Chapter 3 Review (Day 37)

This is what Theoni Pappas writes on page 279 of her Magic of Mathematics:

"Michael Barnsley first considered using randomness as a basis for modeling natural shapes. Hence he invented The Chaos Game."

We already knew yesterday that Pappas continues the Chaos Game on page 279. The inventor of the game, Michael Barnsley, is a British mathematician. He's in fact still alive -- this year he turns 71.

You're still waiting to find out what shape we were able to create using the die and equilateral triangle in yesterday's post. Even though the process is random, a certain shape is more likely to appear than any other shape. The lone picture on this page shows me what this shape is, but of course you can't see this picture at all.

Let me give you a clue as to what the picture looks like. The final sentence on this page is:

"In addition to discovering that the limit of a random process (following a set of rules) produced fractals, Barnsley also devised a random method to reproduce a given shape."

The key word in that phrase is fractals. That's right -- we read Mandlebrot's book about fractals two years ago, and Pappas also mentions fractals in her book. (See my May 10th-12th posts for more information on fractals.)

Indeed, two years ago I wrote:

Part VI of Benoit B. Mandelbrot's The Fractal Geometry of Nature is on Self-Mapping Fractals, and consists of Chapters 18 through 20. In this section, Mandelbrot describes even more methods of obtaining fractals besides self-similarity.

Yet surprisingly, none of those "even more methods" mentioned in Part VI are random processes.

And I actually mentioned the correct answer to today's problem even more recently. Notice that the original three points were the vertices of an equilateral triangle. And as soon as we hear "triangle" and "fractal," one image immediately comes to mind -- Sierpinski's Triangle.

No, Pappas actually never mentions the name Sierpinski in her book, but the picture that she shows us is unmistakably Sierpinski's Triangle. And I mentioned the famous fractal back in my August 17th post, because Michael Serra mentions it in Lesson 0.2 of his Discovering Geometry text.

So if you want to see the fractal, just go back to my second day of school worksheet. Or it's easy just to make a Google image search and find the fractal. Here is a result I found:

https://www.zeuscat.com/andrew/chaos/sierpinski.html

The link above even ties Pascal's Triangle to Sierpinski's using even and odd numbers -- a result alluded to in Ogilvy. So yet again, I find a reason to mention Ogilvy's number theory book.

Pappas doesn't tell us exactly why the random process produces Sierpinski's Triangle. But I think I can demonstrate why a similar random process produces another simple fractal -- the Cantor set.

Recall that Cantor's middle-thirds set is the fractal produced by taking the unit segment and removing the middle third, then removing the middle third of each of the remaining segments, and so on. We know that if the vertices of the original segment have coordinates 0 and 1, then the members of the Cantor set can be written with only 0's and 2's in base 3 (ternary). This is because removing the middle third actually corresponds to removing all the 1's.

Now here's a Barnsley-like process that produces the Cantor set:

1. Label two points H and T.
2. Pick a starting point at random. Flip a coin.
3. Rule: take 2/3 the distance between the starting point and either H or T, depending on whether the coin landed heads or tails.
4. Continue step 3 indefinitely.

I claim that this step reproduces the Cantor set. Let's find out why -- clearly the fraction 2/3 in step 3 has something to do with it, since this is the Cantor middle-thirds set. By "2/3 of the way" between the starting point and either H or T, this means that the next starting point is closer to H or T than to the previous starting point. (I just want to clarify the difference between "2/3 of the way" and "1/3 of the way" here.) You might remember that identifying locations "part of the way" between two points is a Common Core Geometry standard (GPE.B.6).

Now let's suppose that point H has coordinate 0 and point T has coordinate 1, and let's say that our starting point S has coordinate 0.abcd... when written in ternary. What happens when our coin lands on heads? We calculate the next point S' as follows:

S' = (0 + 0 + 0.abcd...)/3 = 0.abcd.../3 = 0.0abcd...

It's easy to divide by three in ternary -- just shift all digits right and add a zero. So flipping heads is equivalent to adding a 0 to the left of the number.

If the coin lands on tails, then we calculate:

S' = (1 + 1 + 0.abcd...)/3 = 2.abcd.../3 = 0.2abcd...

So flipping tails is equivalent to adding a 2 to the left of the number. Thus after flipping the coin repeatedly, we will add lots of 0's and 2's to the left of our number. Therefore we will eventually reproduce the set of all numbers with lots of 0's and 2's in them -- and this is exactly the Cantor set.

