What if there were a message embedded in the digits of pi?

In his book, Contact, Carl Sagan imagined the idea that there could be a message embedded somewhere in the digits of pi. There are lots of ways a message could be embedded, though.

The circumference of a circle can be given by the equation,

\[ \normalsize C = 2 \pi r \]

So pi can be defined as,

\[ \normalsize \pi = \frac{C}{2 r} \]

Or, put another way, pi is the ratio between the circumference and the diameter.

Everybody seems to agree that if you put pi in its decimal form, there are infinitely many digits. I don't know how they figured that out, but since there doesn't seem to be any controvery about it, I'm going to take their word for it. I've always wondered, though, if there's a point at which you get a repeating pattern, and that it repeats from then on out. I don't know if anybody has ever ruled that out or not. I mean if it's an infinite string of numbers couldn't the repeating pattern be really long?

Consider the fraction, 1/3. That one gives 0.33333333. . . But the fraction, 2/7 gives 0.285714285714285714. . . There doesn't seem to be any obvious limit to how long a string of numbers there can be that repeat. It seems possible, then, that somewhere in that infinite expanse of digits, that eventually, you could get a string of numbers that repeat from then on out. That string of numbers could be a million digits long or even 10¹⁰⁰ digits long. I don't know if that's ever been ruled out, but it's something I've always wondered about.

Anyway, back to the topic.

If the digits of pi are infinite, what are the chances that a message could be embedded in there somewhere?

Imagine if pi contained a string of numbers that represented:

I am the God of Abraham Isaac and Jacob

If you convert that to binary code, it would be:

01001001 00100000 01100001 01101101 00100000 01110100 01101000 01100101 00100000 01000111 01101111 01100100 00100000 01101111 01100110 00100000 01000001 01100010 01110010 01100001 01101000 01100001 01101101 00100000 01001001 01110011 01100001 01100001 01100011 00100000 01100001 01101110 01100100 00100000 01001010 01100001 01100011 01101111 01100010

So that's one way it could be embedded in pi. I mean, it seems unlikely, but given infinite digits, and assuming there's no point at which things start to repeat, it seems at least possible that a string of numbers like that could be in there somewhere.

It doesn't have to be in binary, though. It could be in base 3, 4, 5, 6, 7, 8, 9 , or 10. The letter, "I," in binary is 01101001. If you convert that to base 10, it's 105. And you could do that with each letter in the string and end up with a string of numbers in base ten that represent the statement about God. You could also do it in different languages. If you consider the 10 different bases, plus the thousands of different languages there are, there are all kinds of ways for the God statement to be embedded in pi.

I don't want to make this too complicated, so let's assume the string of letters in the statement, "I am the God of Abraham Isaac and Jacob," and forget about the spaces, capitalization, or the commas that should be in there. What are the chances that you could just randomly get, "iamthegodofabrahamisaacandjacob"?

There are 31 characters in that string of letters, and there are 26 letters in the alphabet. So each character could be any of 26 letters. The probability, then, would be 1 in 26³¹. That's about 1 in 7.31 x 10⁴³. Those are slim odds.

But suppose you have 26³¹ chances for it to happen? Then it's not so unlikely that it could happen. If you suppose that there are 26¹⁰⁰⁰ chances for it to happen, it seems inevitable that it would. Since there are infinitely many digits in pi, it seems almost inevitable that somewhere in there, the statement about God would be embedded.

But given how unlikely it is, we shouldn't expect to find it within the first few digits. It may be that we'd have to look through 10⁴⁴ digits before we'd be remotely likely to find it.

What if we found it a lot earlier, though? What if we found it within the first million digits? We know we haven't, but this is just a hypothetical. If that were to be the case, it would be pretty amazing, wouldn't it? But could we draw any conclusions from it? What would the significance be?

Recently, somebody on YouTube said that if there were a message embedded in pi that it would be evidence for God. That's what got me thinking about this whole subject. I had a discussion in the comment section with some people about that because I don't think it would be evidence for God. As amazing as it would be, the reason I don't think it would point to God is because pi is a necessary number. If it happens that the statement about God occured within the first million digits of pi (or anywhere in pi), then it could not have been otherwise. The ratio between the circumference and diameter of a perfect circle is fixed and unalterable by necessity. It's just as necessary as the fact that 2+2=4. There's no possible world in which things are different. God could no more make pi different than he could make 2+2=5. So God can't be credited for embedding the statement if it were in there.

There is a way, though, that a message about God in pi could point to God's existence. So far, I've been talking about pi in an idealized way. I've been talking about a perfect circle in flat Euclidean space. But suppose we were talking about pi in nature? We know, from general relativity, that spacetime has a curvature that varies from place to place. There's more curvature near massive objects than there is when you're in some void far away from any massive objects. Astrophysicists have attempted to determine whether there's any curvature on the scale of the whole observable universe, and it appears to be flat. But they can't measure the curvature of the universe with exact precision. So although it appears to be flat, there's some uncertainty involved. It could be there's some positive curvature, but the curvature is so slight that we can't detect it.

It's similar to how it's hard to tell the earth is round just by looking at your back yard. Even if you had a swimming pool, and the water was perfectly still, you couldn't measure the curvature of the surface of the water and determine that the earth is round. It would appear flat to any measuring device you used because the earth is so big. In the same way, the universe could have some positive curvature that we can't detect within the confines of the observable universe.

Suppose we could, though. Suppose the universe has some positive curvature, be it ever so slight, that would make pi, as it exists in nature, different than the idealized Euclidean pi. And suppose that pi, as it exists in nature, contained the statement about God somewhere within the first 1000 digits.

I think that would be definite proof of God. No conceivable alien could cause the curvature of spacetime to be such that it creates the God message within the first 1000 digits of pi, and given how enormously improbable such a thing would be, the best explanation would appear to be God. I think it would be downright unreasonable to deny the existence of God at that point.

If some given curvature that makes pi spell out the God message proves God's existence, then why wouldn't perfeclty flat spacetime do the same thing? After all, if we're imagining a range of possibilities for the curvature of spacetime, perfect flatness would just be one precise possibility within that range. The universe could have positive curvature, negative curvature, or perfect flatness. Perfect flatness is just as improbable as any other curvature.

It's hard to put my finger on the reason, but I just don't think finding a message in pi given perfect flatness would point to God in the same way that finding a message in pi with some curvature would. Maybe it's because flatness is idealized in some sense. I would still find it pretty amazing if the God message were embedded in the flatness version of pi. I just wouldn't consider it evidence for God.

What would you think if the standard pi contained a message about God somewhere? Would it matter where? Would it matter what the message was? Suppose there were all kinds of words and phrases found in pi, some meaningful, and some gibberish, and one of them happened to be the God message? Would that make an impression on you?

An Aside: There are web pages where you can search for things in the digits of pi. Here's one that allows you to find your birthday (or any other string of numbers) in pi. My birthday begins at the 2,462,816th digit.

I tried searching for "Jesus is God," in a few of those pi search thingies, but none of them would find it for me. I simplified it to:

jesusisgod

In binary, that's:

01101010 01100101 01110011 01110101 01110011 01101001 01110011 01100111 01101111 01100100

In base 10, that's:

106101115117115105115103111100

If anybody else is able to find it, let me know.

EDIT: I found one! This web page found my sequence in the 7.196305715199008 x 10³⁰th position.

Let's see if that's remarkable. My sequence is 30 digits long, and there are 10 possible numbers for each digit. That means the odds are 1 in 10³⁰ that any given sequence 30 digits long would contain my sequence. So yeah, we found it about where we'd expect given the odds. Nothing remarkable about that.