Imagine I have a piece of paper that is folded precisely. If I were using a simple form of code (01=A, 02=B, 26=Z etc.) how reliable would a measurement (in the form of a decimal) be? For example, if the fold comes out at .2008050114192305180919 the length of the paper it would read “THE ANSWER IS…”. Obviously, the longer the message is the harder it would be to get an accurate measurement. Is there a certain point at which it would become technically immeasurable? Sorry if this is worded awkwardly.
Your ability to resolve the location of a fold in a piece of paper is plausibly related to the thickness of the paper. On a piece of paper 10" long and .002" thick, your scheme looks as if it might be able to encode 2 and possibly 3 characters at a time.
The smallest unit of length is the Planck Length, which is ~16 x 10^-34 meters. Many scientists don’t think it’s possible to measure anything shorter than the Planck length.
So, if you have a piece of paper the size of a newspaper, you’ll only get about 34 decimals points of accuracy in your measurement of the fold, no matter what you do. That translates to 17 characters using your system.
A real piece of paper can’t be folded that accurately. For example, a proton is about 10^20 Planck Lengths in size and the celluose molecules that make up paper are even larger. At most you might be able to fold a piece of paper to 10 decimals points of accuracy, or enough to encode at about 5 characters.
This is the carefully-placed-scratch-on-the-stick infinite data compression thing, isn’t it?
It would probably be better to use a piece of metal with a low coefficient of thermal expansion (and temperature would have to be specified anyway. Also, instead of a fold, you could use the length vs. the width.
It would probably become immeasurable when either the metal gets so small you can’t accurately machine it, or when it gets so big that you can’t practically keep it anywhere.
It seems there’d be way easier ways to encode a message, because look at how precise you have to be just for an 11 character message, not counting spaces.
This used to be called a slide rule. 
And if we’re trying to send a message using folded paper, then there are a lot of more efficient ways we could do it. Let’s say that the limit of our ability to localize a fold in a piece of paper is some length we’ll call epsilon, and our piece of paper has length L. Using the OP’s method, we could then encode enough information to be able to distinguish between L/epsilon different messages. But if we can place folds that precisely, let’s put in a lot of folds: For every segment of the paper of length epsilon, we either put a fold or don’t put a fold in that segment. This gives us L/epsilon bits of information, which is enough to distinguish between 2[sup]L/epsilon[/sup] different messages. Unless L/epsilon is pathetically small (in which case, the folded paper is a lousy method of storing information, no matter how we do it), 2[sup]L/epsilon[/sup] is going to be vastly greater than just L/epsilon. So the single-fold method is never the right method to use.
I disagree. Just because the ratio can be a decimal, doesn’t mean it has to be. For example, we could use a piece of material one mile long, and put the fold at one half mile plus one tenth of an inch. That would give us 316799/633600, or .49999842172 . We can then extrapolate to having a paper 100 miles long, etc.
I realize that the example doesn’t give a valid code (49 doesn’t correlate to a number), but my premise still stands. We can go bigger, we don’t have to go smaller. I think our limit is going to be how big we can make the paper, metal, etc.
ETA: In other words, the paper doesn’t have to be the size of a newspaper.
If you put in 0 and 1 marks, then mark your message relative to those, you only have to worry about the relative temperature of different parts of the metal.
Good point.
But seeing as how big I’m planning on making this thing, that’s a real problem 
Paper is notorious for not keeping its dimensions. It will expand and contract with varying humidity, temperature, etc. It will expand and contract differently in different directions and in different places. The pecission you could reliably get in practice would be very, very, low.
Thanks, this is interesting! What if we were to use a metal that is kept at a constant temperature and used length vs width measurements, as per Santo Rugger’s suggestion? If that metal was large enough to fit within an average sized room, would it be possible to measure it accurately with Planck length? Surely something that size with such small measurements would give us the required decimal spaces?
Nothing humans have devised can measure to Planck length or anywhere near it.
You can make a single crystal of silicon and mark a line on that. From its density and atomic mass, and assuming cubic spacing because it’s easy, I get about 3.68 billion atoms per meter. The most accurate mark you’ll be able to make can only be accurate to 1 atom, so you could get about 9.5 digits of accuracy. Calling it ten, you could get five characters encoded using your original method. So on a one meter long single crystal of silicon, you would have “THE A”.
I’m thinking that whatever the object, it can be used to transmit more information by just writing on it.
The limits on data compression are pretty fundamental and unavoidable - as far as I know, any time anyone has ever proposed a method to work around the limits, they’ve got it wrong.