Does Pi terminate or never?

[sol invictus]

The question becomes, can an ultimate creator fashion a universe where the laws of mathematics operate differently? Philosophically speaking I think so. I could probably describe a few myself. They wouldn't be very interesting universes, but that's not necessary for the point. On the other hand, a universe where pi works out mathematically the same except for a carefully crafted variation in the number sequence would function nicely.

What I like about Sagan's idea is that it's logically possible, actually - just extremely unlikely if there's no god. That is, suppose it is the case (as everyone seems to believe) that pi is normal. That means that every finite sequence occurs in it infinitely many times.

So whatever the message is, it's in there somewhere, and all the creator has to do is make sure it occurs near enough to the beginning that we have a chance of noticing it by computing enough digits of pi. Since any sequence can be a significant message in some code, it's just a matter of creating life that will see some relatively early sequence as significant.

Off the top of my head, one example would be a string of a million 1's in the base 10 decimal expansion of pi. The frequency with which that would occur is something like 10^-1,000,000 if pi is normal, so if we find it it would be pretty good proof of the existence of god. But if we had 6 fingers, or 2, or 12, and used a number system with a different base, we might not notice it.

So here's a question for the mathematicians - if the base ten decimal expansion of pi has a string of a million 1s starting, say, at the trillionth digit, would we be likely to notice if we used a base 7 number system (and had computed pi to well past that point)?

 

[Robin]

The question becomes, can an ultimate creator fashion a universe where the laws of mathematics operate differently? Philosophically speaking I think so.

Philosophically speaking, I would say definitely no. Mathematics would be the same in any possible universe or any possible mode of existence.

Of course the symbol system used to work out maths could be different - that would not even require a different universe. We could work out a complete mathematics based on a different set of axioms. But those different systems would also work identically in all possible universes.

 

[Robin]

Do the measurement on a surface that's not flat. By adjusting the amount of curve to the surface you adjust the ratio. Spacetime is curved. All the omnipotent being needs to do is make sure that spacetime is curved to exactly the right degree to make the ratio required.

I am guessing.

But then that wouldn't be Pi, that would be something else.

Pi is defined for a plane - a cartesian product of reals.

 

[orange31]

So here's a question for the mathematicians - if the base ten decimal expansion of pi has a string of a million 1s starting, say, at the trillionth digit, would we be likely to notice if we used a base 7 number system (and had computed pi to well past that point)?

Generally yes, and specifically the answer would be "42".

Thanks Dr K for the math lesson-
----Helpful thought for the math invalids like me, the expression "rational" number simply means ratio , as in the number can be expressed as the ratio of two integers. "Irrational number" has nothing to do with rational/ logical, or being far out like 'imaginary' numbers, etc.
A 'duh' point, but I needed it.

 

[ddt]

Am I able to practice any of these non-standard bases in Excel?

I don't know, I don't use Excel. I tried in OpenOffice Calc to use a non-integer base for the BASE function, and it didn't work - that is, I tried using 2.5 as base and it gave the answer as if I had given it 2 as base. You could program the algorithm on that wiki page into a VBScript function, or whatever it's called.

By the way, phi is irrational because of the use of a circle's radius to determine some of its dimensions, if I read that correctly.

What do you mean with "some of its dimensions"? phi is a number, full stop.

It's irrational because sqrt(5) is irrational. It's obvious that if sqrt(5) were rational, then phi = (1+sqrt(5))/2 were also rational; and conversely, if phi were rational, than sqrt(5) were also rational.

Proof that sqrt(5) is not rational is pretty easy, and goes by contradiction.
Suppose sqrt(5) is rational, and it is equal to p/q where p and q are both (positive) integers. Then simplify this fraction to p'/q' where p' and q' have no (prime) factors in common.
Then we have: sqrt(5) = p' / q' and when we square that equation, and move q' to the other side, we get
5 * q' * q' = p' * p'
The main theorem of number theory states that factorization into prime numbers is unique. So p' * p' must contain a factor 5, and therefore p' must contain a factor 5 and therefore, p' * p' must contain at least two factors 5. That, in its turn, implies that q' * q' must contain at least one factor 5, and therefore q' contains a factor 5. However, we started out with saying that p' and q' had no common factors, and we have now established they both have a factor 5. Contradiction.
Ergo, sqrt(5) is not rational.

A base-phi system might be just what I'm looking for since I have planned out many parts of a new nation-state. The flag is proportioned according to the golden ratio already. So trying a system of weights and measures in base-phi might be the logical next step.

The idea of a system of weights and measures in base-phi is even less practical than imperial weights and measures with its haphazard factors of 3, 12, 14, 16 and what have you.

