In Chapter 18, Mandelbrot begins by describing the transformation known as inversion -- in other words, a circle reflection. I mentioned circle reflections two weeks ago here on the blog as an example of a non-Common Core transformation that the Geogebra software can perform. Well, some of the fractals Mandelbrot mentions are self-inverse -- that is, the image of itself under a circle reflection is itself. His examples of self-inverse fractals include Apollonian nets -- an infinite set of circles, each of which is tangent to some of the others.
Here Mandelbrot generalizes circle reflection to a more general transformation, which he calls a homography, or Mobius transformation. He defines a homography as the product (composite) of an inversion, a symmetry with respect to a line (a Common Core reflection), and a rotation. But as we already know (and will teach next week), a rotation is simply the composite of two reflections in intersecting lines, so mentioning "rotation" here is redundant. As it turns out, not only is every isometry a homography, but so is every similarity transformation! (Don't forget that the only transformations that appear in Common Core are similarity transformations.) This is because a dilation is -- as I mentioned two weeks ago -- the composite of two circle reflections.
In Chapter 19, Mandelbrot mentions another transformation that one can use to produce even more fractals -- namely, squaring, or performing the function z -> z^2 - mu for some complex mu. He calls a fractal that is invariant under this transformation self-squared. One self-squared fractal is so famous that it is simply known as the Mandelbrot set.
Here are links to the Apollonian gasket and the Mandelbrot set:
Last night was the first presidential debate for the Democrats. As I more or less expected, Common Core didn't come up during the debate. Even though, as I mentioned earlier, there exist Democrats who oppose Common Core, opposition to the Core is stronger among Republicans, since the Core was adopted under a Democratic administration.
We move on to Lesson 3-6, which is on constructions. Actually, let's go back to what I wrote about this constructions last year first, and then let me add this year's commentary:
Lesson 3-6 of the U of Chicago text deals with constructions. So, we're finally here. The students will need a straightedge and compass to complete this lesson.
Here's a good point to ask ourselves, which constructions do we want to include here? The text itself focuses on the constructions involving perpendicular lines. Well, let's check the Common Core Standards, our ultimate source for what to include:
.. and now here's what has changed since last year. First of all, I do now have a good constructions worksheet -- the Euclid: The Game worksheet from two weeks ago. I've also since created a worksheet based on the constructions mentioned in today's blog post.
Comparing today's worksheet to Euclid: The Game, we notice that Construction IV: Parallel Lines, is identical to Level 6 of Euclid: The Game. Notice that, as I said above, Construction II: Perpendicular Bisector, allows you to find the midpoint as a "bonus," while Level 2 of Euclid: The Game, actually asks for the midpoint. Construction III requires one to reflect a point, but as far as I know, reflections don't appear in Euclid: The Game at all.
Now here's the thing about Construction I, which is merely copying a segment. This construction sounds trivial with a compass -- and recall how I wrote earlier that this should be considered the simplest construction. Well, in Euclid: The Game, this construction isn't so simple. The problem is that to the ancient Greeks, the compass to be used in construction only allowed one to construct a circle with given center and radius (i.e., Euclid's Third Postulate). It didn't allow one to transfer distance -- this is often called a "collapsible compass" because as soon as the compass is lifted from the paper, it collapses, and so there's no way to recover the distance.
Now as it turns out, we can convert a "collapsible compass" into a regular compass. In fact, this is exactly what Levels 7 through 9 of Euclid: The Game are all about. After completing Level 9 of the game, we unlock a new tool -- the "compass" tool. Once this is unlocked, we can now perform tasks similar to Construction I, since using the compass tool is an abbreviation for performing the more complicated construction using the collapsible compass.
One interesting way of combining the worksheet for today and Euclid: The Game is by having the students play the game first -- including guessing how to solve the higher-level problems -- and then going back to see whether they are right when we cover constructions for real when they reach today's lesson.
Many schools around the country today administered the PSAT exam to sophomores and juniors. So today is a good day for a filler lesson. Last year, the day after I posted the worksheet for Lesson 3-6, I provided an activity where students construct a tic-tac-toe board. In some ways, an activity is a great lesson to post on a day when students are distracted by the PSAT.
Notice that on the day I posted tic-tac-toe last year, I mentioned the collapsible compass. That day, I gave a link to David Joyce's Euclid site, where he describes how the ancient Greek sage was able to use his collapsible compass -- it's his Proposition II. This is the same as Levels 7 through 9:
If the students view the tic-tac-toe board as just a higher level of Euclid: The Game, then might enjoy it, especially if they get to play Tic Tac Toe: The Game on the board afterward. But if constructions are seen as work, then they won't want to do it while other students are taking the PSAT, or after having just finished the PSAT themselves. So teachers should be careful with today's lesson.