Happy Bastille Day, everyone! No, I'm not normally one for celebrating French national holidays, but recall that Bastille Day was a critical date in the life of Evariste Galois -- yes, that young French mathematician who co-founded group theory. Galois led a group of radical republicans in a sort of demonstration at the Place de la Bastille. He was arrested, and that was when his downfall began -- with the events leading to his tragic death nearly a year later.
But I'm not currently reading about Galois -- now it's all about Eugenia Cheng's book. I've made it up to Chapter 7 of How to Bake Pi -- near the end of Part I. Of the latest chapters that I've read since last week's post, Chapter 5, "Generalization," is the chapter that is most relevant to geometry, and so I shall describe this chapter in detail here on the blog.
Chapter 5, like all the other chapters, begins with a recipe. This recipe, for an olive oil plum cake, is one invented by Dr. Cheng herself. She had wanted to bake a cake for all of her guests, but she couldn't find one that satisfied all of the constraints -- many of the guests were on special diets. And so Cheng had to invent the olive oil plum cake. "Everyone said it was delicious, but when they asked me what it was I didn't know what to call it, because it's not really a cake -- it's a generalization of a cake" (emphasis hers).
Well, definition might not be important in cooking, but is very important in mathematics -- as well as the sciences. There is much renewed discussion today about the definition of planet in astronomy -- particularly because the NASA space probe New Horizons is passing by Pluto today. In 2006, just a few months after New Horizons launched, it was decided that Pluto isn't really a planet -- it's a generalization of a planet, a dwarf planet.
And so Chapter 5 is all about generalization in mathematics. Cheng's first mathematical example is the generalization from congruent triangles to similar triangles. Congruent triangles, as we already know, are those that are the same "shape" and the same "size" -- there exists an isometry mapping one to the other. Similar triangles only need to be the same shape, not necessarily the same size -- there only exists a similarity transformation mapping one to the other, not necessarily an isometry. So similarity is a generalization of congruence.
Notice that often in geometry, the specific is actually simpler than the general. And so we consider consider congruence to be simpler than similarity, and so we teach congruence first. But notice that it's the general concept of similarity that matters more in mathematics. In Common Core math we use similarity to derive the slope ratio, prove the Pythagorean Theorem, and define the trig ratios sine, cosine, and tangent -- yet congruence doesn't really appear outside of Geometry class.
So in the Integrated Math I class that I designed, I end up spending more than half of the year in geometry just trying to get to similarity, and then I can finally use similarity to derive slope. I'd actually be willing to skip some of the material to get to similarity faster, but similarity is such a difficult concept and I don't want the students' first semester grades to drop. This is why, if this were an eighth-grade course at a trimester middle school, I'd skip some geometry material and start similar triangles at the start of second trimester. This allows us to cover more algebra in the second and third trimesters while still protecting the first trimester grades.
I have actually seen one Integrated Math text that teaches the general concept of similarity before the specific concept of congruence. This is the Carnegie Learning Integrated Math II text. "Similarity Through Transformations" is Chapter 4, and "Congruence Through Transformations" is Chapter 5. I point out that if we prove the general first, the specific is almost trivial. Notice, for example, that if we already have similarity via AA, congruence via ASA (and AAS) is trivial to prove -- the two triangles are already similar by AA, and the similarity ratio must be 1. George Birkhoff -- the creator of the Ruler Postulate -- also proposed SAS Similarity as a postulate. But I fear that this approach will be confusing to students, especially eighth graders (and we notice that this is a Math II text, therefore above eighth-grade level).
Let's get back to Cheng's book now. She now describes the concept of a proof by contradiction -- known as an "indirect proof" in the U of Chicago text. Going back to her favorite cooking metaphor, Cheng "proves" that one needs to boil water to make tea by trying to make the tea without boiling the water and noticing how disgusting the resulting liquid is. Likewise, Cheng proves that if n is a whole number and n^2 is odd, then so is n. She does so indirectly by assuming that n is even and showing that since the product of two even numbers is even, n^2 would be even as well. This contradicts the fact that n^2 is odd -- therefore n must be odd. Cheng follows this up with another proof involving squares -- namely the classic proof that sqrt(2) is irrational.
