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Valheru
08 Jun 2009, 09:27 AM
WARNING: Massive Spoiler Ahead

To those of you who have read Carl Sagan's novel Contact will recall that, towards the end, during Ellie's discussion with the E.T.I. avatar, she directs a question of what fills THEM with religious/numinous awe.

The answer she is given is that, if one divides a circle's circumference with its diameter, the result is a value that she is already familiar with: π. What fills THEM with awe, is that if you calculate the fractional digits long and deep enough, the digit series switches to a massive run of ones and zeros that is statistically anomalous. In other words, there is a message "hidden" inside the mantissa of π, not unlike the message sent to Earth that allowed us to build "The Machine" in the first place.

The ETI is somewhat cryptic about the discovery, merely stating that they are "working on it".

When Ellie returns to Earth, and the entire project is discredited, she "hermits" herself to the Project Argus Radio installation, and works on algorithms to calculate the mantissa of π. Watched all the time, she smuggles copies of the program out to Palmer Joss, who has it implemented and whereby the "message" hidden is finally "cracked".

As it turns out, the message is encoded in "eleventary" (base 11) :D, and plotting the bits on a square grid draws a perfect circle. The idea is that the universe is designed, seeing as π is a fundamental "magic number" that is integral (sorry) to the universe itself. That it contains a message itself implies that our universe is intentional and "designed", and that the message itself is a message from a pantheistic "god".

Spoiler semi-ends

Now, I've never read any commentary of Mr. Sagan on this little piece, but this punch-line of the novel filled me with a bit of disappointment. My gut feel is that pi being a non-irrational-perhaps-non-infinite (at the very least in base 10 or 11) is highly unlikely. That it contains a message - even less likely.

What I can grok though, is how the mathematical magic of pi's relation to "circularity" might create a sort of echo when you attempt to represent it with digits. Anybody who has used a HEX editor and looked at the raw data behind graphics might understand what I'm trying to convey - there is a suggestion of the end product even though it's not the representation of the end product at all.

My feeling is that Mr. Sagan is multiplying entities, despite the beauty and elegance of the rest of the book. Is it plausible that our universe is intentional and designed? Certainly it is, but then it's subject to the same creaçionista implosion of logic - i.e. who designed the designer, and so on

To me... any "intentionality" to our universe is implicit, not explicit, which would otherwise make the word "intentional" a non sequitur.

To end off, I'd like to share this little entry of the Jargon File (http://catb.org/jargon/html/index.html), which is an entry to the IOCC (International Obfuscated C Code Contest). The IOCCC is a competition amongst hackers to create the most elegant-yet-hard-to-understand program in C. This particular entry is deep, dark, black voodoo magic. It calculates the value of π as a return value by looking at its own surface area. The larger you make the "program", the more accurate the return value is. The program instructions are in the shape of a circle. In other words... a circle encodes a value, and yet a series of digits encode the circle... and the value of pi...... :argh:

Behold:


#define _ -F<00||--F-OO--;
int F=00,OO=00;main(){F_OO();printf("%1.3f\n",4.*-F/OO/OO);}F_OO()
{
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}

Eudaimonist
08 Jun 2009, 11:25 AM
As I see it, Carl Sagan's point was that "spiritual" awe and wonder may be found in the natural universe, and even in mathematics. He wanted to spiritualize science.


eudaimonia,

Mark

Valheru
08 Jun 2009, 01:19 PM
I think he was highly successful in accomplishing that!

lpetrich
08 Jun 2009, 02:16 PM
The problem with that hypothesis is that the value of pi is fixed by logical necessity, meaning that it cannot be designed.

For example, one can find pi with

pi = 4*arctan(1) = 4*(1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...)

This is not a very efficient way of calculating pi, but it illustrates the principle.


