
Folding Paper in Half (2015) - ibra
http://fermatslibrary.com/s/folding-paper-in-half
======
kazinator
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.

~~~
greggman
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)

~~~
kazinator
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.

------
_Codemonkeyism
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?

~~~
kazinator
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.

~~~
_Codemonkeyism
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 ...".

~~~
kazinator
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.

------
MichaelBurge
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](https://www.youtube.com/watch?v=KuG_CeEZV6w)

~~~
dsr_
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.

~~~
pluma
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](https://en.wikipedia.org/wiki/Pattern_welding)

------
zirkonit
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.

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al_biglan
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!"

~~~
Rangi42
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...](http://math.stackexchange.com/questions/581958/solving-cubic-
equations-with-origami)

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

~~~
BrandonM
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.

