Folding Paper in Half (2015) | Hacker News Reader

Folding Paper in Half (2015)

fermatslibrary.com

116 pointsibra20 comments

The idea that you can only fold paper eight times in one direction rings absurdly false, since the problem screams of being obviously related to the thickness of the paper in relation to length.

An informal, physical/graphical proof that there is no upper limit given sufficient length is simply this. Imagine a piece of paper that has already been folded N times:

           --------\
          |         \
          |______    |
          /------`   |
          |          |
          \---------/
             
         ->|      |<- flat section

All you have to do is add more material to the flat section, extending the length of each layer by an equal amount, and then you can fold it again:

           ------------------------\
          |                         \
          |_____________________     |
          /---------------------`    |
          |                          |
          \-------------------------/
             
         ->|                      |<- flat section

You can add as much as you want. Suppose the above is an inch thick and we extend it to be a mile long.

I.e. don't approach this as a problem of the subdivision of a fixed length and width, but rather as the length being open-ended: we can go back and add more.

Is that really true? I mean the radius of the fold is going to mean the outside most layer is going to have to get larger and larger relative to the inner most layer. If it's 1 mile thick then the outer layer will need to be 3.14 miles longer than the inner layer. Won't that have some kind of effect? Is it all proportional or will the length of that difference eventually make it so the outer piece can't stretch/bend/flex over that great a distance (in reality vs magically friction-less paper)

Indeed, basically you're questioning whether this is literally folding in half. But I don't see that constraint anywhere. Just there have to be so many folds, and a flat section. So in half is being used colloquially.

Isn't that what the linked paper says? Its form suggests that the author had a similar insight to this one.

I wonder why in the title this is a "mathematical" limit, not a physical one. Because there are formulas and calculations?

The formula expresses the folding limit based on the physical variable t (thickness).

Does the title confuse "Mathematical" with "Theoretical" limit?

The physical limit is actually more stringent because of the way materials behave under bending.

This is mathematical if we assume geometry to be subsumed under mathematics: it's a geometric problem involving stripes and arcs of stripes.

Don't want to nitpick, but the second part of the title says "how many times you can fold a piece of paper in half" not "geometric problem involving stripes and arcs of stripes".

I might think it's either a "Mathematical limit about the geometry of arcs of stripes" or "Theoretical/Physical limit of how many times ...".

That's true; of course the question can be a departure point into material science: what are the issues in bending a sheet of aluminum (or any other material) N times, or whatever, including the stretching and compression that takes place at the bends and so on. The context of it here being a pure geometric exercise is established early on.

Here's what happens if you ignore the mathematical limit and fold it again with a hydraulic press:

https://www.youtube.com/watch?v=KuG_CeEZV6w

What's shown is not a "folding" action, which is reversible, but a "mixing" action.

The fabled Japanese sword steels are produced with a similar non-reversible mixing action, increasing the carbon content of the alloy each time.

Just because the hype around katanas is a pet peeve of mine: although Japanese swords are better known for this technique it's not exclusive to Japan by far. See https://en.wikipedia.org/wiki/Pattern_welding

Oh my, that Hydraulic Press Channel is quite something!

And I thought the Will It Blend guy was crazy...

I prefer the power hammer [0] over the press, it's a whole different level of crazy. The amount of energy it releases is downright frightening.

[0] https://www.youtube.com/watch?v=mhRNO-MpEf4

You'd find that the n-folded paper is both too tick and to narrow to achieve a n+1 fold. You could still put it in the hydraulic press and apply plenty of force with it, but the points where you apply the force will be too clumped together; the press would not fold the paper, just squeeze it really, really hard, and probably deform it or tear it apart at places.

We can debate the semantics but that's not "paper" anymore.

Current world record is 13 folds.

For 14 folds, the minimum according to the paper is 373x373 feet, for the 15 folds (“5-meter thick”) – a bit over 1000x1000 feet.

This is a great way to teach math. Model something that is interesting but only needs to rely on some basic principles (geometry, series and limits in this case). Best quote of the piece: "Britney found it interesting to realise that when we fold a piece of paper, we are actually finding a solution to a quadratic equation!"

You can even solve cubic equations with origami[1], which is what allows you to double a cube and trisect an angle by adding folding to the usual tools of compass and straightedge. (It's still not enough to square the circle.)

[1]: http://math.stackexchange.com/questions/581958/solving-cubic...

This is beautiful work and a very nice paper, especially for an undergrad.

Note that she was not a university undergrad, but a grade-11 high school student, probably 16 or 17 years old. In the US, each year is a grade, and grade-12 is the final year of high school (which precedes university).

That only makes the work and paper more impressive, IMO. In my high school, our math teachers had very weak theoretical backgrounds. I suspect that Britney would have received less guidance in high school than would be available at university.

Reply on news.ycombinator.com