The most famous early calculator of pi, Archimedes, had discovered that pi is bounded by

xn/sqrt(1 + xn2) < pi/(2n*m) < xn

for some m, where the initial value x0 is given by

x0 = tan(180d/m)

and where each new xn is calculated from

xn+1 = xn/(1 + sqrt(1 + xn2))

I've crunched the numbers for it, and I've found that it converges very quickly. Some simple initial conditions:

m = 3 (triangle): x0 = sqrt(3)
m = 4 (square): x0 = 1

So you can now calculate pi without doing any "trig" -- it's all implicit, and you are essentially calculating

xn = tan(pi/(3*2n))

Ray Moscow
08 Jun 2009, 02:23 PM
Sometimes I think that lpetrich is God, or at least a being of a higher order than us mere humans. :)

Eudaimonist
08 Jun 2009, 02:28 PM
I personally think that pi is a natural and expected result of dealing with a shape in 2+ dimensions, as long as one assumes that those dimensions are perfectly flat and the shape is a mathematical circle (that it has a constant radius). If one assumes these, pi must be what it is. Just what would the universe have to be like in order for pi to not appear designed?

What would have been far more interesting is if the curvature of the universe led to simple shapes that reveal surprising properties that hint at design.


eudaimonia,

Mark

Notta
08 Jun 2009, 03:04 PM
You silly men.

pi = 3

The Bible clearly says so. In 1897, the Indiana House of Representatives unanimously passed a measure redefining the area of a circle and the value of pi. (http://www.snopes.com/religion/pi.asp)

Stand by the Bible (http://gospelofreason.wordpress.com/2007/06/13/god-said-pi-3-stand-by-your-beliefs-dammit/), dammit!!

Ray Moscow
08 Jun 2009, 03:19 PM
God is more of a "ball park" kind of deity, I think.

2 Peter 3:8: But, beloved, be not ignorant of this one thing, that one day is with the Lord as a thousand years, and a thousand years as one day.

OK, so he's off by several orders of magnitude. He's got bigger things to worry about than the value of pi.

His Noodly Appendage
08 Jun 2009, 04:49 PM
There was a little SF story I read once, involving agents traveling between infinitely-parallel universes.

They navigated by means of pi-meters, which measured the local value of pi...

Valheru
09 Jun 2009, 05:31 AM
There was a little SF story I read once, involving agents traveling between infinitely-parallel universes.

They navigated by means of pi-meters, which measured the local value of pi...

You're talking about Greg Bear's Eon and Eternity?

Valheru
09 Jun 2009, 05:37 AM
The problem with that hypothesis is that the value of pi is fixed by logical necessity, meaning that it cannot be designed.

That's EXACTLY the conundrum the ETIs face in the novel - logically it's not possible, but the proof is in the pudding.

That's part of my disappointment - Mr. Sagan knew better than to force the idea by merely writing it into being.

premjan
09 Jun 2009, 05:44 AM
Could be the ratio of circumference to diameter of a circle. That would vary.

Valheru
09 Jun 2009, 05:52 AM
Could be the ration of diameter to circumference of a circle. That would vary.

In Euclidean space, the ratio holds regardless of the size of the circle. It's not a reflection of proportions, but of relations.

I think the value of pi is only meaningful in Euclidean space anyway, though, but I could be wrong.

Codec
09 Jun 2009, 09:08 AM
I think in some ways its a failure of imagination. We don't know how to travel faster than the speed of light, or how to go back in time (both of which are touched on is cosmos), or change the value of pi. If you don't explore these ideas and test them, they will remain limits.
Whilst the EIT had managed the first two, they were still in awe of beings that could do the third. I think thats what Carl Sagan was trying to show here, an idea that seems so impossible you might think it nonsense, much like flying in the air might be to people only 200 years ago.

DMB
09 Jun 2009, 02:35 PM
Here's one for number freaks. This was a question in a Cambridge University maths entrance paper many years ago (well, over 40 anyway):

Tom: Pi is the most important number in mathematics.

Dick: No, e is.

Discuss.

macronencer
10 Jun 2009, 12:43 AM
Nobody has mentioned "normal numbers" yet. From what I can remember, pi has not been shown either to be or not to be "normal" (yet). If it were "normal" that would mean that the digits would contain EVERY possible finite sequence imaginable - in which case, not only the circle described but also any other graphic you could think of (including some rather rude ones) would also be present. That would shed rather different light on Sagan's plot. In fact, it could almost imply that it was a mischievous joke: an idea that I rather like!