But now Cheng points out that sometimes, one tries to prove that something is true by assuming that it is false and trying to derive a contradiction -- but instead of deriving a contradiction, they end up discovering something new. She writes that someone once tried to prove that, just as one needs boiling water to make tea, one needs flour to make chocolate cake. But instead of a contradiction (i.e., terrible-tasting cake), the cook inadvertently discovered flourless chocolate cake. Likewise, an attempt to prove that one needs yeast to make bread resulted in the invention of matzoh.
Cheng gives these examples to set up her story of how non-Euclidean geometry was discovered. We know that Euclid came up with his five postulates. It's interesting how Cheng translates Euclid's famous Fifth Postulate:
"If you draw three random straight lines they will make a triangle somewhere, if you draw them long enough, unless they meet each other at right angles."
I can appreciate what Cheng is trying to get at here -- Euclid's Fifth Postulate mentions two right angles, and so Cheng's translation mentions these two right angles as well. Unfortunately, I'm not sure that Cheng's statement is actually equivalent to Euclid's Parallel Postulate. One of the angles could be 60 degrees and the other 120 degrees, and the lines won't form a triangle, even though there are no right angles. Also, the statement doesn't mention the possibility that the three lines could be concurrent and so there would again be no triangle.
I like my own "translation" of the Fifth Postulate which, just like Cheng's, seeks to mention the two right angles that appear in the original Euclid. It is actually the Perpendicular to Parallels Theorem of the U of Chicago text -- except I take it as a postulate rather than as a proved theorem:
In a plane, if a line is perpendicular to one of two parallel lines, it is perpendicular to the other.
One equivalent statement combines my idea with Cheng's idea of forming a triangle. (I left out that the lines are coplanar, since "in a plane" is already implied for Euclid's postulates.):
If you draw three non-concurrent straight lines, two meeting each other at a right angle and the third meeting one of these at an acute angle, they will make a triangle if you draw them long enough.
Of course, we already know what happened. Mathematicians assumed that the Parallel Postulate was false but, instead of deriving a contradiction, they invented flourless chocolate cake -- that is, they invented non-Euclidean geometry. Cheng mentions two types of non-Euclidean geometry -- elliptical geometry and hyperbolic geometry. Recall that the spherical geometry of Legendre is somewhat like elliptical geometry, but not quite.
Cheng says that in elliptical (and spherical) geometry, the angles of a triangle add up to more than 180 degrees, and in hyperbolic geometry, they add up to less than 180 degrees. Of course, in Euclidean geometry, they add up to exactly 180 degrees. We are very close to reaching the propositions in Legendre that discuss the angles of a spherical triangle.
Now Cheng moves on to a new generalization -- that of distance. She points out that if one wants to take a taxicab to get to a point three blocks east and four blocks south, the distance between the two points is seven blocks, even though the Distance Formula gives the distance between the origin and the point (3, -4) as five units. A taxicab simply can't cross blocks diagonally "as the crow flies."
I have another old geometry textbook -- HRW (Holt, Rinehart, and Winston), dated 1997. I've in fact referred to this text on the blog before, although possibly not by name. This is the text that actually gives (as Dr. Franklin Mason once did) the Corresponding Angles Test as a proved theorem. The theorem is stated in Chapter 3, but the proof is postponed to Chapter 12 (the near-equivalent of Chapter 13 of the U of Chicago text) until after indirect proof is taught. This proof by contradiction uses the Triangle Exterior Angle Equality and thus the Fifth Postulate, in the form of the converse, the Corresponding Angles Consequence Postulate. (The old Dr. M proof used the Triangle Exterior Angle Inequality, which doesn't require a Fifth Postulate to prove.) Still, the HRW text is one of the few geometry texts that doesn't give the Corresponding Angles Test as a postulate.