 

[Robin]

What I like about Sagan's idea is that it's logically possible, actually - just extremely unlikely if there's no god. That is, suppose it is the case (as everyone seems to believe) that pi is normal. That means that every finite sequence occurs in it infinitely many times.

So whatever the message is, it's in there somewhere, and all the creator has to do is make sure it occurs near enough to the beginning that we have a chance of noticing it by computing enough digits of pi.

I don't think it is logically possible. If some intelligence could alter the digits of pi to make this sequence close to the beginning then it could make the sequence right at the beginning, thus this intelligence could make a pi that was 2.1419 for example.

Then the immediate operations that produce these digits would also have to be different and the definitions upon which pi was based would have to be different and going right back the axioms would have to be different - it would end up not being pi at all and not the ratio between the diameter and circumference of a plane circle.

 

[sol invictus]

I don't think it is logically possible.

It obviously is.

If some intelligence could alter the digits of pi to make this sequence close to the beginning then it could make the sequence right at the beginning, thus this intelligence could make a pi that was 2.1419 for example.

That's not what I suggested. We know that isn't the case, so you're discussing some sort of bizarre countermathfactual.

Then the immediate operations that produce these digits would also have to be different and the definitions upon which pi was based would have to be different and going right back the axioms would have to be different - it would end up not being pi at all and not the ratio between the diameter and circumference of a plane circle.

But that isn't the case, and it's not what I suggested (or Sagan for that matter).

Again - as far as we know, to the best of our knowledge, pi may contain a sequence of 10^6 1's starting from the trillionth digit of its base 10 decimal expansion.

There is nothing logically inconsistent about that; it may be true and we may discover it to be true in a decade. However, the odds of it being true - assuming pi is normal - are something like 10^{-1,000,000}. Therefore if it does turn out to be true, we might try to interpret it as a proof of the existence of god.

Why? Because god, having noticed that the transcendental number we call pi has this interesting feature in its base 10 decimal expansion, could have created us with 10 digits on our hands and in a world where the ratio of circumference to diameter of a Euclidean circle has special significance, so that we would notice this incredible coincidence and worship her.

This was just an example; you can think of others (and Sagan's was different as far as I can recall). But the basic idea is simply that god created us so that we would find something we interpret as unusual in the expansion of pi. That's clearly not logically inconsistent.

 

[Robin]

It obviously is.

Nothing obvious about it, the way you put it.

But that isn't the case, and it's not what I suggested (or Sagan for that matter).

Again - as far as we know, to the best of our knowledge, pi may contain a sequence of 10^6 1's starting from the trillionth digit of its base 10 decimal expansion.

There is nothing logically inconsistent about that; it may be true and we may discover it to be true in a decade. However, the odds of it being true - assuming pi is normal - are something like 10^{-1,000,000}. Therefore if it does turn out to be true, we might try to interpret it as a proof of the existence of god.

Why? Because god, having noticed that the transcendental number we call pi has this interesting feature in its base 10 decimal expansion, could have created us with 10 digits on our hands and in a world where the ratio of circumference to diameter of a Euclidean circle has special significance, so that we would notice this incredible coincidence and worship her.

This was just an example; you can think of others (and Sagan's was different as far as I can recall). But the basic idea is simply that god created us so that we would find something we interpret as unusual in the expansion of pi. That's clearly not logically inconsistent.

Well here is what you said:

So whatever the message is, it's in there somewhere, and all the creator has to do is make sure it occurs near enough to the beginning that we have a chance of noticing it by computing enough digits of pi.

So you cannot blame me for misinterpreting this to mean that you were saying that the creator could make the digits of pi different to what they are.

 

[Towlie]

Here's something I've always wondered about Pi: Why do we use the circumference divided by the diameter and not the circumference divided by the radius? I began to wonder about that when I noticed how many times I encountered the expression "2π", such as the fact that there are 2π radians in a circle.

 

[Robin]

As I recall Ellie finds a rasterised circle made of 1's and 0's in the first 10^20 digits of pi.

But she is trying various different bases and she can arrange the 1's and 0's found in any row/column combination.

So I wonder what the odds really are that for some base, for some combination of row/column there might be a rasterised circle of some diameter in the first 10^20 digit expansion.

 

[sol invictus]

So you cannot blame me for misinterpreting this to mean that you were saying that the creator could make the digits of pi different to what they are.

"Make the digits of pi different to what they are": as far as I can parse that it's an oxymoron, so I can indeed blame you for misinterpreting what I said that way.

 

[sol invictus]

As I recall Ellie finds a rasterised circle made of 1's and 0's in the first 10^20 digits of pi.

But she is trying various different bases and she can arrange the 1's and 0's found in any row/column combination.

