6 ms·
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
by srean 4mo 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.
- arijun 4mo 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.
- srean 4mo agoYou 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.
- qsera 4mo 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...
- roelschroeven 4mo agoAnd 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 4mo agoIndeed ! 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. https://en.wikipedia.org/wiki/Huzita%E2%80%93Hatori_axioms https://en.wikipedia.org/wiki/Huzita%E2%80%93Hatori_axioms
- saagarjha 4mo agoI'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.
- srean 4mo agoYup and if you need a right angle, combine it with this https://news.ycombinator.com/item?id=48538771 https://news.ycombinator.com/item?id=48538771 by roelschroeven. I often wondered how to ensure that the corners of a sheet of paper make a right angle. You need that to form a square sheet, otherwise the standard trick of folding along the diagonal gives a rhombus, not a square.
- globular-toast 4mo ago> 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.
- srean 4mo agoI have an asterisk in my post addressing that :) Happy to have picqued your mind. BTW, your method was the method of choice for the surveyors of the Nile, from the Egyptian civilization. Paper is hi-tech and was not available until much later, and as you mentioned doesn't scale. But if I have misplaced my ruler ...
- dotancohen 4mo agoOver long distances the string will sag in the middle. That's one of the reasons (not the only reason) construction uses lasers today.
- gacgacgac 4mo agoFolded paper won't?
- josefx 4mo agoFolded paper has some structure, so not as much?
- boothby 4mo agoIf anybody has ever tried folding a very large paper (or, bedsheets, tarps, etc), they'll realize the wisdom of this comment. Our intuition from folding paper on the order of several to tens of centimetres does not scale to arbitrary size and precision. Paper is relatively rigid, but its rigidity is finite and ensuring local-to-global flatness becomes a painstaking endeavour.
- rusk 4mo agoThe 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.
- wongarsu 4mo agoIf 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
- bena 4mo agoThis is actually a tool used in construction. A chamber filled with chalk and a coiled line. You hook the line to one end of your item, pull the chamber across, make it tight, snap the line. https://www.homedepot.com/p/Milwaukee-100-ft-Bold-Line-Chalk-Reel-Kit-with-Red-Chalk-48-22-3986/207005252 https://www.homedepot.com/p/Milwaukee-100-ft-Bold-Line-Chalk...
- UltraSane 4mo agoUsing one of these to snap a perfect reference line is extremely satisfying.
- IAmBroom 4mo ago"Perfect" is doing some heavy lifting here. The string is always a non-straight catenary curve, unless infinite force is used to pull an indestructable string. A laser beam* across the room will show the defect in the string straightness. It's more than good enough to fool human eyes, which are not good at judging slow gradients (such as all the touristy "mystical anti-gravity locations" where balls roll apparently uphill). Therefore, the snap-line is good enough. But not perfect. * Gravity of course still affects the laser beam's straightness, but on a level good enough to fool electron microscopes, so we can give that a pass.
- mjanx123 4mo agoPaper is thin so the stretching needed to bend it is minimal
- srean 4mo agoExactly.
- foobarian 4mo agoOne 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.
- mjanx123 4mo agoThree negatively curved surfaces (saddles) mate despite not being flat. You need to rotate the surfaces also when lapping. (See the famous Robrenz video)
- srean 4mo agoI did not know this, very interesting.
- srean 4mo agoI think wetting and drying of paper is a bit like heat treatment of steel. The fibres find a new local minimum stable position, prompted by the swelling and the shrinking.
- IAmBroom 4mo agoExactly. They were initially rolled, not merely pressed, which implies a longitudinal strain during the pressing moment.
- jayd16 4mo agorub 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.
- oidar 4mo ago> 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. It would be interesting to see how progressively larger pieces of paper handle this. A roll of masking paper would be the easiest way to test.