The HRW text contains twelve chapters. If I were to use HRW as a classroom text, the first four chapters would make up the ideal first semester for me (corresponding roughly to the first seven chapters of the U of Chicago text). But I wouldn't mind including Chapters 5 and 6 (Chapters 8 and 9 of the U of Chicago text) in the first semester if it would get me to Chapters 11 and 12 by the end of the year. Chapter 11, in particular, is an interesting enrichment chapter of fun things to do after the Common Core exams -- and includes some of the things mentioned in Cheng's Chapter 5. Let's look at the layout of HRW's Chapter 11:
Section 11.1 -- Golden Connections
Section 11.2 -- Taxicab Geometry
Section 11.3 -- Networks
Section 11.4 -- Topology: Twisted Geometry
Section 11.5 -- Euclid Unparalleled
Section 11.6 -- Projective Geometry
Section 11.7 -- Fractal Geometry
So Cheng has already covered Section 11.5, on non-Euclidean geometry. Notice that this text, even though the Corresponding Angles Test is a theorem in this text, both the Corresponding Angles Consequence (Section 3.3) and Playfair's Parallel Postulate (Section 3.5) are postulates in this text. In Section 11.5, both spherical geometry and hyperbolic geometry (the Poincare disc, which we don't discuss in detail on the blog) are given in this section.
But now we've reached taxicab geometry in Cheng -- Section 11.2 of HRW. This text defines the term "taxidistance" as the smallest number of blocks a taxi must travel to go from one point to another. If we think of taxicab geometry as a separate geometry from Euclidean geometry, it would be wrong even to call it non-Euclidean geometry, since not even Euclid's First Postulate holds -- both Cheng and HRW give pairs of points such that there is more than one "line" (path of shortest taxidistance) between the two points. Reflections are not isometries in taxicab geometry (except for those which are parallel to the axes) and so SAS Congruence, which depends on reflections, doesn't hold.
Another common name for "taxidistance" is "Manhattan distance." This refers to the fact that a common place to find taxicabs is New York City. But recall that Cheng works at the U of Chicago, and so her examples use Chicago rather than New York. Interestingly enough, the HRW text also gives a map of Chicago, but in Section 1.7 (on translations), not 11.5.
Both Cheng and HRW point out that in the Windy City, eight blocks, or 800 address units, equals one mile. Unfortunately, Los Angeles is not a simple as Chicago -- here one mile is approximately 800 address units when traveling east or west, but when going north or south, one mile is closer to 1600 address units than 800 -- and different suburbs of L.A. sometimes follow different patterns. I like the idea of 1600 address units to the mile, since one mile is approximately 1600 meters. But it's rare for cities to use this conversion in both directions, since this would make the blocks too small without room for many buildings. (In Manhattan there are 20 blocks to the mile when traveling north or south. In my area, nearby Orange County sets 1000 address units to the mile, which makes calculations easier. But of course, certain Orange County cities override this with their own numbering systems.)
Cheng states that there are three rules that every distance must follow:
1. The distance from A to B is zero when A and B are the same place, and this is the only way the distance from A to B can be zero.
2. The distance from A to B is the same as the distance from B to A.
3. The distance from A to B can't be made shorter by going via C.
Rule 3 is, of course, the Triangle Inequality. Euclidean geometry, spherical geometry, and taxicab geometry all satisfy these three rules, but Cheng gives some examples (train distances, GPS systems that don't take the third dimension of height/elevation into consideration, etc.) that don't. She states that a nonnegative distance satisfying all three of these rules is called a metric.
Section 11.4 of the HRW is on networks. This topic doesn't appear in Cheng's Chapter 5, but it does appear in Section 1-4 of the U of Chicago text -- our opening activity on the Bridges of Konigsberg is based on networks. But what I want to discuss is Section 11.4, because it's the next topic of Cheng's Chapter 5 -- topology.
