixed feelings about pi's normalcy, assuming it is. It seems nice that pi rendered in Ascii contains the Complete Works of Shakespeare and the Complete History of the U.S.A. published in the 23rd century. But ... before you ever get to them in pi's expansion, you'll have to wade through gazillions of Hamlet versions where the Prince and Ophelia elope in Act II. And before you get to the correct History, there will be incorrect versions which end suddenly with the Cuban Missile Crisis ... or in which GWB is elected to a third term in 2008.
Quote:
Originally Posted by Frylock View Post
See Borges' The Library of Babel, which I occasionally teach in my intro to philosophy course.
I've always felt that this was a particularly poignant approach to the subject.
Reply With Quote
  #48  
Old 06-20-2013, 11:27 AM
filmore filmore is offline
Guest
 
Join Date: Aug 2002
Quote:
Originally Posted by septimus View Post
BTW, I have mixed feelings about pi's normalcy, assuming it is. It seems nice that pi rendered in Ascii contains the Complete Works of Shakespeare and the Complete History of the U.S.A. published in the 23rd century. But ... before you ever get to them in pi's expansion, you'll have to wade through gazillions of Hamlet versions where the Prince and Ophelia elope in Act II. And before you get to the correct History, there will be incorrect versions which end suddenly with the Cuban Missile Crisis ... or in which GWB is elected to a third term in 2008.
Is pi special in this regard? Is this true for any irrational number? Could the same thing be said for sqrt(2) or sqrt(3)?
Reply With Quote
  #49  
Old 06-20-2013, 11:54 AM
standingwave standingwave is offline
Guest
 
Join Date: Jul 2010
Quote:
Originally Posted by friedo View Post
An irrational number is a number which can't be written as a fraction of integers (a ratio.) Pi, e, and the square root of two are all irrational numbers.

Transcendental numbers are numbers which can not be the solution to a polynomial equation (whose coefficients are rational.) Pi and e are transcendental numbers but √2 is not because x2 = 2 has a pair of easy solutions.
So, if I understand this correctly, are all transcendental numbers irrational but not all irrationals are transcendental? (This EE never really got into number theory. I see an RLC circuit—essentially a differential equation—and I throw Laplace transforms at it until it submits. ) Because a picture is worth a thousand words I went looking for a venn diagram and found this. Accurate?
Reply With Quote
  #50  
Old 06-20-2013, 12:14 PM
Colophon Colophon is offline
Guest
 
Join Date: Sep 2002
Quote:
Originally Posted by Bryan Ekers View Post
And I was actually wrong, before. In base ½, the value of ½ is represented by 10, of course. Silly mistake on my part. Chew on this, though: in that base 11 represents the value of 1½
Can you actually have a meaningful number system with a base of less than 1?

In base n, the value of each position leftwards from the "decimal" point is 1, n, n2, n3, and so on, e.g.

123 in base 10 = (1 x 102) + (2 x 10) + (3 x 1) = 12310

123 in base 6 = (1 x 62) + (2 x 6) + (3 x 1) = 5110

123 in base π = (1 x π2) + (2 x π) + (3 x 1) ≈ 19.15310

How do you extend this to base ½? The value of each position would decrease as you moved leftwards, and increase as you moved rightwards:

Code:
 X    X    X    X    X    X    .     X       X       X

½^5  ½^4  ½^3  ½^2   ½    1        1/½   1/(½^2)  1/(½^3)...
Seems to me that is basically like a mirror image of binary, with numbers greater than 1 being written to the right of the "decimal point".

Based on the above, you would write 9 (base 10) as 1.001 (base ½). Am I right?

I.e., take the binary number, reverse the bits and stick a point after the initial bit.

1210 = 0.011½.

Last edited by Colophon; 06-20-2013 at 12:17 PM..
Reply With Quote
Reply



Bookmarks

Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is Off
HTML code is Off

Forum Jump


All times are GMT -5. The time now is 11:22 AM.


Powered by vBulletin® Version 3.8.7
Copyright ©2000 - 2014, vBulletin Solutions, Inc.

Send questions for Cecil Adams to: cecil@chicagoreader.com

Send comments about this website to: webmaster@straightdope.com

Terms of Use / Privacy Policy

Advertise on the Straight Dope!
(Your direct line to thousands of the smartest, hippest people on the planet, plus a few total dipsticks.)

Publishers - interested in subscribing to the Straight Dope?
Write to: sdsubscriptions@chicagoreader.com.

Copyright © 2013 Sun-Times Media, LLC.