Thursday, July 30, 2015

Spherical Geometry (Legendre 477-480)

Some bloggers write on their blogs regularly. Others, unfortunately, update their blogs only once in a blue moon. Well, those in the latter category should be updating soon, because in many time zones, tonight there will be indeed a blue moon.

But what, exactly, is a blue moon? We think back to my New Year's Eve calendar post. Clearly "blue moon" has something to do with the moon. In that post, I mentioned how the solar year cannot be divided evenly into lunar months. And recall that a "lunar month" is the period of time from one new moon to the next -- or equivalently, one full moon to the next, since blue moons have something to do with full moons.

In particular, the solar year is not exactly twelve lunar months. We see what happens in the Islamic Calendar, where the years consist of exactly 12 months -- each year ends up being slightly shorter than a solar year. In order to obtain solar years, some of the years must have 13 full moons -- which means that one month must contain two moons. And so we define the blue moon to be the second full moon during the calendar month.

We notice that a lunar month is about 29 1/2 days. This means that, if a month were to contain two full moons, the first one would occur on the first or second day of the month, and the second would occur on the 30th or 31st day of the month. And this is exactly what happens in July 2015. A full moon occurred on July 1st, and the second full moon, the blue moon, occurs tonight -- officially after midnight in most time zones, so the blue moon appears on July 31st on calendars.

How often do blue moons occur? Recall that I mentioned the ancient Greek astronomer Meton, who discovered that we must add seven months to a lunar calendar every 19 years in order to obtain a lunisolar calendar. These extra seven months can be interpreted as the blue moons, so we conclude that a blue moon occurs once every 19/7 years -- about once every three years. (A better estimate is that it occurs once every e years.) So this is what "once in a blue moon" actually means. Notice that blue moons occur somewhat more often than leap years in the Gregorian calendar.

There is another definition of blue moon that sometimes appears -- this one is based on dividing the year into seasons rather than months. A solar year, with its four seasons, normally contains 12 full moons, so that divides naturally into three full moons per season. But if there are 13 full moons in the year, one season would have to contain four full moons. Historically, it is the third of these four full moons that would be the blue moon.

Notice that since the seasons don't line up with the months, a blue moon using the monthly definition can never be a blue moon using the seasonal definition. In particular, tonight's full moon is the second of three full moons during Summer 2015, so it can't be a seasonal blue moon.

When is the next seasonal blue moon? One way to determine this is to look at future Easter dates, since Easter is based on the full moon after the spring equinox (a seasonal marker). We notice that Easter falls on March 27th, 2016, and then on April 16th, 2017. Whenever there is an early Easter followed by a late Easter, we know that the late Easter must occur 13 full moons after the early Easter, and so a seasonal blue moon must occur between the two Easters.

If we consult a 2016 calendar, we see that both a full moon and the summer solstice occur on June 20th, 2016, with the full moon about 12 hours before the solstice. This means that the June 20th full moon is actually the fourth full moon of spring, and so the third full moon of spring -- May 21st, 2016 -- must be the seasonal blue moon.

We notice that the seasonal blue moon occurs about one year after the monthly blue moon. There is a reason for this -- we start by noting that the reason for a monthly blue moon in July 2015 is the fact that the first full moon occurs on the first of the month. If we count 12 lunar months (i.e., one Islamic year, 354 days) from July 1st, 2015, we obtain June 20th, 2016 -- the summer solstice -- and it's the full moon on that day that causes May 21st, 2016 to be a seasonal blue moon. In other words, seasonal blue moons generally occur about a year after monthly blue moons due to the coincidence that the difference between 12 lunar months and one solar year (i.e., 11 or 12 days) is nearly equal to the difference between any equinox/solstice and the first day of the following month (in this case, the difference between June 20th and July 1st).

Here's one final difference between monthly and seasonal full moons -- there can never be a monthly full moon in February, since that month is shorter than a lunar month (even if February contains a Leap Day, it is still about 12 hours shorter than a lunar month). On the other hand, February is one of only four possible months for a seasonal blue moon, since seasonal blue moons must occur about one month before an equinox or solstice, and February is one month before the spring equinox (just as May 21st, 2016 is about one month before the summer solstice).

Meanwhile, I finally finished Eugenia Cheng's How to Bake Pi. In the last two chapters, Cheng explains that categories have initial and terminal objects. For example, in the category that I mentioned in my last post:

Parallelogram --> Rectangle
          |                    |
          |                    |
         V                  V
Rhombus       --> Square

the object "parallelogram" is an initial object, while "square" is a terminal object.

