Dice and Compass Games
By the way, I wasn’t the only one to write about that dice problem the other day. Jim Doherty, with the MrDardy blog, also spoke about it. He’s actively teaching, and hopes to report what his classes made of it. He writes regularly about the teaching experience and the experiments to try to make it better.
This did get me into a fun bit of Twitter chatter about the odds of bloggers writing about the same question like this. I can’t imagine the question having a real answer, though. We both wrote about it because we saw the same initial question on Twitter. But we saw it because we both try following stuff in the mathematics blogosphere. Among other things, that seeks out and connects fun problems like this. And it’s a problem easy to write up.
In a bit more of mathematical puttering-about news, here’s a pleasant little tool for making geometric constructions. It’s got compass-and-straightedge, as well as protractor-and-ruler, features. I admit I’m not sure I have a practical use for it, but it’s pretty and fun.
And you can do amazing things with compass-and-straightedge constructions. For my money, the most amazing thing is quadrature. That’s starting from some other shape and constructing a square with the same area. There are shapes it’s easy to do this for: rectangles, triangles, polygons of all sorts. There are shapes it’s impossible to do this for: circles, most famously. And then there are shapes you’d think would be impossible but aren’t, such as certain lunes. These are crescent-moon shapes. If circles are impossible (and they are), wouldn’t you think a shape with edges are the arcs of two different circles would be impossible too? And yet, they’re possible, for at least the right lunes.
Here’s one. Draw a half-circle. Let’s say, for convenience, that you’ve drawn the upper half of one. Now draw the vertical line from the center of the circle to its top point. Then draw the line connecting the leftmost corner to the top corner. This will be the hypotenuse of a right triangle with two 45-degree angles.
Next, draw the half-circle that fits on that hypotenuse, and that points outward, past the edge of the original half-circle. The lune of interest is the one between the original half-circle and the new one. And you can, using only compass and straightedge, produce a square with exactly the same area as that curved shape. If that’s not remarkable enough, it’s the same area as that triangle we had to start out. But we can not, using compass and straightedge, make a square that’s the same area as that little wedge between lune and triangle.
The quadrature of the triangle isn’t too hard to work out, if you start from scratch. (If you don’t know how to start, try starting with the area of a rectangle instead.) The lune, I’ll admit, I didn’t figure out by myself, but it’s not absurd. That the remaining wedge is impossible you won’t prove on your own. I’m not sure how I would explain it, not in only a few essays.
And with that hook, I’d like to toss in one last appeal for any requests for the Winter 2016 Mathematics A To Z. Before you pull out calendars on me and work out how long three-a-week essays might last, remember that I live in a state that typically gets a long winter. Letters are filling up, but many are still open. And last time around I had to really dig to find a good y- or z- term. If you want a sure in, those are good letters to think up.