Read Online The Unimaginable Mathematics of Borges' Library of Babel by William Goldbloom Bloch - Free Book Online Page A

hexagon is differently
configured, or that at least one hexagon contains exact copies of other books
in the Library. We can't imagine that Borges considered this—or would have
cared—when he assigned numbers to the quantity of shelves on a wall or the
number of books per shelf in the Library.
    Also, it may
seem easy to juggle and tweak the numbers of shelves and books to make each
hexagon hold, say, 625 = 5 4 books. After all, as written in the
story, each hexagon holds 640 books, and 625 is very close to 640. But this is
an opportunity to admire the power of Euclid's unique factorization theorem: if
each of the four non-doorway walls has the same number of shelves, and if each
shelf holds the same number of books, then each hexagon must hold
     
    (4 walls) x ( m shelves per wall) x ( n books per shelf) = 4 mn books.
     
    The prime factors 2 2 = 4 will always be there; neither adjusting the number of shelves per wall, nor
the tally of books per shelf will budge those 2s, which means that 4mn can
never cleanly divide 25 1,312,000 .
    How, then,
might we arrange matters so that the total number of distinct volumes may be
evenly distributed throughout the hexagons? One possible solution is to expand
the alphabet to 25 letters and, as Borges did, include the space, the comma,
and the period to round the total up to 28 = (2 2 )  7 orthographic symbols. Then, if the other (admittedly
arbitrarily chosen) numbers for each book stay the same, there will be 28 1,312,000 distinct books.
    Next, hire
infinitely many cabinetmakers to rebuild the bookshelves in the hexagons, so
that each of the four walls holds four shelves, and each shelf holds 49 books.
Then a total of 4  4  49 = 784 = (2 4 )  (7 2 ) books furnish each hexagon, and since
     

     

     

     
    after the renovation, the 28 1,312,000 books exactly fill (2 2,623,996 ) (7 1,311,998 ) hexagons.
     

     
    For this last section, the aim
is to explain concisely why we are currently, and for the foreseeable future,
unequal to the task of determining the median of the prime numbers expressible
in 100 digits. The median of the set of primes expressible in 100 digits is, in
a sense, the "middle" of all of those primes. To compute the median,
arrange the numbers sequentially from the smallest to the largest prime less
than 10 100 (which is called one googol).
     

     
    Now, if there are an odd
number of primes in the list, the median is the absolute middle of the list. If
there are an even number of primes in the list, the median is the average of
the two primes appearing in the middle of the list. (The average of these two
numbers is guaranteed to be an integer, for the sum of two odd numbers is even,
and we conclude the calculation of the average by dividing by two.)
    The only way
to find the median would be, in one way or another, to account for the complete
list of prime numbers expressible in 100 digits. Including 0, there are exactly
one googol numbers expressible in 100 digits. By the famous prime number
theorem—which we'll outline in a moment—there are more than 10 97 prime numbers smaller than 10 100 . This number may sound manageable,
but 10 97 is trillions of times larger than the number of subatomic
particles in our universe. There simply isn't any imaginable way to list and
keep track of 10 97 numbers, which precludes the possibility of
finding the median. 1
    The prime
number theorem was first conjectured in various forms by Euler and others
beginning in the late eighteenth century and was finally proved about a hundred
years later in 1896 by Hadamard (and independently that same year by Poussin).
Part of the beauty of the prime number theorem is that it provides an excellent
estimate of how many primes there are that are smaller than 10 100 without explicitly naming a single one!
    The prime
number theorem says that if n (n ) is equal to "the number of primes
less than or equal to n," then as n grows very large,
     

     
    where ln( n ) is the
natural log function.