One of Cheng's favorite examples is clock arithmetic. We know that even though 11 + 2 = 13, on the clock, two hours after 11:00 is 1:00 -- that is, 11 + 2 = 1 on the 12-hour clock. Here is a YouTube video from Square One TV that describes clock arithmetic in more detail:

As Cheng points out, there can be any number of hours on the clock, not just 12. For each natural number n, there is a group with n elements, called the cyclic group of integers modulo n. I mentioned earlier on the blog that we can think of the chromatic scale in music as being isomorphic to the group of integers mod 12 (Z/12Z) -- but we could also invent a scale isomorphic to the integers mod 19 (that is, Z/19Z) instead.

Cheng writes that in category theory, if two groups are objects, an example of an arrow between them can be a function that "treats the group operation sensibly." Since I took group theory at UCLA, I know exactly what word she's trying to avoid using here -- homomorphism.

An example of a homomorphism is from Z/24Z to Z/12Z -- that is, it maps the 24-hour clock to the 12-hour clock in the obvious way. We perform this homomorphism every time we try to convert from military time to civilian time. Another homomorphism maps Z/12Z to Z/60Z -- and we perform this homomorphism every time we convert the number the minute hand is pointing at to actual minutes. I point out that this is a homomorphism because it respects the group operation of addition -- we can either add first (1 + 2 = 3) and convert this to minutes (minute hand on 3 = 15 minutes), or we can convert to minutes first (5 minutes and 10 minutes) and then add (5 + 10 = 15 minutes), and we obtain the same answer.

I can see why Cheng would want to avoid using the word homomorphism -- that word can be confused easily with homeomorphism, which is a map between two topologically equivalent figures. Notice that both homomorphisms and homeomorphisms are morphisms -- the arrows in category theory.

Cheng's book was certainly a delight to read, and I highly recommend it to anyone who wants a brief introduction to category theory -- with some dessert recipes thrown in as a bonus!

Of course, I can't say that I'm an expert in category theory yet. I often had to reread some of what Cheng wrote in order for me to understand it -- and I feel that I'm at an advantage only because I've taken a class in group theory. How much more difficult, then, would it be for, say, one of you who hasn't taken group theory to understand Cheng's book? How much even more difficult would it be for one of our students to understand her book?

This is one reason that I like to read math books that challenge me, as Cheng's book did. Just as I find category theory a challenge, my students find High School Geometry a challenge. It's too easy for me to think back to my own days as a student and finding most of Geometry easy (though I admit that even I found some of final chapters, such as trig and especially circles, a bit difficult). But category theory -- now that's something that I find as difficult as my students find Geometry. I can understand better why my students are having trouble with geometrical similarity when I compare it to my own struggle to understand category theory.

In fact, sometimes I wonder whether I, if I ever get my own classroom, should teach myself some difficult subject on the side, such as category theory. Or it doesn't even have to be math -- it could be something like a foreign language. As a Californian, I have many Spanish speakers in my classes, so if I try to understand what they are saying in Spanish, I am doing something that's as hard for me as Geometry is for them. In that way, I can sympathize with my students better.

This is another thing that I've learned about myself in the year that I've started this blog. I've been told that one of my strengths is my ability to teach English learners. When I speak, I repeat myself -- often it's because I can't think of anything to say. Ironically, this can be a strength when I am in a classroom of English learners, because they need that repetition. This is why many of the worksheets that I post here on the blog begin with a vocabulary section.

On the other hand, my repetition is a weakness when I am in a class of English natives and stronger math students. I wonder whether my problem is that as a student growing up, I've been way ahead of most of my fellow students in math, so I subconsciously think of most of my students as having low levels of math and gear my instruction towards them.

This also explains why I can't fully agree with the traditionalists who want to push everyone into Calculus classes and set up all math classes from eighth grade on to prepare them for Calculus. I know that the types of students that I teach would get D's, if not F's, in every single math class from eighth grade on if they were forced through the Calculus door.

Today is a spherical geometry day -- and here we go again! Look at how long this post is, and I've yet to say a single word about spherical geometry.

In the news today there is a story about the remains of a plane found on Reunion Island. I thought this was interesting because Reunion Island may be the closest human inhabited island to the antipodes of my California home. Recall that almost the entire continental United States -- including all of California -- has its antipodes in the Indian Ocean. If I decided that I wanted to go on a vacation and travel as far away from California as I possibly can, Reunion Island would be a great choice.

Here is a link to Reunion Island on the Degree Confluence page:

Notice that much of this page is in French, as Reunion is a French possession. The author writes that there is no actual confluence on the island -- all seven confluences are in the ocean, but all are close enough to the island for it to be visible on the horizon (which is a requirement).

