Why does paper fold so well?
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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.
I don't think that's sufficient--tinfoil doesn't stretch, but it doesn't fold nearly as neatly as paper.
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.
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.
thank you!
My best (worst) HN days are like that :)
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.
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.
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 ...
I've never tried other brands.
[that 50,000$ ufo on a tripod by the highway, dont try to steal one, man do they ever know exactly where they are]
yeah i like team yellow too :) the beam spread is what you want to check out if your serious about > 100 ft distance
^ this is a very commercial website, but example of higher end equipment and technique
the rest is trigonometry, withe the GPSS defining points relative to the baseline.
https://www.gps.gov/trilateration
https://apps.dtic.mil/sti/html/tr/ADA165932/index.html
https://app.studyraid.com/en/read/15241/528230/determining-d...
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.
https://www.homedepot.com/p/Milwaukee-100-ft-Bold-Line-Chalk...
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.
If the purpose is also to measure the distance between the 'pegs' and one uses the length of the cable in between, then it can be a problem. That's why survey chains are expensive.
If we get real picky, no physical method will really be accurate because straight line is a mathematical abstraction. It can only be approximated in the physical world, much like a circle.
Light paths come closest, although they 'bend', they bend in a way that is 'straight' with respect to space-time.
[0] https://www.tajimatool.com/product_category/mt/#chalk-rite
I love youtube manufacturing videos, saw one where they made a wooden Japanese one that has like a hopper of soot where the string is pulled through.
Consider the planets. They noticed that among thousands of stars these five move funny. Of course it helped that most of them are very bright and don't twinkle. On a clear sky even Sirius often doesn't twinkle though.
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.
Still a sheet of paper is already made smooth and consistently flat. I'm not sure how well it works from hand made paper.
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.
We know that Solids CANNOT be compressed. So what's actually being folded is the air gaps.
Which is why you can't easily fold a piece of tungsten. It has less air gaps.
Metallic film folds quite nicely.
For me this kind of simple, straightforward, educational content is part of the reason I’m proud to pay my TV licence.
Air pockets become water pockets = neutral.
Fibers denser than water = sink.
Big backlog of folds to approve seems like it's ignored. I'm in the middle of fixing some rendering issues hence I haven't approve them yet. Rendering fixes and fold approvals should be up in a few weeks.