In topology, we generalize isometries and similarity transformations even further. We know that two figures are congruent if there exists a isometry mapping one to the other, and similar if there exists a similarity transformation mapping one to the other. Well, in topology, we find out that two figures are topologically equivalent if there exists a homeomorphism mapping one to the other. Here, we see that a homeomorphism is simply a continuous map with a continuous inverse. (I'd like to say that homeomorphisms preserve Betweenness, except that Betweenness is usually defined to imply Collinearity, which is not necessarily preserved.)
I actually mentioned homeomorphisms earlier on the blog in May -- when trying to determine whether there exists a 2D homeomorphism whose composite with itself yields a reflection (since we already know that no such isometry or similarity transformation exists). The mapping we came up with that day was not a homeomorphism because we had to cut the plane to perform the transformation. For our homeomorphisms, we are not allowed to cut anything. So there is no homeomorphism mapping a figure without a hole to a figure with a hole. The rule that the inverse must also be continuous is there because there does exist a continuous mapping from a figure with a hole to a figure without a hole -- simply make the hole shrink until it disappears. But its inverse is not continuous -- we can't make a hole appear unless we cut.
The most usual example of two topologically equivalent figures is a doughnut, or torus, and a coffee cup or mug, since each has exactly one hole. Both Cheng and HRW give this example, but neither gives the old joke -- a topologist is a mathematician who can't tell the difference between a doughnut and a coffee cup. (The Simpsons book that I read earlier this year also makes a topology joke -- Homer Simpson makes a doughnut topologically equivalent to a sphere by simply taking a bite out that doughnut! In reality, a bite is a cut and therefore not allowed.)
An argument can be made that the Swiss mathematician Leonhard Euler was the first topologist -- and in fact, his Konigsberg bridge problem was the first topology problem. Another topology problem associated with Euler is the formula V - E + F = 2, which states the relationship among vertices, edges, and faces in a polyhedron. As it turns out, this formula holds on both the Euclidean plane and spherical geometry, but not on a torus. (The HRW text gives another theorem, the Jordan Curve Theorem, which also holds on a plane and a sphere but not a torus.)
Cheng ends Chapter 5 with "A Generalization Game." She gives the five figures square, rhombus, parallelogram, trapezoid, and quadrilateral and states that these are given from the most specific to the most general. Notice that for this to work, Cheng must use the inclusive definition of trapezoid -- otherwise a trapezoid is not really a generalization of a parallelogram (since a parallelogram would no longer be a specific type of trapezoid.)
Chapter 6 of Cheng is called "Internal vs. External." In this chapter, Cheng distinguishes between external motivations for studying math (e.g., to solve a problem or earn a good grade) and internal motivations for studying it (math for its own sake). Unfortunately, most math studied in school is externally motivated -- students must learn it because the Common Core says so. Cheng gives the the ultimate example of internally motivated math studied for its own sake -- category theory.
The last chapter from Eugenia Cheng that I read was Chapter 7, called "Axiomatization." But this discusses axioms about natural numbers and real numbers, not geometry. She does discuss the definition of a "group," which was first studied by that guy who was imprisoned on Bastille Day, Evariste Galois. Cheng writes that definitions are just as important as axioms -- her example is that whether human beings have a soul depends on the definition of "soul." (Likewise, whether Pluto is a planet depends on the definition of "planet.") That's all that I'll say about this chapter here on the blog, save to mention that the recipe is for jaffa cakes. (In fact, she mentions jaffa cakes to make a point, namely that the basic ingredients of jaffa cakes depends on what one considers to be "basic," and the same is true for axioms, or postulates. I've run into that problem several times when designing my geometry course and deciding which axioms/postulates to include.)
I've spent so much time on Cheng's book that I still haven't reached spherical geometry -- but of course, Cheng briefly mentions non-Euclidean geometry in her book. I begin today's discussion of spherical geometry by stating that I've made another attempt to reach the confluence 34N, 118W. At the link that I mentioned in my last post, it stated that I had to travel on the winding streets in the order Turnbull Canyon Drive, Edgeridge Drive, Skyline Drive, Descending Drive, Oak Canyon, and finally Athel Drive.
