
Folding fractions - mmastrac
http://plus.maths.org/content/folding-numbers
======
Avshalom
There's an easier way to do this by the way:

Fold an arbitrary length perpendicular to the axis you want, then fold that
length again... To X folds. Fold diagonally from top corner to the last fold
then fold at the the intersections.

    
    
       ___    ___    ___    _____
      |   |  |___|  |__/|  |_|_|/|
      |   |  |___|  |_/_|  |_|/|_|
      |   |  |___|  |/__|  |/|_|_|
      |   |  |   |  |   |  | | | |
      |   |  |   |  |   |  | | | |
      |   |  |   |  |   |  | | | |
      |   |  |   |  |   |  | | | |
      |___|  |___|  |___|  |_|_|_|

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murbard2
Folding a third is not nearly as impressive as angle trisection, which is
impossible with a straight ruler and compass, but possible using origami.

[http://en.wikipedia.org/wiki/Angle_trisection#Using_origami](http://en.wikipedia.org/wiki/Angle_trisection#Using_origami)

~~~
cbd1984
> angle trisection, which is impossible with a straight ruler and compass

To be clearer: It is impossible to come up with a general algorithm to
_exactly_ trisect an _arbitrary_ angle using only a straightedge (not a ruler)
and a compass in a finite number of steps.

And the straightedge isn't a ruler because it can't be used to measure
distances. Neither can the compass. This is an axiom system, and therefore the
rules are absolute.

[http://en.wikipedia.org/wiki/Compass-and-
straightedge_constr...](http://en.wikipedia.org/wiki/Compass-and-
straightedge_construction)

~~~
murbard2
Yes, I meant straight edge, sorry I had an ESL moment.

~~~
pbhjpbhj
You're not wrong _per se_ \- a ruler usually has graduations but _could_ just
be a straight-edge. However, as the GP says, "ruler" implies a straight-edged
measuring stick (aka "measure" [n]).

Ruler derives, I gather, from Latin _regula_ which just means a straight stick
or bar and in turn derives from terms referring to keeping straight (literally
or figuratively).

tl;dr measuring stick ⊂ ruler.

~~~
murbard2
My native tongue is French, where it is "règle et compas", I translated règle
tu ruler, but added straight in front vaguely remembering that the English
expression had the word "straight" in it.

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dmayle
This is really useful for origami, when you want to fold unusual splits into
your paper.

You can also use it in the opposite direction, so instead of bringing the
point up to a specific mark on the top edge going from left to right (e.g. 1/2
on top gets you 2/3 on the right that you divide in two to get 1/3), you plan
it so that your right edge get's crossed at 2/n (e.g. fold the paper so that
the y length is 1/2, then your point will hit the 1/3 mark at the top of the
paper). This is useful when you want to go from an easy fraction like 1/8 to
1/7, as opposed to going from 1/4 through 1/5 and 1/6 to get to 1/7.

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chris_wot
I love using doing things like this.

Possibly somewhat off-topic, but the proof for the equation sin(A + B) = sin
A.cos B + cos B.sin X is quite a nifty diagram that can be found here:

[http://en.wikipedia.org/wiki/List_of_trigonometric_identitie...](http://en.wikipedia.org/wiki/List_of_trigonometric_identities#mediaviewer/File:AngleAdditionDiagramSine.svg)

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GregBuchholz
It seems like the link to the book is broken.

Origamics: Mathematical Explorations Through Paper Folding
[https://books.google.com/books?id=zJR2Rr_wuFQC](https://books.google.com/books?id=zJR2Rr_wuFQC)

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Avitas
This is one of those interesting items that is truly deserving of the headline
"Neat Trick."

It reminds me of a card trick that I learned as a child that appeared to work
by magic, but instead worked every time because it involved something complex
going on involving math.

~~~
dsjoerg
Ruler Manufacturers Hate Him!

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allthatglitters
The is a delightfully entertaining article! Does anyone have a recommendation
on the Kazuo Haga book, "Origamics"? vs David Mitchell's "Paper Crystals"?
Origamics is a bit pricey... worth it?

~~~
GregBuchholz
For books of questionable worth, I like to go to my local library, and get an
interlibrary loan. I find this indispensable for mathy books, where it can be
hard to judge the necessary prerequisites.

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hammock
Always wondered how to do this and never figured out how. My solution was to
roll up the paper into a circle and slowly crease at the right roll amount.
Reminds me of my old compass and straight edge lessons...

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amelius
Interesting. But, it must be said that he was just lucky that the x^2 terms
canceled :)

