Chapter 5 Review (Day 58)

Today I subbed in a high school self-contained special ed class. Obviously, the class is full of aides, and there isn't much of any math taught today. So there is no "A Day in the Life" today.

Chapter 15 of Ian Stewart's The Story of Mathematics is called "Rubber-Sheet Geometry: Qualitative Beats Quantitative." Here's how it begins:

"The main ingredients of Euclid's geometry -- lines, angles, circles, squares, and so on -- are all related to measurement."

Today's chapter is all about something geometric that doesn't involve measurement -- topology. As Stewart reminds us, a topologist can't tell the difference between a doughnut and a coffee-cup.

"Even today we mostly use topology indirectly, as one technique among many. You don't find anything topological sitting in your kitchen -- not obviously, at least."

Stewart starts out with Euler, whose two contributions to pre-topology both appear in our U of Chicago text. The first is Euler's formula (actually discovered by Descartes) for the number of faces, edges, and vertices in a polyhedron. It turns out that the formula mentioned in an Lesson 9-7 exercise is incomplete -- there needs to be a term for the number of holes in the polyhedron:

"In fact, if a polyhedron has g holes, then F + V - E = 2 - 2g. What, exactly, is a hole? This question is harder than it looks."

Stewart's main example is an empty picture frame, which contains one hole. Therefore, a picture frame can't possibly be topologically equivalent to a sphere:

"For a rigorous proof of the impossibility, we need look no further than the fact that for this polyhedron, F + V - E = 0."

Then Stewart jumps to the other famous Euler problem -- the Bridges of Konigsberg, Lesson 1-4 of the U of Chicago text (and my old first day of school lesson):

"The city of Konigsberg, then in Prussia, was situated on both banks of the river Pregelarme, in which there were two islands."

I don't need to repeat the Konigsberg problem, since I've already blogged so much about it. Stewart, states the word valency to explain why the network isn't transversable -- all four nodes of the network have odd valency. Meanwhile, he discusses why both problems are related to topology:

"One counts faces, edges, and vertices and the other counts valencies; one is about a universal relation between three numbers, the other a relation that must occur if there exists a tour."

But neither depends on the exact shape of the polyhedron or map -- just on the number of holes. So both are considered part of topology. Meanwhile, two other mathematicians discovered another topologically significant shape -- a surface with only one side. Of the two mathematicians who found it -- Johann Listing and Augustus Mobius -- one of them is more famously associated with it:

"Listing published it in Der Census Raumlicher Complexe, and Mobius put it in a paper on surfaces. For a long time Euler's ideas on polyhedra were something of a side-issue in mathematics began glimpse a new approach to geometry, which they called 'analysis situs' -- the analysis of position."

Stewart moves on to discuss a new construction of the complex numbers -- the Riemann sphere:

"Amazingly, this construction fits beautifully into the standard calculations in complex analysis, and now equations like 1/0 = infinity make perfect sense."

Riemann also discovered how to deal with multi-valued functions such as square root -- replace the sphere with a more complicated surface:

"The geometry of this surface tells us a lot about the square root function, and the same idea can be extended to other complex functions. The description of the surface is rather indirect, and we can ask what shape is it?"

This leads to a theorem that tells us what all possible 2D surfaces are -- the Classification Theorem:

"With the proof of this theorem, the topology of two-dimensional spaces -- surfaces -- could be considered known."

Stewart tells us that on a sphere, every closed loop can be continuously shrunk to a point, like the Equator being shrunk until it becomes the North Pole:

"In contrast, every surface that is not equivalent to a sphere contains loops that cannot be deformed to points."

This leads us to the famous Poincare conjecture -- the 3D analog of the above statement. A few years ago our side-along reading was all about the Poincare conjecture, so I don't need to give all the details about the conjecture yet again. Here's some of what Stewart writes:

"The sphere has constant positive curvature. A torus with two or more holes can be represented as a surface of constant negative curvature. So the topology of surfaces can be reinterpreted in terms of three types of surfaces can be reinterpreted in terms of three types of geometry, one Euclidean and two non-Euclidean, namely, Euclidean geometry itself, elliptic geometry (positive curvature), and hyperbolic geometry (negative curvature)."

The author reminds us that in 3D, there are more than three (indeed, there are eight) geometries:

"The Poincare conjecture would be an immediate consequence, because the condition that all loops shrink rules out seven geometries, leaving just the geometry of constant positive curvature -- that of the 3-sphere."

And here's a little of what Stewart writes about Grigori Perelman, who finally proved the conjecture:

"If the manifold manages to simplify itself completely after only finitely many singularities have arisen, each piece will support just one of the eight geometries, and reversing the cutting operations (surgery) tells us how to glue those pieces back together to reconstruct the manifold. The Poincare conjecture is famous for another reason: it is one of the eight Millennium Mathematics Problems selected by the Clay Institute, and as such its solution -- suitably verified -- attracts a million-dollar prize."

