But there are several interesting fractals in this section of the book. Trees and their branches can be approximated with fractals -- and I once read a webpage on how one can use a fractal formula to program a computer to draw a tree. Mandelbrot also writes that veins and arteries in the body are also fractal, which is how they are able to transport blood to every part of the body.
Today marks the start of Unit 3 of my geometry course, which is on Rotations. According to my pacing plan, this unit will last until the end of the quarter.
But there are several things I must say about this unit. First of all, we won't actually see any rotations until next week, because first I must cover some of the material in Lessons 3-5 and 3-6 that we skipped over in order to start Chapter 4. Then after we introduce rotations, recall that I'm now including the triangle congruence theorems SSS, SAS, and ASA here in the rotations unit, which means that I'm moving some Chapter 7 material earlier.
Yet even today's Lesson 3-5 needs to be changed slightly from last year. First of all, this lesson includes our first parallel test -- the Two Perpendiculars Theorem. It made sense last year to include parallel tests here in the Rotations unit because we were using Hung-Hsi Wu's 180-degree rotations to prove the parallel tests, but this year we're not doing it that way. Still, there's no harm in keeping the theorem here.
But that takes us to our second problem. We actually proved the Two Perpendiculars Theorem last year by using the U of Chicago's definition of parallel -- the one that defines a line as being parallel to itself. I said earlier that I would not use that definition this year -- so this means that unfortunately I can't keep the proof as written.
Last year, I wrote that we can prove the Two Perpendiculars Theorem indirectly. In fact, on the blog I actually posted the indirect proof. This is what I wrote:
Assume that lines l and m are both perpendicular to line n, yet aren't parallel. Then the lines must intersect (as they can't be skew, since we said "coplanar") at some point P. So l and m are two lines passing through point P perpendicular to n. But the Uniqueness of Perpendiculars Theorem (stated on this about two weeks ago) states that there is only one line passing through point P perpendicular to n, a blatant contradiction. Therefore l and m must be parallel. QED
Then I made a big deal out of how allowing a line to be parallel to itself can convert an indirect proof into a direct proof. Here's how: the indirect proof assumes that the lines intersect at some point P and that leads to a contradiction, so therefore the lines are parallel. The direct proof says, assume that the lines intersect at some point P and that leads to the lines intersecting at every point. Therefore, the lines intersect either nowhere or everywhere, so therefore they are parallel.
We already agreed not to allow a line to be parallel to itself anymore. And so I can't simply post last year's Lesson 3-5 material where it's stated that a line is parallel to itself. Instead, here's my plan:
-- Notice that there were two pages labeled "Two Perpendiculars Theorem" last year. One of them introduces the theorem and hints at the indirect proof, while the other shows a direct proof using the parallelism of a line with itself. And so I just drop the page with the direct proof, so that all the students see is the indirect proof. We note that we usually don't write indirect proofs in two-column format anyway (although I once saw a text that did write indirect proofs in two columns). And so in a way, the first page for the Two Perpendiculars theorem is sufficient if all we want to show the students is the indirect proof.
-- Now the page "Guided Notes: The First Parallel Tests" can also be dropped. It actually gave the students hints on how to complete the direct proof, but sans the direct proof page, the guided notes are now superfluous as well (and besides, that page didn't print well and is fading red anyway).
-- The Line Parallel to Mirror Theorem is a special case. The direct proof that I provided didn't actually invoke the parallelism of a line with itself. But nonetheless, it's another proof that would look much better as an indirect proof than a direct proof. I hinted at this indirect proof last year but never actually wrote it. Here's what it looks like:
Given: l | | m, l' the reflection image of l with mirror m
Prove: l | | l'
Assume that there exists some point Q on l such that Q', the reflection image of Q, also lies on l. Now Q and Q' can't be the same point, because otherwise the reflection image of a point can only be itself when that point lies on the mirror m -- which would mean that Q lies on both l and m, an instant contradiction as we are given that l | | m.
So Q and Q' are distinct points on l. In other words, line QQ' is just another name for line l. Now by the definition of reflection, m is the perpendicular bisector of
I decided to keep both original Line Parallel to Mirror Theorem pages anyway, even though one may wish to change the proof page.
-- So we're now sneaking in indirect proofs here. Indirect proofs appear in the U of Chicago text in Chapter 13, but we're not covering Chapter 13 in order anymore. So I could move up the worksheet for Lessons 13-3 and 13-4 to today instead. The only problem is -- just as we saw with the 13-1 and 13-2 worksheet -- there are too many references to material from Chapters 5 to 12 on my indirect proof worksheet, because it originally appears in Chapter 13. Teachers can still dig up my 13-3 and 13-4 worksheet but would have to skip over questions on triangle angle-sum and other concepts that haven't been covered this year yet.
