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        riddles >> putnam exam (pure math) >> Square Folding Curiosity
        
(Message started by: Perfection on Mar 13^th, 2004, 11:30pm)

Title: Square Folding Curiosity
Post by Perfection on Mar 13^th, 2004, 11:30pm

So today I was just you know, chillin', hangin' out and what-not and I came up with an interesting question

If I were to take a 1x1 piece of paper (negligible depth, infinite flexibility, cannot be ripped) what is the largest distance I can get between 4 points if it is required that they all be equidistant?

So I knew it must somehow reach all vertices of a tetrahedron, and since I knew that two equilaterial triangles sharing two points meet those requirements I realized the largest one that could be made on the paper had the diagonal for altitudes and met at the other one giving me a rhombus with side lengths of sqrt(2/3). Here's the fun part, is this the true solution or is there some way to make the 4 equistant points farther out with the 1x1 square.

Title: Re: Square Folding Curiosity
Post by grimbal on May 27^th, 2004, 6:04pm

I think you can have the 4 points at the summit of a tetrahedron with side 1. It is a little difficult to explain,though.
Make the construction you did, but make the common base of the triangles with length 1. Fold 90° down along the base, fold up along the sides of the triangles and at the ends of the diagonal. You should have the 4 corners on a perfect tetrahedron.

Title: Re: Square Folding Curiosity
Post by SWF on May 31^st, 2004, 5:54pm

In the figure below the paper is white on one side, yellow on the other.
1) Fold along diagonal to make triange
2) Fold back and forth on black line (I did not calculate exact position- might be drawn out of proportion).
3) To get from position in third figure to that in forth figure, you need to open up the paper and flip the portion with D on in inside out.
ut of proportion).
4) Open up until distance from B to C is 1, and if the position of fold in step 2 was correct, D will be distance 1 from A, B, and C.

Title: Re: Square Folding Curiosity
Post by Perfection on Jun 6^th, 2004, 8:40pm

Okay, now I'm confused.

Would it be possible for one you fine folks post an unfolded versian showing where all the fold lines are?

That would make me very happy!