Monday, October 26, 2015

Halloween Geometry

Link to PSet

Since Halloween's nearly here, I thought a problem set with some spooky shapes would be fun. There's no answer key for this one because these are more like exercises than real problems: We walk through two fairly simple compass and straightedge constructions to make some Halloween-y moons and ghosts that have some special properties. If you end up making either of one of these (or both) feel free to send them my way and I'll post some student images here on the site!

A practical note: I suggest in the PS that you may want some graph paper for Exercise 2 in particular. If you don't feel like buying yourself a whole pad, you can find some patterns to print out a sheet at a time here. This site also has hexagonal graph paper and some other fun options we may use in later problem sets.

We're really only introducing two new topics here, but they may lead you to some fun ways to play around with Euclidean constructions. Once you know how to bisect lines and arcs, there's a lot of neat stuff you can do to play around making symmetrical shapes of your own design. Those shapes may also be a fun starting point for making your own tessellating shapes, per Exercise 2. I don't really talk about tesselations in depth here (I figured this could be more of a "follow-the-steps" kind of problem set), but we'll almost certainly return to them in future assignments and talk about their properties in more detail. For now, I'd say that trying to work out why the ghost works (especially how his final zig-zag needs to be drawn) is a good way to build some intuitions about tesselations.

If you like tesselations (and who doesn't?), you may want to try playing around with the two shapes below, which are called the "Kite" and the "Dart." These are prototiles for an aperiodic tiling of the plane, which means that try as you might, you won't be able to come up with a way to make a repeating pattern with these.

If you want to find out more about aperiodic tiling, go check out the wikipedia page for Penrose Tiling, which will point you to more fun ways to cover the plane. Happy Halloween, and happy constructing!

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