Sunday, July 27, 2014

Source 4: Jennifer Silverman

My final source, for now, is Jennifer Silverman, an independent math consultant:

On her page, Silverman states seven steps to building a geometric system based on the Common Core Standards. At first, we notice that Silverman's Step 4 is clearly based on rotations, and so this is not much different from Wu's presentation. But Steps 5-6 jump out at me:

5. Lines equidistant at 2 places are equidistant everywhere.
6. Equidistant lines are defined as parallel.

This is an interesting definition of parallel. In fact, I once saw another website (which I've now lost and may no longer exist) which showed an animated image of two parallel lines cut by a transversal and a translation, along the direction of the transversal, which made corresponding angles coincide (as opposed to the rotation making alternate interior angles coincide). Even though Silverman's webpage is based on rotations, I wondered whether there could be another geometric system based on parallel lines, corresponding angles, and translations, using her Steps 5-7.

But there's a problem with Silverman's definition of "parallel." To see why, we must go all the way back to Euclid.

Anyone remotely familiar with Euclid is aware of his Fifth Postulate, the one on parallels. It is independent of the other four postulates -- that is, no matter how hard we try, we can't prove it as a theorem (unlike many of the other so-called "postulates" in modern texts). We now know that there are several non-Euclidean geometries. In particular, we have:

Hyperbolic Geometry: many parallels to a given line through a point not on that line
Euclidean Geometry: exactly one parallel to a given line through a point not on that line
Elliptic Geometry: no parallel to a given line through a point not on that line

Now, as it turns out, some of the results involving parallel lines require the Parallel Postulate to prove, while others don't. In particular, the statement that if two lines are parallel, then alternate interior (or corresponding) angles are congruent require the Parallel Postulate, but the converse does not require it. We see this with Euclid himself, where Propositions 27-28 don't invoke his Fifth Postulate, while Proposition 29 does require the Fifth.

Most modern texts don't distinguish between the former and the latter, but authors like Dr. M and Wu do make the distinction. Mathematicians prefer to use as few postulates as possible when proving theorems -- this is considered more elegant. Indeed, Euclid himself does this -- he proves his first 28 Propositions using only the first four Postulates, then only beginning with Proposition 29 does he invoke the Fifth Postulate. So I should try to prove "if corresponding angles are congruent, then the lines are parallel" without using the Parallel Postulate.

Now here's the problem: the statement "lines equidistant at 2 places are equidistant everywhere" (Silverman's Step 5) requires the Parallel Postulate. It's only true in Euclidean geometry -- indeed, only in Euclidean geometry do there even exist lines that are equidistant everywhere. And so Silverman's Step 6, "equidistant lines are defined as parallel," is not equivalent to the standard definition of parallel unless one assumes the Parallel Postulate. As it turns out, this isn't a problem in elliptic geometry, where there are neither parallel nor equidistant lines (so technically speaking, lines are parallel iff they are equidistant is vacuously true). The problem is in hyperbolic geometry, where there are infinitely many lines parallel to a given line through a point not on the line, but none of those lines are everywhere equidistant from the given line.

One way to avoid this is to combine Steps 5 and 6: "lines equidistant at 2 places are parallel" (where "parallel" means the standard definition -- using this as a new definition remains invalid). As it turns out, this is provable without the Parallel Postulate. Its proof involves a shape called a "Saccheri quadrilateral" -- a figure with opposite sides parallel (so it's a parallelogram) and two right angles, yet isn't necessarily a rectangle (unless one adds the Parallel Postulate back).

Indeed, let's return to translations -- what I was originally trying to accomplish here. We look at the definition of translation as given by the U of Chicago text -- notice the similarly between this definition and that of rotation given yesterday:

U of Chicago:
"A translation (or slide) is the composite of two reflections over parallel lines."

This definition is invalid unless we assume the Parallel Postulate. Indeed, Wu's definition of "translation" also mentions the Parallel Postulate. So without the postulate, we're having trouble even defining "translation"!

Notice that in elliptic geometry, there are no parallel lines and hence no translations. In hyperbolic geometry, there is a definition of "translation" that's similar to the U of Chicago definition. There, a translation is the composite of two reflections over ultraparallel lines. As it turns out, most pairs of parallel lines are "ultraparallel" in the sense that through a point not on a given line, there are infinitely many ultraparallel lines but only two parallel lines that aren't ultraparallel. If the reflecting lines are parallel but not ultraparallel, the composite is called a "horolation." So in hyperbolic geometry, there are five different isometries (horolation, translation, rotation, reflection, and glide reflection) whereas in Euclidean geometry there are only four. (In elliptic there are only three.)

And now we see a problem -- are we really, in a high school course, going to start discussing ultraparallel lines and Saccheri quadrilaterals for the sole purpose of proving "if corresponding angles are congruent then the lines are parallel" without using the Parallel Postulate? ...especially when we'd soon introduce the Parallel Postulate anyway to prove the converse (and ultraparallel lines become parallel lines, and Saccheri quadrilaterals become rectangles)? ...especially when we can use rotations instead (and rotations, being the composite of reflections in intersecting lines, have nothing to do with parallel lines and so nothing to do with the Parallel Postulate)?

Of course we won't. And so I'll stick to using the Wu rotation method -- after all, the student I was tutoring quickly understood the rotation proof, and the goal of teaching is to present the material so that the students understand it.

It's a shame, though. One thing I like about the translation method is that it unites the two definitions of "corresponding angle" commonly used in geometry classes -- two lines cut by a transversal form corresponding angles, and congruent triangles have corresponding angles (and sides) that are congruent (CPCTC). Here, "corresponding angles" would be angles that are the preimage and image of some isometry, whether it be a translation for the parallel lines or a translation (or reflection, etc.) for the congruent triangles. But unfortunately, we can't use this method in a high school course.

Now I want to avoid being a hypocrite. I just criticized Silverman for using a definition of "parallel" that isn't equivalent to the standard definition unless one assumes the Parallel Postulate, then turn around and endorse definitions of "translation" that require a Parallel Postulate. And the definition of "translation" given by the U of Chicago in terms of composites of reflections is similar to the one that hyperbolic geometers use -- just make the reflecting lines "ultraparallel."

But it's much more important to avoid confusing high school students with concepts such as "ultraparallel" lines. And besides, notice that I will move rotations up to Chapter 4, but keep translations in Chapter 6. In the intervening Chapter 5, the U of Chicago text gives a brief description of non-Euclidean geometry. This will be a good place to mention the Parallel Postulate, in the form first proposed by the 18th century Scottish mathematician John Playfair (which is why both Joyce and Dr. M call it "Playfair's postulate"): "given a line and a point not on it, at most one line parallel to the given line can be drawn through the point." Notice the phrase "at most," which is used to rule out hyperbolic geometry. As it turns out, elliptic geometry is actually ruled out by other postulates besides Euclid's Fifth Postulate. Then defining translations as the composite of two reflections in parallel lines is OK because by that point we've already assumed Playfair.

In my next post, I will try to provide a few visual previews of what the geometry on this blog will actually look like.

No comments:

Post a Comment