srean
2 hours ago
Oh 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.
roelschroeven
15 minutes ago
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.
srean
7 minutes ago
Indeed !
One can create an axiomatic system of geometry through such coincident folds (as an alternative to straight-edge and compass) and it turns out to be more powerful than the Euclidean system.
One can construct cube roots, trisect angles. Depending on the choice of paper folding axioms one can go beyond cube roots and k-secting angles to the entire set of algebraic numbers.
arijun
2 hours ago
> 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.
qsera
10 minutes ago
"Paper folds in a straight line" and I was like "duh! what else?" Until I read this comment, and it bought back all the memories where I tried to fold other things like plastic sheets and tin foils and how they never ended in straight line...damn. I never noticed...
srean
2 hours ago
You are perhaps commenting about the force needed to fold, the persistence of the folded shape. My comment is about the shape of the crease once it has been folded.
Most metals are stretchier than paper. If it is thick it will resist folding, but once you have folded it, that is, the two flat boundary surfaces have coincided, the crease would be a straight line if the surfaces cannot stretch.
How much force you will need to exert to form a fold depends on material properties but the geometrical nature of the crease is dictated by stretching.