Saturday, March 17, 2012

origami envelope redux

A while ago I posted about how the traditional origami envelope could make a good object for simple math investigations. You can find instructions for the envelope on Origami USA's website. Like I mentioned, there are a lot of nice aspects to the envelope, such as the shapes in its crease pattern and its rotational symmetry.  One thing to look at is the relationship between the size of the envelope and the paper you start with.  It turns out that there is a nice relationship between the interior rectangle in the crease pattern and original paper (the area of the front/back of the envelope is just this rectangle with two of its corners trimmed off).

If you assume that the sides of the original paper are of length a and b, with a greater than b, you can begin to work your way towards finding out the dimensions of the resulting rectangle by following along with the folds, and using the Pythagorean theorem (actually, you just need a very special case of Pythagoras - for triangles with 45 degree angles)

Following the folds, subtracting and adding from some lengths what you deduce from other folds leads to some slightly intimidating looking expressions.

Thankfully, these simplify down to what I found to be a surprising result. The interior rectangle has an area equal to one quarter the area of the original rectangle. More surprising to me was that each side of the smaller rectangle depended only on one of the sides of the original - the longer side of the interior rectangle is 1/sqrt(2) of the shorter original side, and the shorter side of the interior rectangle is 1/(2sqrt(2)) of the longer original. 

I'm sure there are probably easier ways to see this relationship. If you construct it in GSP or other dynamic geometry package you can experiment easily with increasing the side lengths.