Ice Skating (approximately)!

Link to PSet
Answer Key: Pending

It's been a while since I posted a new problem set (and I still owe everyone an answer key for HW8), but hopefully I can make up for that with the return of Boofernaut and Boofinski. This time, the two of them are doing some ice skating together and Boofernaut is trying to figure out how to make the best copy (or approximation) that she can of the paths her brother makes on the pond.

The goal of this assignment is to get kids thinking about how to make a model of something complicated with simpler tools. We tackle this in one dimension by trying to make piecewise-linear approximations of curved paths and then in two-dimensions by using LEGO bricks to make a model of a closed contour. In both questions, we ask kids to think about what changes in their approximations as they get to use a lower vs. higher resolution model (more lines in 1D, or making the shape bigger/smaller in 2D) as well as how they decide what approximation is the best. How do you choose what details to leave out and which ones to model?

I also ask kids to think about the "inverse problem" in each case: If you only have the approximation to start with, what do you think the original curve or shape was? This is particularly fun because there is definitely not just one right answer - there are endless original curves that could lead to the same simpler model. Inverse problems are everywhere in science and mathematics (we deal with them in my lab when we try to use EEG measured on someone's scalp to make guesses about where in the brain activity is coming from), and hopefully this is a nice introduction to the challenges of making inferences from sparse data.

We ask you to use some LEGO bricks to solve the second set of problems here, so I thought I'd include a link to one of my favorite 3D shape approximations - the LEGO sphere! The instructions at this link are not mine, but we did manage to follow them a while ago to make a pretty sweet approximation of a half-dome out of standard bricks (we didn't have enough unused bricks to make the whole thing). Enjoy!