HOWEVER... nobody has proved pi is normal, as I said. So this isn't necessarily the case, although many mathematicians seem to have their money on normality, so it's a fair bet.

Having said all that, for a sufficiently small sequence (say, the expression in binary of a circle in a 32x32 grid), the possibility of its being included in the sequence by chance (rather than because pi is normal) would be expected to be a little higher.

Make of all that what you will, but in my view it does at least raise the possibility that the "god" hypothesis is incorrect.

macronencer
10 Jun 2009, 12:51 AM
Here's one for number freaks. This was a question in a Cambridge University maths entrance paper many years ago (well, over 40 anyway):

Tom: Pi is the most important number in mathematics.

Dick: No, e is.

Discuss.

I like pi e - yum!

Well of course the famous equality linking the five most important numbers in maths (0, 1, i, pi and e) is Euler's Identity:

e ^ (i pi) + 1 = 0

When I was at university the maths department noticeboard one day contained the slogan:

"Maths lecturers are number - e ^ (i pi)"

Some student had written next to this:

"pi i-ed"

;)

His Noodly Appendage
10 Jun 2009, 04:36 AM
There was a little SF story I read once, involving agents traveling between infinitely-parallel universes.

They navigated by means of pi-meters, which measured the local value of pi...

You're talking about Greg Bear's Eon and Eternity?

Possibly. I really can't remember.

DMB
10 Jun 2009, 09:03 AM
Here's one for number freaks. This was a question in a Cambridge University maths entrance paper many years ago (well, over 40 anyway):

Tom: Pi is the most important number in mathematics.

Dick: No, e is.

Discuss.

I like pi e - yum!

Well of course the famous equality linking the five most important numbers in maths (0, 1, i, pi and e) is Euler's Identity:

e ^ (i pi) + 1 = 0

When I was at university the maths department noticeboard one day contained the slogan:

"Maths lecturers are number - e ^ (i pi)"

Some student had written next to this:

"pi i-ed"

;)

I am totally in love with Euler's identity. It almost gives me orgasms.

Eudaimonist
10 Jun 2009, 10:52 AM
Here's one for number freaks. This was a question in a Cambridge University maths entrance paper many years ago (well, over 40 anyway):

Tom: Pi is the most important number in mathematics.

Dick: No, e is.

Discuss.

I would have said "i" is most important, but go figure. :)


eudaimonia,

Mark

DMB
10 Jun 2009, 01:04 PM
Here's one for number freaks. This was a question in a Cambridge University maths entrance paper many years ago (well, over 40 anyway):

Tom: Pi is the most important number in mathematics.

Dick: No, e is.

Discuss.

I would have said "i" is most important, but go figure. :)


eudaimonia,

Mark

But Euler brings them all in!

Ray Moscow
10 Jun 2009, 01:07 PM
Weren't such things what Spinoza considered the "eternal"? Their contemplation was our participation in "eternity".

Works for me.

Eudaimonist
12 Jun 2009, 08:18 AM
But Euler brings them all in!

So I see. Very cool.


eudaimonia,

Mark

HinduWoman
12 Jun 2009, 12:01 PM
Sometimes I think that lpetrich is God, or at least a being of a higher order than us mere humans. :)

Ipetrich is Sysadmin so that means a higher order of humans! ;)

HinduWoman
12 Jun 2009, 12:05 PM
Sometimes I think that lpetrich is God, or at least a being of a higher order than us mere humans. :)

Ipetrich is Sysadmin so that means a higher order of humans! ;)

You realize that this means we both know there is a God/Higher Order, but are:eek: just denying it out of stubbornness?

Valheru
15 Jun 2009, 06:34 AM
Sometimes I think that lpetrich is God, or at least a being of a higher order than us mere humans. :)

Ipetrich is Sysadmin so that means a higher order of humans! ;)

You realize that this means we both know there is a God/Higher Order, but are:eek: just denying it out of stubbornness?


The SysAdmin is eternally constrained by the vBulletin version number.