Here Today; Pi Tomorrow

It’s pi day! Be irrational!

Earlier this week I mentioned that “this coming Saturday is a doubly special date (especially this year).” One of the things that makes it special is that it is pi day — 3/14 (at least for those who put the month before the day). What makes it extra-special this year is that it’s 3/14/15— a pi day that comes around only once per century. (Super-duper extra-special pi day, which happens only once in a given calendar, happened way back on 3/14/1529.)

I’ve written before about the magical pi, and I’m not going to get into it, as such, today. I’m more of a tau-ist, anyway; pi is only half as interesting. (Unfortunately, extra-special tau day isn’t until 6/28/31, and the super-duper extra-special day isn’t until 6/28/3185!)

What I do want to talk about is a fascinating property of pi.

The movie is pretty good. The book is way better!

If you’ve read Carl Sagan’s science fiction book, Contact, you may recall the part about a message hidden deep within the digits of pi.

When Ellie Arroway and the team of travelers meet the “Vegans” they learn that there are messages hidden in pi and e and other transcendental numbers.

The Vegans don’t tell them what the messages are, just that they exist.

At the very end of the novel, a computer program Ellie set to calculating digits of pi discovers the first, very simple, message.

Here is a quote from the end of that final chapter:

“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 further in. It doesn’t matter what you look like, or what you’re made of, of 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.

“She found what she had been searching for.”

It’s a striking, lovely, lyrical, mystical idea. That hidden deep inside fundamental, simple, universal objects, such as the ratio of a circle’s outside and its diameter, is a message.

Pi cake! (The formula is one of the ways of calculating pi.)

A message that says, “This all has meaning.”

A message that could only be put there in the making of the universe.

Guess what.

It turns out that this is probably true.

The (perhaps disappointing) thing is that — outside of any mathematical mysticism about the real, let alone transcendental, numbers themselves (and there is plenty of fodder for such ruminations) — this is both expected and normal.

In fact, “normal” is exactly the right word.

If the digits of pi — which we know go on forever without repeating — have the mathematical property of being normal (and we think they do), then that raster pattern of a circle does exist somewhere within it.

“One number to bring them all and in the digits find them.”

But that’s just the beginning.

So does every GIF, JPEG, and PNG, ever created. So does every image file that could be created. So do the entire works of Shakespeare in every language on Earth. So does every book ever written or which could be written. So does any sequence of numbers you can name.

An infinite normal string contains every possible finite string somewhere within it. It may take longer than the age of the universe to find it, but it’s there.

Let me repeat this: If the infinite series of digits comprising pi are mathematically normal — and we think they are — then every possible finite sequence exists somewhere within it.

That means every image file, every audio file, every text file, every PDF file, every file of any type or encoding. They all exist somewhere in pi.

Pizza pi!

This may boggle the mind (but so do many other true things). It may be hard to believe that a highly structured sequence could exist in a “random” series of digits, but as with many things, human intuition tricks us here.

I’ve heard of an experiment that used college kids flipping a coin thousands of times and recording their results. The kids did this on their own time, and were not observed.

As you might expect, some cheated thinking they could just write down a series of “random” tails-heads results.

They were all caught.

The problem was that humans don’t credit a sequence such as “tails, tails, tails, tails, tails, tails, tails, tails” as random. It certainly doesn’t seem to fit in with our impression of randomness.

But if you flip a coin often enough — and these students were asked to do thousands of flips — you will get sequences of eight tails in a row.

Every episode is also in pi!

That’s just a trivial example of apparent structure in a really quite short random sequence. With an infinite random sequence, much larger sequences of structure emerge.

Here’s the kicker: We think almost all real numbers (the irrational ones, not just the transcendental ones) have the property of being mathematically normal. And given that the real numbers are uncountably infinite, nearly all numbers are real.

Therefore, almost all numbers contain every secret ever told, every universal truth, every lie, every book, every text, every image, every sound, every video, every spreadsheet, every building plan, every finite thing that can be described with numbers.

And that is something to think about on pi day!

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pi and i

You may be wondering, if 3/14/15 is extra-special pi day, why did I say this coming Saturday was “doubly special” (and not just extra special). The answer to that involves a birthdate that — delightfully — happens to coincide with pi day.

The part about “(especially this year)” also applies to this, but you’re going to have to wait for tomorrow for the details!

Suffice to say that I’m about to take you on a special journey!

Until then, tomorrow and every day, get real and be irrational!