Tuesday, March 28, 2017

Lesson 12-8: The SSS Similarity Theorem (Day 128)

This is what Theoni Pappas writes on page 87 of her Magic of Mathematics:

"Having been influenced by perspective in the art of the Renaissance and desiring to help artists, 17th century architect/ engineer/ mathematician Girard Desargues initiated the study of projective geometry. He wrote a number of theorems (Desargues' Theorem) that are still essential in the study of projective geometry."

Of course, I had to check out Desargues' Theorem. Here's a link to Cut the Knot:

http://www.cut-the-knot.org/Curriculum/Geometry/Desargues.shtml

Let A1B1C1 and A2B2C2 be two triangles. Consider two conditions:
1. Lines A1A2, B2B2, C1C2 joining the corresponding vertices are concurrent.
2. Points ab, bc, ca of intersection of the (extended) sides A1B1 and A2B2, B1C1 and B2C2, C1A1 and C2A2, respectively, are collinear.
Desargues' Theorem claims that 1. implies 2. It's dual asserts that 1. follows from 2. In particular, the dual to Desargues' theorem coincides with its converse.
Curiously, Desargues' theorem admits an intuitive proof if considered as a statement in the 3-dimensional space, but is not as easy in the 2-dimensional case, where it is often taken as an axiom [that is, a postulate -- dw].

In researching Desargues' Theorem, I also stumbled upon the concept of homography. All the transformations mentioned in this section of Pappas as well as Lesson 1-5 of the U of Chicago. Of course, we're supposed to imagine a homography as mapping (or projecting) a figure from one plane to another, but we can more easily consider a homography mapping figures in the same plane. In this case, homography has a simple definition -- it's a 1-1 correspondence that preserves B-C, betweenness and collinearity.

Clearly all isometries are homographies. All similarity transformations are homographies. Last year, I wrote about the concept of an affine transformation -- Lesson 6-1 of the U of Chicago text gives a simple affine transformation, S(x, y) = (2x, x + y). Well, all affine transformations are homographies.

Isometries and similarity transformations map squares to squares. Affine transformations map squares to parallelograms. As Pappas writes, homographies map squares to any quadrilateral -- indeed, given any quadrilateral, there exists a homography mapping a square to it.

Homographies are often understood using matrices, just as isometries, similarity, and affine transformations often are, but this is tricky. I mentioned how McDougal-Littell Integrated Math II text (which I didn't buy -- I have only the Math I text) uses transformation matrices. Mostly likely, the text uses matrix multiplication for reflections and rotations, but matrix addition for translations. This is unsatisfying as it seems that we should be able to use multiplication for all isometries. But the problem is that multiplying any 2x2 matrix by the vector (0, 0) gives (0, 0), yet translations clearly don't map the origin to itself.

The trick is to pretend that the xy-coordinate plane is actually the plane z = 1 in 3D. Then a particular matrix multiplication can actually map (0, 0, 1) to (h, k, 1) and represent a translation. It turns out that all homographies in the plane can be represented by 3x3 matrices -- the only restriction on the matrix is that its determinant be anything but zero. For the homography to be an isometry, though, the condition on the matrix is complicated -- it's necessary that the determinant of the matrix be 1 or -1, but that's not sufficient. For example, trigonometry is required to find the matrix of a rotation whose magnitude isn't a multiple of 90 degrees.

OK, that's enough about homographies. I just write about them in order to point out that there's a whole world of transformations beyond isometries and similarity transformations -- affine transformations and homographies are the next two steps up the ladder. So in particular, isometries and similarity transformations aren't things invented by the Common Core to torture traditionalists.

This is what I wrote last year about today's lesson:

Let's get to our Geometry lesson. We are now working on the AA~ and SSS~ theorems, which complete our study of similarity. There are several ways we can prove these at this point. We can use the original dilation proofs given in the U of Chicago text (Lessons 12-8 and 12-9), or we can use the one similarity theorem we already have (SAS~) plus the corresponding congruence theorems (ASA and SSS, respectively). [2017 update: last year I reversed Lessons 12-8 and 12-9 but this year I'm preserving the original order, so no, we can't use SAS~ to prove SSS~.]