alt Hacker NewsOh no! woe is me, they don't highlight my absolutely, ridiculously favourite fact/curiosity about a sheet of smooth paper:
If you fold it clean, the crease is a straight line. In fact I don't know of any other good way of obtaining a straight edge from scratch quickly, meaning without transporting one existing straight edge to another (*).
I remember spending a lot of enamored time coming up with different geometrical proofs of this fact. Perhaps the only time I have come close to jumping out of the proverbial bath tub.
The underlying reason is that paper does not stretch (**) (but, paradoxically, it does bend fine. It's a paradox because bending needs stretching).
I have to restrain myself from grabbing strangers off the streets to ask -- how cool is that.
Three other demonstrations that never fail to nerd-snipe me like this are Dirac's belt trick, that straight woven cloth rips usually at 90 degrees, and the working of a teeny tiny metacircular interpreter.
(*) Rope stretching is a close competitor, but the tension needs to be really really high and it is difficult to run a pencil along it to mark a straight line, lest you distort the st. line.
(**) of course, it does, but a tiny amount.
Coming back to straight line folds, this property holds beyond just Euclidean space, it holds for Riemannian geometry and probably for any continuous metric space.
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One thing I find interesting about paper is that wetting and drying it turns it uneven. Even when drying it under a press.
And then another ridiculous process not involving paper, but super cool nonetheless is creating a flat surface by grinding 3 not-flat objects against each other in round-robin manner.
I'm a fan of tearing paper along a crease rather than cutting it for this reason, since the tear is straight and using scissors will invariably be all over the place.
If you want to use a rope to get a straight line, your best bet is to turn the rope itself into the pencil. Coat it in chalk or other powder, then put it under tension and snap it on to the desired surface
> The underlying reason is that paper does not stretch
I don't think that's sufficient--tinfoil doesn't stretch, but it doesn't fold nearly as neatly as paper.
Paper is thin so the stretching needed to bend it is minimal
The shortest distance between two points is a straight line?
A sheet of paper approximates a Cartesian plane probably more closely than most things we can fold
Therefore a fold will always be in line with the theoretical 2D plane and thus will be the shortest (straight) line.
rub 3 surfaces together and they end up smooth and flat. Fast in the across the world sort of way but not fast like folding a sheet of paper.
Still a sheet of paper is already made smooth and consistently flat. I'm not sure how well it works from hand made paper.
> In fact I don't know of any other good way of obtaining a straight edge from scratch quickly
A string made taut between two points is surely a better way? And works at much bigger sizes too (people build walls and foundations using this technique all the time). The paper is less useful in practice because any paper you find is probably straight and square anyway.
Still, I had fun thinking about this as I definitely hadn't considered it before.
And once you have created such a straight line, you can fold the paper again such that the first crease lines up on both sides of the new crease, and then you have a right angle.