#1




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 (10^{20}) 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 10^{20} but we seem to closing in. Last edited by standingwave; 06192013 at 03:12 AM.. Reason: forgot to include the wiki link 
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#2




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 x^{2} = 2 has a pair of easy solutions. As for the rest of it, I got no idea. 
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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 base11 digits mixed together randomly. The chances that a random base11 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 10,000long string of only 1's and 0's by looking through just 10^20 base11 digits.
The plot then has Ellie put those 1's and 0's into a square (in the example I'm giving, a 100by100 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. 
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"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 11dimensional hyperrectangle. Silly aliens. If they wanted to send a twodimensional image, just make the length the product of two primes.

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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. 
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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. 
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The representation doesn't matter. Last edited by KarlGauss; 06192013 at 08:49 AM.. 
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Dec: 3.14159265358979323846 ... Hex: 3.243F6A8885A308D31319 ... 
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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.)

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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; 06192013 at 10:32 AM.. 
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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 
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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? 
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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.) 
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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. 
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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." 
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Someone else earlier said it had to be nonrandom based on the probability of only getting 1's and 0's in a set length. Well... perhaps it was nonrandom 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? 
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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............ 
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I struggled with integral calculus (second semester) but, strangely enough, I warmed to differential equations and infinite series. 
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Right. I understand that irrationality is being formed as a fraction, but I was speaking more to the idea that the code being nonrandom necessitates a pattern.

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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; 06192013 at 04:26 PM.. 
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Google Books has a copy of Contact, with much of it readable onscreen. Chapter 24 (page 430) has the piece that I remember:
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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". 
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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 fanwanking. 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.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 timeframes. 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. 
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In base pi, pi is 1.0000........

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It's possible to have nonpositiveinteger number bases:
http://en.wikipedia.org/wiki/Nonsta...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. 
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This is true; however it becomes impossible to accurately count how many circles you have.

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#36




You are technically correct. The best kind of correct.
In basepi, 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. 
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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. 
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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; 06202013 at 09:15 AM.. 
#39




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; 06202013 at 09:16 AM.. 
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1 ... π  Profit!!! 
#41




Well, for basepi, 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. 
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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 10_{10} is greater than pi^3 (or 100_{π}) See more discussion on the XKCD forums here: http://forums.xkcd.com/viewtopic.php?f=17&t=36351 Last edited by Colophon; 06202013 at 09:53 AM.. 
#43




Too late to edit, but I meant of course that 10 in base 10 is greater than pi SQUARED, not cubed.

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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; 06202013 at 10:35 AM.. 
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Egads. When come back, bring pi.

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"Your mathematicians have made an effort to calculate it out to..."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; 06202013 at 11:24 AM.. Reason: typo 
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In base n, the value of each position leftwards from the "decimal" point is 1, n, n^{2}, n^{3}, and so on, e.g. 123 in base 10 = (1 x 10^{2}) + (2 x 10) + (3 x 1) = 123_{10} 123 in base 6 = (1 x 6^{2}) + (2 x 6) + (3 x 1) = 51_{10} 123 in base π = (1 x π^{2}) + (2 x π) + (3 x 1) ≈ 19.153_{10} 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)... 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. 12_{10} = 0.011_{½}. Last edited by Colophon; 06202013 at 01:17 PM.. 
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