Why does paper fold so well?

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.

arijun2• 1 month 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.

srean14• 1 month 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.

qsera• 1 month 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• 1 month 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.

srean14• 1 month 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.

https://en.wikipedia.org/wiki/Huzita%E2%80%93Hatori_axioms

saagarjha• 1 month ago

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.

srean14• 1 month ago

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.

globular-toast• 1 month 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.

srean14• 1 month ago

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 ...

dotancohen3• 1 month ago

Over long distances the string will sag in the middle. That's one of the reasons (not the only reason) construction uses lasers today.

rusk• 1 month ago

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.

wongarsu• 1 month ago

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

bena• 1 month ago

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...

mjanx1232• 1 month ago

Paper is thin so the stretching needed to bend it is minimal

srean14• 1 month ago

Exactly.

foobarian• 1 month ago

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.

mjanx1232• 1 month ago

Three negatively curved surfaces (saddles) mate despite not being flat. You need to rotate the surfaces also when lapping. (See the famous Robrenz video)

srean14• 1 month ago

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.

jayd16• 1 month ago

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.

oidar• 1 month 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.

arijun2• 1 month ago

Unless I'm missing a transcript somewhere, this is missing an [audio] tag.

leebeef• 1 month ago

This is exactly the sort of hard-hitting journalism that makes me proud to pay my TV licence.

tomalpha• 1 month ago

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.

ern• 1 month ago

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.

Peterz_shu• 1 month ago

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.

gruez• 1 month ago

But you can fold aluminum foil just fine? That certainly doesn't have "a lot of air gaps within its fiber"

dotancohen3• 1 month ago

Actually I've never had success cleanly folding aluminium foil. There are always imperfections and creases.

IAmBroom4• 1 month ago

Pretty sure the material's innate stiffness plays a role there, too.

Metallic film folds quite nicely.

fuzzythinker• 1 month ago

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.

Why does paper fold so well?

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