Manooh and I have had all kinds of conversations about pi. Actually I wrote the song lyrics that I posted yesterday because she and I had been discussing the fact that pi contains, somewhere in its digits, all the melodies that have ever been written or ever will be written.
It’s true for poems too. For example if you write A as 01, B as 02, all the way up to Z as 26, space as 27 and so on for other punctuation, then any poem you can think of (as well as all the poems you can’t think of) will show up sooner or later as a string of digits in pi, if you go out far enough (2515212701180527031205220518). And if you write music as a string of digits, then you can also find all the melodies.
Manooh, quite rightly, felt that a song needs music. Since I had used the first thirty one digits for the lyrics of my little tribute to pi, she responded by using the digits immediately after those for the melody. The result was the following lovely song, which she promptly recorded and posted, and has now graciously given me permission to link to here, for your listening pleasure:
A song regarding pi
The song is gorgeous. It sounds like Kurt Weill.
Hey,
I don’t think pi actually has to contain EVERY sequence of digits, just because its decimal expansion is infinite and non-repeating.
For example, 0.101100111000… also has an infinite and non-repeating expansion… but it obviously doesn’t contain EVERY sequence of digits.
Or is there something special about pi that lets you prove it contains ANY sequence of digits I might pick?
Cheers,
Vlad
The way you’ve defined it, only melodies with discrete notes can be found in pi. But we can do even better with the Riemann zeta function. Any piece of an analytic function can be approximated arbitrarily well by the Riemann zeta function. This means that any sound waveform can be found somewhere in the domain of the Riemann zeta function.
In response to Vlad: Actually it was a bit of a trick, since almost every number contains all sequences of digits. It is generally believed that pi is a normal number (as is almost every number), a number for which all sequences of N digits occur with equal probability, for all positive N in all base representations, although it has not yet been proven that pi is normal.
In response to Dan: Yes! That is a way cool observation! Now if we could only find an easy way to explain Riemann zeta functions to people who don’t already know about them… 🙂
There’s the book “Prime Obsession” that attempts to explain the Riemann hypothesis for the layman – so maybe there’s an account in there. Turing started building a physical device to compute the Riemann zeta function so there may be a nice mechanical picture.
Here’s the updated link to the song: http://manooh.com/files/pi.mp3