I originally wanted to avoid indirect proofs because they are tricky for students. But we've seen that those with a preference for strongly proof-based Geometry courses don't discriminate between direct and indirect proofs -- Dr. Franklin Mason covers indirect proofs early in his text (Lesson 2.6), and Dr. David Joyce criticized the Prentice-Hall text for not giving a proof of Postulate 1-2 (the Line Intersection Theorem), even though the derivation from Postulate 1-1 is actually an indirect proof. So today would be a great day to show the students some more mild indirect proofs.
This is what I wrote last year on today's lesson. I had to cut much of it out due to the changes on indirect proof -- so instead I only include a discussion of the other main result of Lesson 3-5 -- the Perpendicular to Parallels Theorem. As of today, I haven't decided whether I'm going to treat Perpendicular to Parallels the same this year as I did last year:
Officially, I'm doing Section 3-5 now, but then again, not really. Let's consider the contents of this particular section:
-- The definition of perpendicular has already been covered. I'm moved it to Section 3-2 when I defined right angles, because I wanted to get it in before jumping to Chapter 4 on reflections, since reflections are defined in terms of the perpendicular bisector.
-- The Perpendicular to Parallels Theorem is an interesting case. Last week, I mentioned that there are certain interesting and important theorems that are derived from Perpendicular to Parallels -- and these include the properties of translations as well as some of the concurrency theorems. I said that it might be better just to include this as apostulate and use it to prove those other theorems. And as I think about it more and more, I like the idea of including this as a postulate, then using it to prove those other theorems as well as Playfair's Parallel Postulate, and then finally using Playfair to prove the other Parallel Consequences following Dr. Franklin Mason.
Now as I imply in this post, most of what I write on this blog is derived from mathematicians like Dr. M and Dr. Wu, who have written extensively about Common Core Geometry. But my plan to include Perpendicular to Parallels as a postulate appears to be original to me. I've searched and I have yet to see any text or website who will derive all the results of parallel lines from a Perpendicular to Parallels Postulate. Then again, what I'm doing here is, in some ways, as old as Euclid.
Let's look at Playfair's Parallel Postulate again:
Through a point not on a line, there is at most one line parallel to the given line.
This is a straightforward, easy to understand rendering of a Parallel Postulate, and Dr. M uses this postulate to derive his Parallel Consequences. But let's look at Euclid's Fifth Postulate, as written on David Joyce's website:
That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
No modern geometry text would word its Parallel Postulate in this manner. For one thing, even though the use of degrees to measure angles dates back to the ancient Babylonians, Euclid never uses degrees in his Elements. So the phrase "less than two right angles" is really just Euclid's way of writing "less than 180 degrees." Indeed, in Section 13-6, the U of Chicago text phrases Euclid's Fifth Postulate as:
If two lines are cut by a transversal, and the interior angles on the same side of the transversal have a total measure of less than 180, then the lines will intersect on that side of the transversal.
But let's go back to the Perpendicular to Parallels Theorem as stated in Section 3-5:
In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.
Now count the number of right angles mentioned in this theorem. The transversal being perpendicular to the first line gives us our first right angle, and the conclusion that the transversal is perpendicular to the second line gives us our second right angle. So we have two right angles -- just like Euclid! So in some ways, making the Perpendicular to Parallels Theorem into our Parallel Postulate is making our postulate more like Euclid's Fifth Postulate, not less.
Of course, if we wanted to make our postulate even more like Euclid's, we could write:
If a plane, if a transversal is perpendicular to one line and form an acute angle (that is, less than right) with another, then those two lines intersect on the same side of the transversal as the acute angle.
But this would set us up for many indirect proofs, which I want to avoid. So our Perpendicular to Parallels Postulate is the closest we can get to Euclid without confusing students with indirect proofs.
So this is exactly what I plan on doing. Since we're in Section 3-5, the section that has Perpendicular to Parallels given, I could include it here -- we don't need to worry about how to prove it since I want to make it a postulate. But I already said that I want to wait until Chapter 5 before including any sort of Parallel Postulate. And so the new postulate will be given in that chapter.
Returning to Lesson 3-5:
-- The Perpendicular Lines and Slopes Theorem must also wait. Common Core Geometry gives an interesting way to prove this theorem, but the proof depends on similar triangles, which I don't plan on covering until second semester.