## Wednesday, January 30, 2019

### an origami surprise

For a recent origami-based math activity, I gave students printed instructions for two origami models: a pinwheel, and an open-top masu box (both from Origami USA).

They were to learn how to fold the models and answer some questions about the results:
Assuming that the paper has length of one unit, without measuring can you determine the perimeter and area of the pinwheel, and the volume and surface area of the box?
I could honestly tell them: I did not know the answers, so they would have to explain to me how they found the results.

Opening up the finished models to reveal the pattern of folds provided a good strategy for getting to the answers. Considering how the folds divided the paper (into sixteen squares) and using the Pythagorean Theorem to calculate the lengths of diagonal folds allows you to get all the lengths you need.

The multiform pinwheel has a crease pattern like this:

And the masu box has the following crease pattern:

When it came time for the answers to the math problems I had posed, I had a mild surprise: two of the quantities that I had asked for came out to the same value - the area of the pinwheel and the outside surface area of the box were identical (3/8 units - interestingly just under half of the paper is exposed, the rest is folded in).

Taking another look at the crease patterns, you can see how the image of the box can be transformed into the pinwheel, demonstrating without calculations that the areas are the same: