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. :rolleyes:
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.”
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?
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.
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.
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.
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.
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.
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
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!
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.