Somehow, I ended up taking Turnbull Canyon to Edgeridge directly to Athel. On the right side, I found the pullout that was mentioned at the link. It is stated that this pullout is only 100 meters away from the confluence! This is close enough to make an official visit -- but of course I did not take any pictures of the area. I would never actually participate in the Degree Confluence Project unless I had a GPS -- and even then I'd probably go to a rarely visited confluence, not 34N, 118W. According to the link, 35N, 120W is a Southern California confluence visited very rarely, but according to the links of the three previous official visits, 35N, 120W is even steeper than 34N, 118W! (Recall what I wrote about GPS and height/elevation earlier in today's post!)
At 34N, 118W I was hoping for mile markers, but they stopped once I left Turnbull Canyon. I can say that Edgeridge Drive appeared to be near mile marker 1, on the right side. At the pullout on Athel I saw the house that is mentioned in the following link:
I don't know the address of that house -- and even if I did, that wouldn't help anyone find it since I doubt the 800 = 1 mile, 1600 = 1 mile tricks would work on these winding roads that are as far from a perfect grid as one can get.
Let's finally get to Legendre's Elements of Geometry and discuss the next four Propositions that we are to cover.
469. Every plane perpendicular to the radius at its extremity is a tangent to the sphere.
Demonstration: Let FAG be a plane perpendicular to the radius AO at its extremity; if we take any point M in this plane, and join OM, AM, the angle OAM will be a right angle, and thus the distance OM will be greater than OA; consequently the point M is without the sphere; and as the same might be shown with respect to every other point of the plane FAG, it follows that this plane has only the point A in common with the sphere; therefore it is a tangent to this surface (440).
Here Proposition 440 is simply the definition of a plane tangent to a sphere. As it turns out, this proof is nearly identical to the one given in Section 13-5 of the U of Chicago text, which gives the case of a line tangent to a circle. In both proof, we form a right triangle with the radius and any point in the plane or line whose tangency we are trying to prove. Since the hypotenuse of a (Euclidean) right triangle is longer than the leg, it follows that the endpoint of the radius is the only point that lies on the circle or sphere.
470. Scholium. It may be shown, in like manner, that two spheres have only one point in common, and are consequently tangents to each other, when the distance of their centers is equal to the sum or difference of their radii; in this case, the centers and the point of contact are in the same straight line.
The text considers two spheres to be tangent to each other when they intersect in one point. One of Cheng's more advanced examples of two topologically equivalence figures is the complement of two interlocking circles and a sphere with a torus tangent to it in its interior.
471. The angle BAC, which two arcs of great circles make with each other, is equal to the angle FAG formed by the tangents of these arcs at the point A; it has also for its measure the arc DE, described from the point A as a pole, and comprehended between the sides AB, AC, produced if necessary.
Demonstration: For the tangent AF, drawn in the plane of arc AB, is perpendicular to the radius AO (110); and the tangent AG, drawn in the plane of arc AC, is perpendicular to the same radius AO; therefore the angle FAG is equal to the angle of the planes OAB, OAC (349), which is that of the arcs AB, AC, and which is designated by BAC.
In like manner, if the arc AD is equal to a quadrant, and also AE, the lines OD, OE, will be perpendicular to AO, and the angle DOE will be equal to the angle of the planes AOD, AOE; therefore the arc DE is the measure of the angle of these planes, or the measure of the angle CAB.
Legendre's 471 is the first to describe how to define the measure of a spherical angle. There are two ways given here. One is to note that each side of the spherical angle is an arc, each arc lies in a plane passing through the center of the sphere, and these two planes meet at a dihedral angle, so the measure of that dihedral angle is defined to be the measure of the spherical angle. The other way is to do so by means of poles. If we imagine drawing an angle whose vertex is at, say, the North Pole, we can find its measure by considering the great circle whose pole is the North Pole -- the Equator. Extending the sides of our angle all the way to the Equator, we define the measure of the angle to be the arc length of the Equatorial distance between the two sides (taking the radius of the globe to be 1).
