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.
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.
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.
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.
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.
"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...
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.
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
This 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.
Early humans fill me with awe with their ingenuity.
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.
The Tajima ones [0] are phenomenal, though the hook leaves a longer blank stretch than I'd like. They make a super nice snap knife too. Highly recommend Tajima for anything they make. Annoyingly, they don't sell a rip saw, only crosscut.
"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.
Yes it will be a catenary but it is not a problem if the purpose is to mark the ground.
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.
You can usually put enough tension on the string to make any droop negligible. But yes modern laser levels are a better if less tactile option in some cases.
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.
Three negatively curved surfaces (saddles) mate despite not being flat. You need to rotate the surfaces also when lapping. (See the famous Robrenz video)
I 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.
> 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.
> 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.
yeah its pretty hard to get new help to pull hard without fear of breaking the tool, coincidentally the beam width of a laser will increase over long distance, similar to line slack sagging over long distance.
The DeWalt that I've been using recently had a bright green line pencil thin, in the sun, easily ten meters from the device. That's far enough to do an entire patio - ten meters in each direction. I don't know about longer needs, such as sidewalks, but for building construction the DeWalt was excellent.
If 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.
I'm not from the UK, but the soft power of BBC Radio 4 in the late 90s and early 2000s (the Real Player era) made the UK seem like an advanced nation to my young and intellectually curious self. If lived in the UK at the time, I'd have been immensely proud of the quality of the programming.
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.
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.
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.
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.
https://en.wikipedia.org/wiki/Huzita%E2%80%93Hatori_axioms
I'm not getting any work done today
thank you!
LoL. Most welcome.
My best (worst) HN days are like that :)
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.
Yup and if you need a right angle, combine it with this
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.
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.
> 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 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...
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.
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
This 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...
It's technology that goes back thousands of years, here's one from the Iliad: https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext....
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.
Early humans fill me with awe with their ingenuity.
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.
The Tajima ones [0] are phenomenal, though the hook leaves a longer blank stretch than I'd like. They make a super nice snap knife too. Highly recommend Tajima for anything they make. Annoyingly, they don't sell a rip saw, only crosscut.
[0] https://www.tajimatool.com/product_category/mt/#chalk-rite
Using one of these to snap a perfect reference line is extremely satisfying.
"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.
Yes it will be a catenary but it is not a problem if the purpose is to mark the ground.
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.
You can usually put enough tension on the string to make any droop negligible. But yes modern laser levels are a better if less tactile option in some cases.
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.
Three negatively curved surfaces (saddles) mate despite not being flat. You need to rotate the surfaces also when lapping. (See the famous Robrenz video)
I did not know this, very interesting.
I 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.
Exactly. They were initially rolled, not merely pressed, which implies a longitudinal strain during the pressing moment.
Paper is thin so the stretching needed to bend it is minimal
Exactly.
> 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.
> 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.
Over long distances the string will sag in the middle. That's one of the reasons (not the only reason) construction uses lasers today.
yeah its pretty hard to get new help to pull hard without fear of breaking the tool, coincidentally the beam width of a laser will increase over long distance, similar to line slack sagging over long distance.
The DeWalt that I've been using recently had a bright green line pencil thin, in the sun, easily ten meters from the device. That's far enough to do an entire patio - ten meters in each direction. I don't know about longer needs, such as sidewalks, but for building construction the DeWalt was excellent.
I've never tried other brands.
the really serious guys use GPS based equipment, and its serious stuff, being able to center up on a 1/64 inch brad tacked into the asphalt.
[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
https://www.gpsalaska.com/
^ this is a very commercial website, but example of higher end equipment and technique
How do you calibrate the GPS
there is a sattelite network, and a terrestrial reference station infra
Yeah but how do we know there distances accurately
usually laser ranging is used to establish a baseline, rather than using a theodolite and chain like OG surveyors.
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...
It depends what you need that line for -- if you're projecting onto a wall, that sag matters. If you're projecting onto the ground, it doesn't.
Folded paper won't?
If 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.
Then there is the puzzle of coming up with folds to intentionally increase the rigidity of the surface. Essentially, using Theorema Pizzarium
Folded paper has some structure, so not as much?
I 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 ...
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.
It has a lot of air gaps within its fibers.
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.
But you can fold aluminum foil just fine? That certainly doesn't have "a lot of air gaps within its fiber"
Actually I've never had success cleanly folding aluminium foil. There are always imperfections and creases.
Pretty sure the material's innate stiffness plays a role there, too.
Metallic film folds quite nicely.
Nice old video about this: https://www.youtube.com/watch?v=EKEavnS10HI
It was this paper folding video (and the guy's unapologetic Finnish accent) that launched Hydraulic Press Channel to fame.
https://www.youtube.com/watch?v=KuG_CeEZV6w
The Mythbusters proved you can fold it up all the way to 11: https://www.youtube.com/watch?v=65Qzc3_NtGs
Unless I'm missing a transcript somewhere, this is missing an [audio] tag.
This is exactly the sort of hard-hitting journalism that makes me proud to pay my TV licence.
It’s always hard to tell whether a comment like this is supportive or sarcastic so this is either me agreeing or disagreeing with you…
For me this kind of simple, straightforward, educational content is part of the reason I’m proud to pay my TV licence.
I'm not from the UK, but the soft power of BBC Radio 4 in the late 90s and early 2000s (the Real Player era) made the UK seem like an advanced nation to my young and intellectually curious self. If lived in the UK at the time, I'd have been immensely proud of the quality of the programming.
And why does it sink. Its basically squished wood!
I really can't imagine getting paper to sink any more easily than wood?
Try toilet paper. Let it into the water vertically as to not catch air pockets.
It floats, unless it is burdened with matter. Even when fully wet.
because the fibers are denser than wood and hydrophilic - so the pockets between the fibers fill with water that displaces the air.
Air pockets become water pockets = neutral.
Fibers denser than water = sink.
Because the air is squished out I suppose
Shameless plug - https://foldmation.com
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.
A lovely little podcast on paper physics for origami.
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