Help Figuring Out this Stick Thing

December 6, 2007 8:18 PM

I recall something that I believe I heard Carl Sagan say either in an episode of Cosmos or in one of his earlier books, that one could store enough information to hold an entire encyclopedia by putting a notch in a stick, if only one could measure where the notch lies to some absurd astronomically precise figure. Am I imagining this?

Could have something to do with Anaximander, a pre-Socratic Greek philosopher.

This one has been bugging me for decades...

posted by macinchik to Science & Nature (13 answers total)

Are you interested in rediscovering the quote or merely understanding the concept involved? I imagine it's a bit of both, so I'll just take a stab in the dark and illustrate one way this could work. Of course, if I'm totally off base here, I apologize!

Consider the encyclopedia in question is a giant text file on a computer. Being digital data, this encyclopedia can be represented as a sequence of bytes with different values. These values can be amalgamated into one gigantic number that represents that unique sequence of bytes.

For the sake of simplicity, if you were to say the maximum amount of "data" the stick could store in this one dimensional sense were merely 4 bytes (clearly WAY less than the size of the encyclopedia, of course).. that's 32 bits (4 * 8), and there are 4294967296 different combinations there. Therefore, if you could make a notch on the stick with a resolution of 1/4294967296th the length of the stick, you could represent any 4 byte pattern with that.

Unfortunately, for any realistically sized stick, I'd hazard a guess that (2 ^ multiple billions) is going to outnumber the amount of atoms available in a length, although my knowledge of the atomic world is practically nil so perhaps someone could correct that.
posted by wackybrit at 8:28 PM on December 6, 2007

I don't know that I ever saw the episode in question, but the underlying theory is wackybrit's explanation, with a little more information. Encode the whole encyclopedia into numbers, as he says. Then put a decimal place in front of that. You now have a very long fraction. Mark the stick so that the ratio of the two halves is that fraction.

That way, you don't have to rely on a particular length unit. Not that this is anything more than a thought experiment, but still.
posted by notsnot at 8:40 PM on December 6, 2007

Good idea notsnot! :)

One icky thing about this whole concept, however, is that if you could make notches with such resolution, you're losing out on the significant increases in the amount of data storage you could get by merely making a few more notches (!)
posted by wackybrit at 8:44 PM on December 6, 2007

Another way of looking at it: say you have an encyclopedia text of 100 megabytes after compression. (would be about a gig after expansion, which seems reasonable for a consumer-level encyclopedia.)

That can be represented as a number that's 100 megabytes long -- that's all the string of bytes is, after all, just a number.

The minimum measurable distance would appear to be one Planck length, which is about 10^-35 meters... very, very tiny, far smaller than we can accurately measure now.

I can't get Google to deal with numbers this big, and running bc on Unix just spits out more digits than I can handle, so as a quick handwavy guess, if indeed Planck lengths are as small as you can reliably measure, then you'd need a stick bigger than the universe to accurately represent the number.
posted by Malor at 8:52 PM on December 6, 2007

Best answer: I don't know whether Sagan discussed it, but the principle is illustrated in comic form in Martin Gardner's Aha! Gotcha: Paradoxes to Puzzle and Delight, apparently now reprinted in a single volume with Aha! Insight. I can't figure out if it's possible to link directly to a given page in Amazon's "search inside" function, but you can find it by searching inside on "amazing code" and following the link to p. 48; see also discussion on the following page.
posted by DevilsAdvocate at 9:22 PM on December 6, 2007 [1 favorite]

Doing Malor's math: Representing 100mb of data as a number gives you a number that's 2^100000000 long. This is about 10^30000000, which is, at very least, 29999000 orders of magnitude larger than the number of atoms in the universe. Of course, if you had truly arbitrary precision, there's nothing to stop you from using this method, but thanks to quantum theory, you don't.
posted by goingonit at 9:44 PM on December 6, 2007

You might be interested in reading this discussion on the blue from a few days ago.

Also, previously
posted by jpdoane at 9:59 PM on December 6, 2007

I think Simon Singh mentions this in The Code Book.
My spaghetti version of this scheme here.
posted by weapons-grade pandemonium at 10:15 PM on December 6, 2007

A notch doesn't have to be perfectly perpendicular to the stick. It could rest on an angle. It also doesn't have to be a straight line. Its shape and position relative to the stick could also represent data.
posted by limon at 12:29 AM on December 7, 2007

This sounds remarkably like an explanation of Shannon's channel capacity theorem that I recently read.

C = B log (1 + SNR)

C is how much information a channel can hold.
B is the bandwidth of the channel.
SNR is the signal to noise ratio; that is, the ratio of the strength of the signal carrying information relative to the strength of noise.

I was reading a recent communications textbook by B.P. Lathi (a great author) - I think it was this one. He tried to explain this theorem intuitively. One of its results is that if there is no noise (N=0, so SNR goes to infinity), then the channel capacity is infinite. Another way to think about noise is that the signal sent from the transmitter will be received exactly by the receiver, to an arbitrary degree of precision.

This is an electrical engineering textbook, so his argument was voltage levels instead of notches on a stick, but it's the same concept. He said that if there was no noise, you could choose an encoding scheme to transmit whatever information you wanted. For example, the contents of this comment could be chosen to map to a voltage of 1.32935798740917957987309235601871023123. Since there is no noise in the channel, the receiver will receive this voltage exactly. The decoder can take this voltage and recover the original message.

Alternatively, we could choose to map the entire contents of the library of congress to a voltage of 1203912830.238957385704871537509172340912630587687460870927834....... and so on. Upon receiving such a signal, the receiver decodes it as the library of congress.

It is of course a difficult challenge to make such an encoder/decoder, but that's beside the point; the point is that there is no limit to the amount of information that can be sent if there is no noise.
posted by PercussivePaul at 1:22 AM on December 7, 2007

"I believe it because it’s absurd: the entire Britannica - not to mention the stacks of my old branch as well as the entire Library of Congress - can in theory be encoded by a single notch on a rod… Any text, however long and complex, is a linear stream of characters. Letters, punctuation marks, typographic symbols: fewer than a hundred types. Each of these hundred can be replaced by a unique three digit number… Now the remarkable twist. If I run the triplets together and put a decimal point in front of the number, the result is a rational fraction running to millions of decimal places. But a fraction represents, and is represented by, a portion of any distance, say from one end of a stick to the other… I can incorporate my box of sayings in a discrete point a little farther than 13/100 of the way from tip to base. I might look at the mark from time to time, delighting in knowing that it encased the cycle of every famous saying ever to flesh out my calendar."

-Richard Powers, The Gold Bug Variations
posted by Daily Alice at 5:16 AM on December 7, 2007 [1 favorite]

That's an interesting point, limon. That gives at least three independent numbers to work with: the ratio of the sections of the stick divided by one end of the notch, the ratio of the sections of the stick divided by the other end of the notch, and the ratio of the stick's length to the length of the notch (which can be made arbitrarily long by changing the angle at which it approaches the endpoints).
posted by solotoro at 5:51 AM on December 7, 2007

Response by poster: This site is amazing and so is every single one of these responses. Thank you very much.
posted by macinchik at 6:24 AM on December 7, 2007