## Thursday, May 23, 2019

### The Library of Babel and the digits of Pi

One of the most famous stories by Jorge Luis Borges is The Library of Babel (La Biblioteca de Babel). This library contains all possible books. For this statement to make sense, we must specify the definition of a book. For Borges, each book is made of one million three hundred and twelve thousand characters, chosen among all the possible permutations of that length and built with a set of 25 basic characters (the space, 22 letters, and two punctuation marks).
The number of books in the Library is huge, for the number of permutations of a certain number of symbols grows according to the factorial of their length. The factorial of a number N is obtained by multiplying together all the natural numbers from 1 to N:
N!=1×2×3×...×N
The result of this operation grows disproportionately. Thus, 5! = 120; 10! = 3,628,800; 100!>9×10157. The number of books in the Library of Babel does not grow so quickly, because each book contains many repetitions of the symbols, which decreases the number of possibilities, but consider what will be the factorial of 1,312,000, if the factorial of 100 is a number with 157 digits.
The number of books in the Library of Babel is huge, but it is not infinite. That set of books contains all the possible chains of that length, whether or not they make sense. But that means that it contains all the books that have ever been written; all that could ever be written; all translations of each book to all the existing or possible languages​​...

 Jorge Luis Borges
The problem is, we don’t know where each book is. Their distribution in the Library is random, or at least the librarians think it is. It is true that one of the possible books (which we also don’t know where it is) must be the accurate catalog of the Library; but it is also true that the Library must contain a huge number of false catalogs. How can we know which one is accurate and which is false, if at any given time we have access to quite a small number of books?
Each book in the Library is a unique copy. But if you destroy one book, it is not a problem, for the Library contains millions of copies that differ from that book only in one character; several billion copies that just differ in two characters; and so on. All those copies can be considered copies of the starting book with one, two... errata.
Borges didn't dream up what I’m going to say next, but the Library of Babel also contains my genome; it contains your genome, kind reader; the genome of all human beings that have ever existed or could come into existence; and the genome of all living beings that have ever existed or could come into existence. How is this possible? Let’s think a little:
 Poster of GATTACA,a film whose titleis a DNA chain
The genome of a human being is a succession of nucleotides that belong to four different types: A (adenine), C (cytosine), G (guanine) and T (thymine). Therefore, a genome can be represented by a succession of repetitions of the four symbols A, C, G and T. Since the genome of a human being contains about three thousand two hundred million nucleotides, it follows that each genome will fit in 2440 books of the Library. In other words, to obtain one genome, we should concatenate the content of those 2440 books. Yes, I know, we must first find all those books, which we don’t know where they are, but the fact is that they are there, somewhere.
What is the relation of the number π, mentioned in the title, with all this? In another post I mentioned that the digits of π seem to fulfill all the conditions established by statistics to define what a random number is. Actually, this has not been proven, but until now all the tests of randomness applied to the digits of π we know (currently 31 billion) have been met. Therefore a reasonable conjecture has been proposed, which states that the digits of π, in fact, probably meet all those conditions.
But then, it must happen that any sequence of the digits 0 to 9 that comes to mind must be somewhere among the digits of π. For instance, the sequence 123 appears for the first time in position 1924 of the decimal digits of π (counted from the right of the decimal point); 1234 appears at position 13,807; 12345 appears at position 49,702; and so on.
Let us now consider that any finite sequence of letters (such as those used to write the books in the Library of Babel) can be encoded using just the ten digits. For instance, we could represent the space by 00; A by 01; B by 02; and so on, including the two punctuation marks. A book in the Library could be coded with 2,624,000 digits (two digits for each letter). And that number, however huge it is, will appear somewhere among the figures of π, if the conjecture mentioned is true. We probably won’t know where it is, but we also didn’t know how to find a specific book in the Library.
The conclusion is clear: the figures of π contain the entire Library of Babel. And therefore, among the figures of π, we could find my coded genome; your genome, kind reader; the genome of all human beings that have ever existed or may come into being; and the genome of all living beings that have ever existed or may come into being. And since π has infinite digits, each of those genomes will appear repeatedly among the digits of π, not just once, but infinitely many times.

The same post in Spanish
Thematic Thread on Mathematics: Previous Next
Manuel Alfonseca