To see your world in a grain of sand, and a heaven in a wild flower. Hold infinity in the palm of your hand, an eternity in an hour. Is it possible? Lets for a moment assume William Blake was speaking literally and a grain of sand is spherical; we can infer that all the examples have a common thread, pi. Pi is a mathematic constant representing the value obtained from dividing the circumference of a circle by it’s diameter; the result being an irrational number. As with all irrational numbers, they continue on infinitely without repeating after the decimal (as far as we know).
If you are a reddit user or general meme consumer, you have possibly come across the concept of pi and the infinite world of digital information. If not lets take a look at the claim.
“Pi is an infinite, nonrepeating decimal – meaning that every possible number combination exists somewhere in pi. Converted into ASCII text, somewhere in that infinite string of digits is the name of every person you will ever love, the date, time, and manner of your death, and the answers to all the great questions of the universe. Converted into a bitmap, somewhere in that infinite string of digits is a pixel-perfect representation of the first thing you saw on this earth, the last thing you will see before your life leaves you, and all the moments, momentous and mundane, that will occur between those two points.
All information that has ever existed or will ever exist, the DNA of every being in the universe, EVERYTHING: all contained in the ratio of a circumference and a diameter.”
Mine blowing concept for sure, but is it plausible, can this help us understand Blake, and why is pi preferred over other irrational numbers.
Unlike many irrational numbers, pi contains a normalized distribution of each number from 1 to 9. This means that 1/10 are 1’s, 1/10 are 2’s and so on. Mathematicians have calculated this fact up to the trillionth decimal place. Furthermore, if pi is a “normal number,” then every finite string of numbers occurs with exactly the frequency you’d expect if the digits were random. With this information, it then becomes possible to translate sequences of ASII or bitmap from pi.
For simplicity, lets only concern ourselves with ASII characters for now. ASCII uses either seven or eight binary digits that combine to form letters, numbers, space, or punctuation marks. Ignoring some technical details about how ASCII is really implemented and assuming 00 to 99 covers the full range of characters; we can go through the digits of pi two at a time and extract some string of symbols. If pi is normal, your life story is in there somewhere.
If that doesn’t convince you, let’s look at a more mathematical analysis of finding ASII encodings. Let’s say you want to find your life story in pi. We’ll assume your life story isn’t going to take up more room than the Bible, around 3.5 million characters. Now we need to make a list of all possible words that are no more than 3.5 million characters long. This will give you some huge, but still finite list of possible words.
Let’s say K is the number of possible words you can find. We then look at the digits of pi in chunks of length K. To the first chunk, assign the first word on our list. To the second distinct chunk (which may overlap with the first; that’s OK), assign the second word, and so on. We “only” have K words on our list, and we have at least K+1 distinct blocks of length K (any irrational number has at least K+1 distinct strings that are K digits long), so we’ll run out of words before we run out of blocks. If we have more distinct blocks than we have words on our list (we will), we can just start over at the beginning of our list of words. There’s nothing wrong with encoding the same word twice. That will happen anyway because some K-digit blocks will show up multiple (and even infinitely many) times. This gives us all possible strings up to some finite length. And there is no upper limit to how long the strings on your list can be. Therefore, your life story and the entirety of existence can be found within the decimals of pi.
Furthermore, maybe William Blake was on to something momentous. Perhaps we can find all that is, was, and will be in a the ratio between a circle’s circumference and it’s diameter; within every circle or sphere. To see your world in a grain of sand, heaven in a wild flower, or eternity in an hour.