Thoughts on the Library of Babel
Each book in the Library of Babel has 1312000 characters (22 letters, space, comma and period). This means that there are 251312000 ~= 1.956 * 101834097 books. This is a really really big number. There are only about 1080 atoms in the observable universe. If we wanted a set of unique books, each dedicated to an atom, each book would only have to be about 57 pages long.
Each hexagon contains 20 shelves of 35 books - that's 700 books. 251312000 / 700 ~= 251311997.964 hexagons. Borges describes them extending in all directions, and also conjectures that the Library is cyclical - keep going far enough in one direction, and you'll come around on the other side. Let us also assume that no matter which direction you go, you'll visit the same number of hexes before repeating (i.e. the space is roughly cubic). Each shelf has 35 books. If we assume that each book, plus the shelf ends, takes up between 1 and 2 inches (a 410 page book near me is about an inch thick, but that doesn't count shelf-ends, and its covers and pages may be thinner than the books here. On the other hand, they might be thicker; this is a divine library, after all.) This gives as a side length of between 35 and 70 inches, which means the distance from face to opposite face is between 59 and 121 inches - roughly five to ten feet. With the stair well in the center, it's no wonder that the inhabitants had to sleep upright in the closets.
The hallway between hexagons is described as narrow - indeed it would have to be! Assuming that both of the closets are roughly two feet wide, this leaves between -1 feet and 1.8 feet for the hallway. If they were only 1.5 feet wide, this leaves us with a hallway between 0 feet and 2.8 feet wide. While one might slide sideways into the sleeping closet, I can't think of any way I'd care to defecate in a room where one of the dimensions is less than a foot and a half.
If the space is roughly cubic, then each of the dimensions is roughly 25437332.6 hexagons in length. Assuming that each hex is only separated from the others by 7 feet, that's 25437333.4 feet in each direction. The universe is 93 billion light years across. That's only 2519.64 feet. The ratio between the diameter of a proton, and the observable universe, is 2530. The ratio between the universe and the Library is 25473313.7. I can think of no analogy to express how tiny the universe is in comparison to the Library.
Those searching for Vindications certainly have their work cut out for themselves.
Each book contains 1312000 characters. There are 25 possible characters. This translates to about 6.092 MB of information per book. If the information that you're looking for contains more than that amount of entropy, no one book is going to provide it for you. One option would be to have books point to other books. After providing a portion of the information, it would then have a pointer to the next piece of information. Would this be feasible, though? You'd need slightly over two characters to identify which book in a hex is important, leaving us 1311998 characters to identify which hex you need to find. If the hex is just the one next door, things are simple. Unfortunately, it might be +/- 25437332.4 hexes away, in three different directions. Assuming that we can magically know where each direction description begins and ends (thus avoiding the need for prefix-free which could save us space on smaller values, but cost us much more for larger ones) we would require 437332.6 characters per direction. Unfortunately, this means 1311997.96 characters to identify the hex. Add in that used to record which book in the hex it is, and we've filled the entire book, just telling us where to find the next one.
This is, of course, no surprise. If this was not the case, one could turn it into a perpetual compression system, by designing an algorithm for distributing the books (i.e. mapping a position in the library into the book held there), and then just providing the the first book. Even if each book only leaked a single bit, you could use a universal code to slowly get out an arbitrary message close 251312000 bits long, all encoded in a 6.1 MB file. This is, of course, impossible. In short, there exist books for which the shortest way of indicating how to read it is to simply provide the book itself.
What does this say for hopes of having a Catalogue (or even a catalogue of catalogues)? If one is to exist, the books it must be either be either multiple times longer than the books it references, or those other tomes must lie relatively close - remember that only 1/25 of the books in the Library lie close enough that only 1311999 characters are necessary to describe them, and in a standard sized book, that would only leave one character left. About one in 25656000 are close enough that you could fit the directions for two of them into a standard sized book.