So I wonder what the odds really are that for some base, for some combination of row/column there might be a rasterised circle of some diameter in the first 10^20 digit expansion.

Essentially 1 - all you'd need to do is find

...010...
...101...
...010...

somewhere. But presumably she found a really big one, or something.

 

[gnome]

But she is trying various different bases and she can arrange the 1's and 0's found in any row/column combination.

Not quite--she was looking for a sequence that had a length that was the product of two primes... it lends itself to only one way of lining them up. This (we hope) is the same way a sequential binary signal could self-identify as a two-dimensional image for signals we send for aliens to receive... one that was the product of three primes could be examined as a three-dimensional diagram or (more likely) as a two-dimensional movie.

 

[gnome]

Here's something I've always wondered about Pi: Why do we use the circumference divided by the diameter and not the circumference divided by the radius? I began to wonder about that when I noticed how many times I encountered the expression "2π", such as the fact that there are 2π radians in a circle.

Honestly it's just a convention I think.

Plus using the diameter we get (e^(pi*i))+1=0, which is pretty nifty.

 

[gnome]

This was just an example; you can think of others (and Sagan's was different as far as I can recall). But the basic idea is simply that god created us so that we would find something we interpret as unusual in the expansion of pi. That's clearly not logically inconsistent.

It isn't, but it is different from what was presented in the book, which implied heavily that Sagan's "God" set the value of PI so that the "message" would be in it. In fact, it appeared in base eleven according to the book.

he was implying intelligence in the creation of the universe, not a personal creator for humans.

 

[gnome]

I don't think it is logically possible. If some intelligence could alter the digits of pi to make this sequence close to the beginning then it could make the sequence right at the beginning, thus this intelligence could make a pi that was 2.1419 for example.

Then the immediate operations that produce these digits would also have to be different and the definitions upon which pi was based would have to be different and going right back the axioms would have to be different - it would end up not being pi at all and not the ratio between the diameter and circumference of a plane circle.

Here's the beauty... by putting it umpteen thousands of digits down the line, so that in any practical scale it would not prevent circles from existing properly... the creator ensures that the physics of it works out the same way--and that we need a certain degree of development (computers) before we can discover the message.

 

[Ron Webb]

The value of pi is what it is, and not even God can change it. It would be extremely unlikely to find a string of a million 1s or a rasterized circle or any other such simple sequence or pattern in the first trillion or so digits, in any base. If we found such a thing I'm not sure what we would make of it, but I wouldn't call it a message from God. I can only say that God would probably be as astonished as us. (And yes, I know it should probably be "as astonished as we", but that just sounds silly.)

However, if we found a more complex coded message, such as the phrase "I am the Lord your God", spelled out in ASCII characters, then I think that would be very good evidence of God. Not that God could have put that particular string there either, but He certainly could have influenced the historical development of the English language and the choice of ASCII character representations so that the string, when we eventually found it, would have the meaning that it does.

 

[Myriad]

So far, the analyses of the compression scheme seem to agree that the number of bits in the file to be compressed would be (as an expected value) comparable to the number of bits needed to express the position of the first chance occurrence of a sequence of digits identical to the file within the digits of pi. Thus, there is no compression.

However, if a lossy compression stream is acceptable, then we could do much better. For instance, suppose in our 6 gigabyte movie we allow a mere 20 single bit errors. The density of possible digit sequences to point increases by a factor of about 48,000,000,000 ^ 60, or 5.7 * 10^640 which means the expected digit offset to find the first one is only 1 / 5.7e640 the distance from the start... giving a compression factor of...

... oops, that only cuts about 2,135 bits off the length of the number giving the offset. Out of 48 billion. A compression factor of a few millionths of a percent. Darn those counterintuitive big numbers.

(Not to mention the minor "takes much longer than the age of the universe to decompress" problem.)

Respectfully,
Myriad

 

[Robin]

"Make the digits of pi different to what they are": as far as I can parse that it's an oxymoron...

Indeed it is, that is my whole point.

When you talked of the making sure the message appears close to the beginning of the digits of pi it sounds like you are talking about altering the digits of pi.

... so I can indeed blame you for misinterpreting what I said that way.

Sure - when somebody misunderstands them immediately assume the blame does not lie with the clarity of your writing.

 

[Robin]

It isn't, but it is different from what was presented in the book, which implied heavily that Sagan's "God" set the value of PI so that the "message" would be in it. In fact, it appeared in base eleven according to the book.

he was implying intelligence in the creation of the universe, not a personal creator for humans.

But it would not matter how a God created the Universe, it would not change the value of Pi by a single digit.

No God could alter Pi.