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.

envelope doodle

prisoner's dilemma

"The essence of mathematics resides in its freedom."
- Georg Cantor 

In a recent article in Prospect magazine, David McConnell writes of his experience teaching basic mathematics to prison inmates. He explains,  "I once decided to teach maths to prisoners. Surprisingly, many of them embraced prison discipline to study for the General Educational Development (GED) test...." Like his students, McConnell's own learning path wasn't a direct one:

I’d re-learned maths myself as a kind of tourist or traveller. Instead of the disconnected and seemingly arbitrary techniques I’d studied in the odd moments between recess and leaf collecting and dinosaur books, I found maths a unified country, sober and dazzling, when I returned to it as an adult.

Both in his return to math as an adult, and in his reflections on math's appeal for those that have lost their freedom, McConnell's piece reminded me of an article about Wole Soyinka written by Anushree Majumdar in the Indian Express a couple of years ago:

As a school student, Nigerian author Wole Soyinka loathed mathematics... Little did the Nobel Laureate know at the time, that mathematics would save him from losing his mind, during his imprisonment in 1997. “When I was imprisoned, I was thrown into solitary confinement. I had been placed under trial but it was a barren existence. I invented games in my head. I began doing mathematics again. I’d scratch on the floor of the cell with a stone, working  out permutations and combinations, using different formulae. Hours would pass  but it nearly drove me crazy too ... I had to create an interior life to survive”

A literary example of using mathematics as a means of hanging on to sanity (unsuccessfully) under imprisonment (and torture) is found in George Orwell's 1984. In 1984, Winston Smith clings to the one truth that he feels cannot be unmade, that 2 + 2 = 4.

Against this narrative of mathematics as liberation or resistance stands another that sees the rigor and certainty of math as an oppressive force. As Dostoyevsky's underground man says: " twice two makes five is sometimes a very charming thing too." (See the Wikipedia article on 2+2=5 for more.)

three views of K(6,12)

some complete bipartite graphs