week[2]_foldingQuestion

This week we studied conjunctions and disjunctions. This is pretty much union and interaction so this part is not that hard for me. The more interesting thing is, I see instructor posts folding problem on next week slide and I decide to talk about it here this week.

Here is what I got from the Wikipedia.

The napkin folding problem is a problem in geometry and the mathematics of paper folding that explores whether folding a square or a rectangular napkin can increase its perimeter.

The rule is one can consider sequential folding of all layers along a line. In this case it can be shown that the perimeter is always non-increasing under such folding, thus never exceeding 4

Robert J. Lang, who is an American physicist, devised two different solutions. Both involved sinking flaps and so were not necessarily rigidly foldable. The simplest was based on the origami bird base and gave a solution with a perimeter of about 4.12 compared to the original perimeter of 4. The second solution can be used to make a figure with a perimeter as large as desired. He divides the square into a large number of smaller squares and employs the 'sea urchin' type origami construction described in his 1990 book, Origami Sea Life. The crease pattern shown is the n = 5 case and can be used to produce a flat figure with 25 flaps, one for each of the large circles, and sinking is used to thin them. When very thin the 25 arms will give a 25 pointed star with a small center and a perimeter approaching N2/(N − 1). In the case of N = 5 this is about 6.25, and the total length goes up approximately as N.