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        riddles >> easy >> Constructing a trifold
        
(Message started by: 0.999... on Dec 2^nd, 2010, 5:16pm)

Title: Constructing a trifold
Post by 0.999... on Dec 2^nd, 2010, 5:16pm

I am not exactly skilled at Geometry, so this problem might seem obvious to some, but at any rate I enjoyed it.
Suppose we are given a (rectangular) sheet of paper and choose a side. How does one fold that side in thirds?

Note that we are not given any means of measurement other than folding of the given sheet of paper, and
the exact result must be obtained in a finite number of folds.

I think that it is understood what I mean by a fold, but if necessary, I can make that more precise.

Title: Re: Constructing a trifold
Post by icecoldcelt on Dec 3^rd, 2010, 6:07am

1. Fold into quarters
2. Open paper
3. [hide]Cut away 1 of the quarters[/hide]
4. [hide]Finished[/hide]

;D

Title: Re: Constructing a trifold
Post by towr on Dec 3^rd, 2010, 8:59am

I can do it with seven folds. But I don't have time to make a nice graphic at the moment. (And it's assuming you can make a nice fold through two known points, which may not be ideal.)

Title: Re: Constructing a trifold
Post by towr on Dec 3^rd, 2010, 3:05pm

[See attachment]

Title: Re: Constructing a trifold
Post by Noke Lieu on Dec 6^th, 2010, 3:27pm

Far more elegant than mine (as expected) though I'd throw down the challenge of actually doing it on paper to see whose is most accurate.

It's easier to describe thinking about rope...

Person A grabs end A. Person B grabs end B.
They both grab the rope (loosely) in another spot near their respective ends, and swap ends A&B. They then walk backwards, allowing the rope to adjust.

Unfortunately, this technique is going to be inaccurate due to the curvature inherent prior to folding.... By how much?

Title: Re: Constructing a trifold
Post by Grimbal on Dec 7^th, 2010, 6:32am

I can do it in 5. (~~too lazy~~ too busy to make a picture...)
[hide]
- fold in halves -> vertical line A.
- fold a diagonal, lower left to upper right -> line B
- fold a diagonal, upper left to lower center (use A) -> line C
- fold the right border to the intersection of B and C -> vertical line D
- fold the left border on line D -> line E

D and E fold the paper in 1/3s.
[/hide]

Title: Re: Constructing a trifold
Post by towr on Dec 7^th, 2010, 8:57am

Nice.

Title: Re: Constructing a trifold
Post by 0.999... on Dec 7^th, 2010, 5:38pm

Indeed, very elegant.

I had at first attempted to construct a central equilateral triangle with the desired side length, which I solved as long as it fits. Then, I realized, as it appears towr did, that the compass and straight-edge construction would translate over.

Here's an alternative five-fold solution when the side, A, you wish to fold in thirds is the shorter side. Denote the sides of the paper B,A,C,D in counterclockwise order.
[hide]Make half of a square by taking the corner BA and folding it onto C. (i.e. produce a diagonal H from corner AC to B with 45 degree angle of elevation and keep the fold the produced it.)
Fold the (45 degree) corner HC straight upward to where BA is, and open it.
You will have creased a portion of the perpendicular bisector of A, call it A'. Similarly for C, call it C'.
Take the corner AC to intersect C', to produce a fold that extends from the corner BA, and keep it there; fold BA to the point of intersection between B and A'.
Open it while holding BA there; fold AC to the location of BA.
These two flaps mark two points along B which divide it into thirds.[/hide]

Given the longer side [hide]the method is almost exactly the same except in the second fold where it goes all the way to the top (of course this is only true if the longer side is less than twice as long as the shorter)[/hide].