Stewart ends the chapter by showing us what topology means in the real world:

"In retrospect, the main difficulties in developing topology were internal ones, best solved by abstract means; connections with the real world had to wait until the techniques were sorted out properly."

The sidebars in this chapter include Cauchy's Proof of the Descartes-Euler Formula and the Mobius "band" (often called "Mobius strip" in American English), biographies of Jules Henri Poincare and Grigori Perelman, what topology did for them, and what topology does for us.

Today is the review for the Chapter 5 Test. This is what I wrote last year about the worksheet for today's review:

There are many important concepts covered on this test -- isosceles triangles, quadrilaterals, and parallel lines. Here are the types of questions that appear on this test -- all of these coming from the SPUR section of Chapter 5 of the U of Chicago text.

• The first two questions direct the student to draw an example of the figure using ruler, compass, or protractor. I don't expect most classrooms to have a compass -- and if the students don't have a ruler and protractor, close enough is good enough. For students with a straightedge and compass, the necessary constructions are perpendicular lines (for the scalene right triangle) and congruent line segments (for the isosceles acute triangle -- a single arc is sufficient).
• Question 3 requires use of the Trapezoid Angle Theorem and algebra. Once again, this test will require the most algebra of any first semester chapter.
• Questions 4-5 require use of the Parallel Consequences and algebra. The first question is the Alternate Interior Angles Consequence while technically, the second one is actually Same-Side Exterior Angles. Savvy students should be able to figure it out by using, for example, the Linear Pair Theorem followed by Corresponding Angles. One way I remember is that any two angles formed by two parallel lines and a transversal are either equal or supplementary -- and one can eyeball it to see whether the angles are acute or obtuse to tell which ones are which.
• Questions 6-7 require use of the Isosceles Triangle Theorem -- neither one is straightforward, though, as either the Linear Pair or Vertical Angles Theorems are necessary. Once again, the students can look for acute and obtuse angles to find which ones are equal or supplementary.
• Questions 8-9 require use of the Triangle-Sum Theorem. The first one only needs arithmetic, while the second needs algebra.
• Question 10 requires use of the Quadrilateral-Sum Theorem. Once again, linear pairs, vertical angles, and algebra are needed.
• Question 11 requires use of the Quadrilateral Hierarchy Theorem. Students must arrange the various shapes from most general to most specific.
• In Questions 12-13, students must identify the quadrilaterals. The second one is a rhombus, not necessarily a square, since nowhere is it stated that the angles are right angles. Some might note that here I appear to contradict myself -- didn't I just say that in Question 5, students are supposed to assume that angle 1 is obtuse and angle 7 is acute in order to determine that they are supplementary, but here in Question 13 students should not assume that any of the angles are right angles? The difference is that in Question 5, identifying acute and obtuse angles is used as a mnemonic to remember actual theorems such as Linear Pair or Corresponding Angles Consequence, but in Question 13 nothing warrants knowing that there are any right angles.
• Question 14 is a true-or-false question about the Quadrilateral Hierarchy. If the statement is false, the students should draw a square that isn't a parallelogram. But they shouldn't -- because the statement happens to be true.
• In Question 15, students must determine whether the quadrilateral has any symmetry lines. As it turns out, the figure is a kite, so by the Kite Symmetry Theorem, it has a symmetry line.
• Question 16 is another true-or-false question about the Quadrilateral Hierarchy, but this time students are not directed to draw a counterexample. As it turns out, the statement is true -- that the bases have a common bisector is mentioned in the Isosceles Trapezoid Symmetry Theorem.
• In Question 17, students are to evaluate a conjecture. The U of Chicago text asks students to rank on an A-E scale whether they believe the conjecture to be true or false -- but I don't do this on a test, or otherwise every student would just choose C ("I'm not sure!"). So the students' only choices are true and false. As it turns out, the conjecture is true, and it can proved simply by knowing that half of 180 is 90. But students don't have to prove it -- proofs can wait until...
• ...Questions 18-19. The keys to the first question are the Alternate Interior Angles Test and the definition of trapezoid. The keys to the second question are two definitions -- those of circle and kite. Notice that the first proof underscores the preference for inclusive rather than exclusive definitions. Under the exclusive definition, EFHG could be a parallelogram -- and the way it is drawn, it almost looks like a parallelogram. We can't prove that it's a parallelogram -- but we can't prove that it's not a parallelogram either, which means that under the exclusive definition, we can't prove that it's a trapezoid either! The second question isn't a problem for the exclusive definition --OQPR is clearly nonconvex while all parallelograms are convex. But even if point O had been drawn outside of circle P, we could still prove that it's an inclusive kite, but it could be a parallelogram (hence a rhombus) and not an exclusive kite.
• Question 20 has the students draw part of the Quadrilateral Hierarchy. It goes without saying that inclusive definitions should be used, so a rectangle is an isosceles trapezoid.