Believe it or not, this works even if the original point isn't on segment HT, or even on line HT! Let's assume now that the endpoints are now H(0, 0) and T(1, 0), and our starting point S(x, y) isn't on the x-axis -- that is, that its y-coordinate isn't zero. Then each time we perform step 3, the new y-value is exactly 1/3 of the old y-value. So the sequence of points quickly approach the x-axis, in that for any value epsilon > 0, all points will eventually be within epsilon of the x-axis -- and so as far as the eyes are concerned, the points might as well be on the x-axis. And the x-coordinates will still be stuffed with 0's and 2's, and so they are in the Cantor set. If we start on line HT but not between H and T, then after sufficiently many steps, we'll wind up between them. And if we start on a point that's not in the Cantor set (because its coordinate has a 1), eventually the 1's will be pushed so far to the right by 0's and 2's that once again, our eyes will convince us that we're in the Cantor set.

OK, let's get to the U of Chicago text. Chapter 3 is one of the two shortest chapters in the text -- it has only six sections. All the other chapters from 2 to 6 have seven sections, and we've seen that with seven sections, there's time for two review days rather than one.

But providing a third review day for Chapter 3 is awkward. It's one thing to review on Monday for a Wednesday test, but it's another to review the previous Friday for a Wednesday test. It's more logical to study two days before the test than five days before, unless it's a big test like the final (or the state test, or the PSAT, or the bar exam, and so on). With a mere chapter test, studying the previous Friday is counterproductive as students may forget over the weekend.

Oh, and by the way, earlier I mentioned the PSAT as an example of a test for which students may prepare five or more days in advance. It has come to my attention that many schools will be administering the PSAT on Wednesday, October 11th -- and this might affect the day on which the Chapter 3 Test can be given.

Originally, this was going to be my plan for the last few days of Chapter 3:

Friday, October 6th: Activity Day (Day 37)
Monday, October 9th: Chapter 3 Review Day 1 (Day 38)
Tuesday, October 10th: Chapter 3 Review Day 2 (Day 39)
Wednesday, October 11th: Chapter 3 Test (Day 40)

This follows the pattern that we've established earlier this year -- Chapter 1 Test on Day 20, Chapter 2 Test on Day 30, and so on.

But Day 40 is exactly the day that the PSAT will be given. I've said it before on the blog that it is cruel to expect students to take the PSAT for four hours and then give them another test the same day.

Furthermore, notice that this is Day 40 according to the district whose calendar I'm following. As it turns out, the high schools in this district will actually observe a minimum day on Wednesday. The students will take the PSAT four hours, eat lunch, and then are dismissed to go home! All freshmen will be taking the PSAT 8/9 on Wednesday, and all seniors will have special activities that day. And so no student will attend any classes that day. So I couldn't give the Chapter 3 Test on Wednesday even if I wanted to.

Other districts might still hold classes after the PSAT, but it would be inconsistent for me to follow the day count of a district and not follow all of its testing days -- including finals and the PSAT. And besides, it's absurd to insist that students test on Wednesday when we have an extra day anyway due to the short six-lesson chapter.

And so instead, we take advantage of the shorter chapter to have two review days for the test (just like all the other chapters from 2 to 6), since the spare day will be used for the PSAT:

Friday, October 6th: Chapter 3 Review Day 1 (Day 37)
Monday, October 9th: Chapter 3 Review Day 2 (Day 38)
Tuesday, October 10th: Chapter 3 Test (Day 39)
Wednesday, October 11th: PSAT Test (Day 40)

Now for Chapter 2, we used the first review day for the worksheet and the second day for a special activity based on Geometry teacher Shaun Carter's blog -- but that was when the test was on a Wednesday and review days were Monday and Tuesday. With a Tuesday test, it makes more sense for the review worksheet to be on Monday (again because of the weekend).

And so today I post Shaun Carter's lesson. This looks like interesting way to teach parallel lines and corresponding angles. But unfortunately, he doesn't provide a worksheet for this lesson. Instead there is only a Twitter post:

https://twitter.com/theshauncarter/status/910927986542686209

I created a worksheet for this lesson based on what Carter has on his board in this Tweet. I like the idea of using color-coded Post-its just as Carter does. Notice that this lesson serves as review not just for Lesson 3-4 on parallel lines, but also Lesson 3-2 on linear pairs and vertical angles.