[DannyB2]

Not consecutively repeating patterns.

But if you take any length pattern of digits, it would repeat an infinite number of times.

Let's take a one digit pattern, say '5'. Since the digits of pi continue forever, there would be an infinite number of '5's.

Now consider a longer pattern '53'. Since the digits of pi continue forever, there would be an infinite number of '53's. In fact, each 53 will be from one of the infinite number of '5's in the previous pattern '5'.

Now consider a longer pattern '537' . . .

. . . to continue . . .

It was long ago when I read Contact (the book), so I hope I don't misremember this too badly. At the end of the book the main character was given a budget, lab, resources, etc. They were working on looking for a message in the digits of Pi. Eventually her beeper beeped and they had found one! It must be woven into the fabric of the universe.

I think any sequence of digits that had any kind of message you are going to eventually find in Pi. Just like, if you look long enough you'll find a 5. If you keep looking you'll eventually find a 53. Keep looking, you'll eventually find a 537. Etc.

[wool_gather]

This is the concept of [a "normal" number][0] (which is a strange choice of word, I think). Apparently pi is not proven to be normal. But the implication as I understand is that, yes, any given sequence that you'd care to look for is there somewhere.

[0]:https://en.wikipedia.org/wiki/Normal_number

[stephencanon]

Almost all numbers are normal, so it seems like a perfectly reasonable choice of terminology =)

[On the other hand, almost none of the numbers someone on the street might name are normal, so ...]

[pmiller2]

Offhand, I would guess that “normal” is the most overloaded word in mathematics. We should probably use it less.

[wool_gather]

Ah, I didn't realize that, thanks! That makes sense now that you say it.

[stephencanon]

Almost all real numbers are a lot of things, though:

   - uncomputable
   - irrational
   - transcendental
   - ...

So it's not really that good of a justification. =)