This is, of course, assuming that The Catalogue is not the Book of Sand.
The real kicker, of course, is that ANY book could be a pointer to the catalogue, but it would take far longer than we can imagine to check even just one.
Each hexagon contains 20 shelves of 35 books - that's 700 books. 251312000 / 700 ~= 251311997.964 hexagons. Borges describes them extending in all directions, and also conjectures that the Library is cyclical - keep going far enough in one direction, and you'll come around on the other side. Let us also assume that no matter which direction you go, you'll visit the same number of hexes before repeating (i.e. the space is roughly cubic). Each shelf has 35 books. If we assume that each book, plus the shelf ends, takes up between 1 and 2 inches (a 410 page book near me is about an inch thick, but that doesn't count shelf-ends, and its covers and pages may be thinner than the books here. On the other hand, they might be thicker; this is a divine library, after all.) This gives as a side length of between 35 and 70 inches, which means the distance from face to opposite face is between 59 and 121 inches - roughly five to ten feet. With the stair well in the center, it's no wonder that the inhabitants had to sleep upright in the closets.
The hallway between hexagons is described as narrow - indeed it would have to be! Assuming that both of the closets are roughly two feet wide, this leaves between -1 feet and 1.8 feet for the hallway. If they were only 1.5 feet wide, this leaves us with a hallway between 0 feet and 2.8 feet wide. While one might slide sideways into the sleeping closet, I can't think of any way I'd care to defecate in a room where one of the dimensions is less than a foot and a half.
If the space is roughly cubic, then each of the dimensions is roughly 25437332.6 hexagons in length. Assuming that each hex is only separated from the others by 7 feet, that's 25437333.4 feet in each direction. The universe is 93 billion light years across. That's only 2519.64 feet. The ratio between the diameter of a proton, and the observable universe, is 2530. The ratio between the universe and the Library is 25473313.7. I can think of no analogy to express how tiny the universe is in comparison to the Library.
Those searching for Vindications certainly have their work cut out for themselves.
Each book contains 1312000 characters. There are 25 possible characters. This translates to about 6.092 MB of information per book. If the information that you're looking for contains more than that amount of entropy, no one book is going to provide it for you. One option would be to have books point to other books. After providing a portion of the information, it would then have a pointer to the next piece of information. Would this be feasible, though? You'd need slightly over two characters to identify which book in a hex is important, leaving us 1311998 characters to identify which hex you need to find. If the hex is just the one next door, things are simple. Unfortunately, it might be +/- 25437332.4 hexes away, in three different directions. Assuming that we can magically know where each direction description begins and ends (thus avoiding the need for prefix-free which could save us space on smaller values, but cost us much more for larger ones) we would require 437332.6 characters per direction. Unfortunately, this means 1311997.96 characters to identify the hex. Add in that used to record which book in the hex it is, and we've filled the entire book, just telling us where to find the next one.
This is, of course, no surprise. If this was not the case, one could turn it into a perpetual compression system, by designing an algorithm for distributing the books (i.e. mapping a position in the library into the book held there), and then just providing the the first book. Even if each book only leaked a single bit, you could use a universal code to slowly get out an arbitrary message close 251312000 bits long, all encoded in a 6.1 MB file. This is, of course, impossible. In short, there exist books for which the shortest way of indicating how to read it is to simply provide the book itself.
What does this say for hopes of having a Catalogue (or even a catalogue of catalogues)? If one is to exist, the books it must be either be either multiple times longer than the books it references, or those other tomes must lie relatively close - remember that only 1/25 of the books in the Library lie close enough that only 1311999 characters are necessary to describe them, and in a standard sized book, that would only leave one character left. About one in 25656000 are close enough that you could fit the directions for two of them into a standard sized book.
This is, of course, assuming that The Catalogue is not the Book of Sand.
The real kicker, of course, is that ANY book could be a pointer to the catalogue, but it would take far longer than we can imagine to check even just one.