Encyclopedia on a toothpick (idea)

The entire Encyclopædia Britannica is in the digits of pi; I'm just not telling you at what digit it starts.

Haruki Murakami's puzzle is a thought experiment that could under no circumstances be carried out; at least, not in the way it is described. Nitpicking perhaps, but the following analysis perhaps adds some further insight about scale, infinity, and precision.

There are two practical reasons why Murakami's experiment would fail. First of all; atoms are of finite size. Marking the toothpick would mean a removal of material (at least one atom). It would be impossible to arbitrarily mark the toothpick at any desired location, without splitting the atom in two.

The size of an atom is determined by its atomic radius or, since we're dealing with molecules and not atoms, by its covalent radius. For instance, consider the following nanotoothpick consisting of two strands of carbon, over a length of ten atoms.

   C-C-C-C-C-C-C-C-C-C 
    \ \ \ \ \ \ \ \ \ \
     C-C-C-C-C-C-C-C-C-C

The reason for the double-strand nanotoothpick is that it could be marked by removal of a carbon atom (provided that you could actually synthesize such a molecule reliably):

   C-C-C-C-C-C-C-C-C-C 
    \ \ \   \ \ \ \ \ \
     C-C-C   C-C-C-C-C-C

     0 1 2 3 4 5 6 7 8 9

The covalent radius of carbon is approximately 0.77 Angstrom. Thus, the length of this theoretical toothpick is an amazing 1.54 nm. The toothpick above is marked at the "3" position. The entire toothpick can encode 10 positions. Written as a fraction of the entire length of the toothpick these positions are: 0.0, 0.1, ..., 0.9. Thus, we need 10 atoms to mark the toothpick up to one digit accuracy.

If we extend this analysis to larger toothpicks to encode more digits, we arrive at the second reason why Murakami's experiment is only a thought experiment. To mark the toothpick up to two digits accuracy, the carbon strand needs to be 100 atoms long. In this case, the fractional positions are: 0.00, 0.01, ..., 0.99.

The number of atoms required for each successive digit increases exponentially: for n digits accuracy, 10ⁿ atoms are required. Even though the atoms are very small, the length of the toothpick increases dramatically.

The 1999 edition of the Encyclopædia Britannica contains approximately 44 million words. The 2002 is even bigger; give or take 56 million words. For arguments' sake, let's assume the average encyclopedia contains 50 million words, with an average word length of 5 characters. That's a total of 250 million (2.5 × 10⁸) characters. We're not counting spaces. Let's also assume that we need two digits to encode each character (lowercase & uppercase characters, digits, and special characters). This would mean that we need 500 million (5 × 10⁸) digits to encode the entire encyclopedia. We could probably pack this a little more efficient, but considering the following analysis, it really doesn't matter:

A toothpick that could contain the entire Encyclopædia Britannica by the location of a single mark would require a length of:

10^{500 000 000} atoms

That is ten to the power of five hundred million, or a one followed by five hundred million zeros. To put that in perspective: a toothpick the size of our galaxy would be approximately 10³⁰ carbon atoms long. That's still 499 999 970 orders of magnitude smaller, but there is really no way to visualize numbers this large^*.

Let's go the other way, and calculate how many digits we could encode on a toothpick of (say) 3 inches long. This toothpick would be a carbon strand 5 × 10⁸ atoms in length. This carbon strand could encode log(5× 10⁸) = 9 digits precision. Just enough to spell out one single word.

A more feasible (ahem) way to encode the Encyclopædia Britannica onto the toothpick would be to use multiple markings as a binary code. Every 8 carbon atoms along the length would act as one byte. In this scheme, we would have 5× 10⁸ / 8 = 62.6 MB of data. Just enough to fit the entire text of the encyclopedia.

* Ok, Jongleur seems to be better at visualizing these large numbers and calculated that a chance of 1 in 10 to the power of 500 million would be equal to winning the Powerball lottery in all 20 states that offer it with the same lucky number and getting attacked by sharks on 6 occasions in one week, every week for the next 5000 years.