My dim understanding of the issues leads me to consider a conflict. Pi is more or less considered to be more or less random, or some flavor of random, notwithstanding known patterns of Pi. "Random," to me, sounds a lot like "unorganized."
A book is definitely organized. A larger book is highly organized (entropically speaking). So while you probably can find the same sequence of words in a two-word or ten-word or other small book in Pi, at some point you get a book that's too highly organized to appear in Pi.
However, Pi is also infinite, so it's infinitely possible to find any sequence. (This sounds really hand-wavy to me).
But since Pi is infinite, then isn't it also infinitely unorganized?
"Random," to me, sounds a lot like "unorganized." A book is definitely organized.
Search for "microstate vs macrostate" and keep checking links until you hit an explanation that you like.
Alternatively, let me give it a go: Enumerate all 10-character strings. The string "Hi, there!" appears once, just as often as "l9.gn;omeh" (which also appears once). We call these "microstates". However, if we label 10-char strings green if they look like valid English and red if they don't, then clearly most of our table is going to be colored red. Red and Green are "macrostates". Unlike microstates, red and green do not have equal probability if we choose an element of the enumeration at random.
p(red)>>p(green)>>p("Hi, there!")=p("l9.gn;omeh")
OK, now you were probably picturing a "entropy trace" computed over windows of digits of pi, like you would get out of binwalk. ''Entropy'' is a macrostate, not a microstate. Even though all sequences of digits within a window are equally likely, if you choose a sequence at random it will probably (!!!!) have high ''entropy''. But possibly not. Having high entropy is like being labeled "red" in the 10-character enumeration table: if you pick an entry at random, you're probably going to hit red, but if you keep doing it then eventually you will hit green, and if you do it even longer then eventually you will hit "Hi, there!". Just like if you keep looking at digits of pi, eventually you'll find a block of them where the ''entropy'' is low. In fact, if pi has the properties that Mathematicians conjecture it does, you'll find infinitely many such blocks of anomalously low ''entropy''. If you keep looking long enough, you'll find one with your book in it. These anomalies aren't due to pi being less than perfectly random, they are due to the definition ''entropy'' being statistical in nature. It's only "right" about randomness most of the time.
However, Pi is also infinite, so it's infinitely possible to find any sequence. (This sounds really hand-wavy to me).
Here is the more precise mathematical analog from the article:
PI is believed to be a normal number and therefore all possible finite sequences of characters appear equally often in it
it's still an open question, but the trouble doesn't lie in the definitions, it lies in finding absolute proof, which they haven't done yet.
A book from start to finish could be considered to be 'highly organised'. However, if the book you find is only an infinitely small subset of pi, then it should have no bearing on whether you call pi itself 'highly organised'.
Infinity is very large, and even with something as simple as flipping a coin, you could probably find arbitrary patterns in it if you looked hard enough. 1000 heads in a row? Alternating between heads and tails for 100 flips?
Take a book (which is "organised") and loss-lessly compress it. The compression uses that organisation to be able to save space. The compressed file is now "unorganised".
Comments
Disclaimer: I don't know what I'm talking about.
My dim understanding of the issues leads me to consider a conflict. Pi is more or less considered to be more or less random, or some flavor of random, notwithstanding known patterns of Pi. "Random," to me, sounds a lot like "unorganized."
A book is definitely organized. A larger book is highly organized (entropically speaking). So while you probably can find the same sequence of words in a two-word or ten-word or other small book in Pi, at some point you get a book that's too highly organized to appear in Pi.
However, Pi is also infinite, so it's infinitely possible to find any sequence. (This sounds really hand-wavy to me).
But since Pi is infinite, then isn't it also infinitely unorganized?
Search for "microstate vs macrostate" and keep checking links until you hit an explanation that you like.
Alternatively, let me give it a go: Enumerate all 10-character strings. The string "Hi, there!" appears once, just as often as "l9.gn;omeh" (which also appears once). We call these "microstates". However, if we label 10-char strings green if they look like valid English and red if they don't, then clearly most of our table is going to be colored red. Red and Green are "macrostates". Unlike microstates, red and green do not have equal probability if we choose an element of the enumeration at random.
OK, now you were probably picturing a "entropy trace" computed over windows of digits of pi, like you would get out of binwalk. ''Entropy'' is a macrostate, not a microstate. Even though all sequences of digits within a window are equally likely, if you choose a sequence at random it will probably (!!!!) have high ''entropy''. But possibly not. Having high entropy is like being labeled "red" in the 10-character enumeration table: if you pick an entry at random, you're probably going to hit red, but if you keep doing it then eventually you will hit green, and if you do it even longer then eventually you will hit "Hi, there!". Just like if you keep looking at digits of pi, eventually you'll find a block of them where the ''entropy'' is low. In fact, if pi has the properties that Mathematicians conjecture it does, you'll find infinitely many such blocks of anomalously low ''entropy''. If you keep looking long enough, you'll find one with your book in it. These anomalies aren't due to pi being less than perfectly random, they are due to the definition ''entropy'' being statistical in nature. It's only "right" about randomness most of the time.Here is the more precise mathematical analog from the article:
it's still an open question, but the trouble doesn't lie in the definitions, it lies in finding absolute proof, which they haven't done yet.
That's actually pretty good, thanks.
A book from start to finish could be considered to be 'highly organised'. However, if the book you find is only an infinitely small subset of pi, then it should have no bearing on whether you call pi itself 'highly organised'.
Infinity is very large, and even with something as simple as flipping a coin, you could probably find arbitrary patterns in it if you looked hard enough. 1000 heads in a row? Alternating between heads and tails for 100 flips?
http://www.wired.com/2012/12/what-does-randomness-look-like/
Take a book (which is "organised") and loss-lessly compress it. The compression uses that organisation to be able to save space. The compressed file is now "unorganised".