472. Corollary. The angles of spherical triangles may be compared with each other by means of the arcs of great circles, described from their vertices as poles, and comprehended between their sides; thus it is easy to make an angle equal to a given angle.
Legendre's 472 extends the idea of spherical angle measures to the angles of a spherical triangle. We are now hinting, just as I promised earlier, the sum of the angle measures of a spherical triangle.
473. Scholium. The angles opposite to each other at the vertex, as ACO, BCN, are equal; for each is equal to the angle formed by the two planes ACB, OCN (350).
It will be perceived, also, that in the meeting of two arcs ACB, OCN, the two adjacent angles ACO, OCB, taken together, are equal to two right angles.
Legendre's 473 is evidently the analog of the Linear Pair and Vertical Angle Theorems of Euclidean geometry, given in Section 3-2 of the U of Chicago text. In both Euclidean and spherical geometry, vertical angles have equal measures and angles forming a linear pair are supplementary.
474. The triangle ABC being given, if, from the points A, B, C, as poles, the arcs EF, FD, DE be described, forming the triangle DEF, reciprocally the points D, E, F, will be poles of the sides BC, AC, AB.
Demonstration: The point A being the pole of the arc EF, the distance AE is a quadrant; the point C being the pole of the arc DE, the distance CE is likewise a quadrant; consequently, the point E is distant a quadrant from each of the points A, C; therefore it is a pole of the arc AC (467). It may be shown, in the same manner, that D is the pole of the arc BC, and F that of the arc AB.
Legendre's 474 assigns to every spherical triangle ABC a related triangle DEF, such that the vertices of one triangle are the poles of the sides of the other, and vice versa. Technically speaking, since every great circle has two poles, there are two possibilities for DEF (but there would be only one such triangle in elliptic geometry since there we identity antipodes as the same point). Notice that the function mapping ABC to DEF is in general not an isometry, but these triangles are related in an important manner, as we see in Legendre's next propositions.
475. Corollary. Hence the triangle ABC may be described by means of DEF, as DEF is described by means of ABC.
476. The same things being supposed as in the preceding theorem, each angle of one of the triangles ABC, DEF, will have for its measure a semicircumference minus the side opposite in the other triangle.
Demonstration. Let the sides AB, AC be produced, if necessary, until they meet EF in G and H; since the point A is the pole of the arc GH, the angle A will have for its measure the arc GH. But the arc EH is a quadrant, as also GF, since E is the pole of AH, and F the pole of AG (465); consequently EH + GF is equal to a semicircumference. But EH + GF is the same as EF + GH; therefore the arc GH, which measures the angle A, is equal to a semicircumference minus the side EF; likewise the angle B has for its measure 1/2 circ. - DF and the angle C, 1/2 circ. - DE.
This property must be reciprocal between the two triangles, since they are described in the same manner, the one by means of the other. Thus we shall find that the angles D, E, F, of the triangle DEF, have for their measure respectively 1/2 circ. - BC, 1/2 circ. - AC, 1/2 circ. - AB. Indeed, the angle D, for example, has for its measure the arc MI, but MI + BC = MC + BI = 1/2 circ., therefore the arc MI, the measure of the angle D, = 1/2 circ - BC, and so of the others.
Legendre's 476 can be rewritten using "tau" instead of circumference -- or, better yet, "pi" instead of semicircumference or "1/2 circ."
As it turns out, Proposition 476 is sufficient to derive the angle sum of a spherical triangle. But I will wait until Legendre himself derives the sum in his Proposition 489 -- we still have a long way to go.
Thus ends this post. Once again, I wish everyone a happy Bastille Day -- well, only the French care about that, so instead I wish everyone a happy Pluto Day.