The coordinates of this confluence is 21S, 55E. The coordinates of my actual antipodal point would be 34S, 62E, which is nearly a thousand miles away. The website also states that there is another nearby island, Mauritius, that is visible from the confluence 21S, 57E. This may actually be closer to my antipodal point than Reunion, since 57E is closer to 62E. If I wanted to go on my near-antipodean vacation, Mauritius may be a better choice than Reunion, especially since Mauritius is a former British colony (now it's independent), and so its inhabitants are more likely to speak English. (What did I just say in this post about learning languages again? Even though I took French in high school, I doubt that I know it enough to have a conversation with someone on Reunion Island, and so I can avoid the foreign language by going to Mauritius instead.)

Reunion and Mauritius have about a million inhabitants each. It is now summer in California, and so it is currently winter in Reunion and Mauritius. Both islands would normally be in a time zone 12 hours ahead of mine, But in reality, both islands are only 11 hours ahead of California because we observe Daylight Saving Time here. (Recall that these islands aren't true antipodes but are actually much closer to the Equator than California. Tropical areas almost never observe DST because the length of the day doesn't change enough to justify DST.)

Speaking of French, I know that these spherical geometry posts are supposed to be all about the French mathematician Legendre and his Elements of Geometry. But I often like to discuss the geometry of the earth in my spherical geometry posts -- including things like degrees, angles, and especially antipodes as such antipodal points are critical in spherical geometry. But now let's get back to Legendre's book. For those who don't remember the link to the Legendre text, here it is again:

We are currently on pages 165-166 of Legendre. Notice that the first thing that appears on page 165 is a demonstration (proof) of the last proposition on page 164, and so this has already been covered. As it turns out, there are only four propositions left on these two pages, Propositions 477 through 480 --but these are some of the most important theorems of spherical geometry. So we definitely want to spend some extra time on these.

Legendre's Proposition 477 is a "scholium" (follow-up) to his 476:

477. Scholium. It may be remarked that, beside the triangle DEF, three others may be formed by the intersection of the three arcs DE, EF, DF. But the proposition only applies to the central triangle, which is distinguished from the three others by this, that the two angles A, D are situated on the same side of BC, the two B, E on the same side of AC, and the two C, F, on the same side of AB.

Different names shall be given to the triangles ABC, DEF; we shall call them polar triangles.

So Legendre tells us that in his 476 and 477, he is defining something called "polar triangles." In particular, given any triangle ABC, there exists a unique triangle DEF such that ABC and DEF are polar triangles.

In 476, Legendre tells us how to find, given triangle ABC, a triangle DEF that is polar to ABC. The name "polar" implies poles -- recall that every circle (including great circles, since the sides of a spherical triangle are arcs of great circles) has exactly two poles. We then let D be the pole of BC, E the pole of AC, and F the pole of AB.

This may be easier to visualize with a specific triangle, Let A and C both lie on the Equator, and point B lie somewhere due north of C. We can easily find the location of point E -- this point must be the pole of AC, and since AC is the Equator, E must be the North Pole. Of course, every line actually has two poles, so how do we know that E is the North Pole and not the South Pole? This is what the "scholium" in 477 tells us -- B and E must lie on the same side of AC (the Equator), and since B is north of C, E must be the North Pole.

Now let's find point D. We see that D is the pole of BC, and B lies north of C. This means that BC is a meridian. We've mentioned that the pole of any meridian lies on the Equator, with longitude exactly 90 degrees away from that meridian. So we know that D lies somewhere on the Equator.

Point F is unfortunately difficult to locate, since it's the pole of AB and this line is neither the Equator nor a meridian. But we already know a few things about triangle DEF. We notice that ABC might be a very small triangle compared to the size of the globe -- A and C could be just a few miles apart on the equator, and B just a few miles north of C. But its polar triangle DEF must be quite large, because one of its vertices D lies on the Equator and another vertex E lies at the North Pole! So ABC and DEF are in general not congruent triangles -- nor did Legendre intend them to be congruent. Indeed, making ABC smaller results in making DEF larger, not smaller.

Now suppose point B isn't merely any point north of C, but it is in fact the North Pole itself. This is a very special case, but it's the only case where it's easy to calculate the poles of all three sides (since we only know how to find the poles of the Equator and the meridians). In particular, we let C be the intersection of the Equator and the Prime Meridian. We want A to lie on the Equator as well, so why don't we place it on my home meridian -- 118W. Then this makes the angle at B measure 118 degrees, while the angles at A and C are both right angles. (That's right -- the angles of the triangle clearly don't add up to 180 degrees, but we're still waiting for Proposition 489 to tell us what the sum of the angles acually is.)

Now we see that E is still at the North Pole, since it's a pole of the Equator AC. (Notice that the points B and E must be on the same side of the Equator AC, but they are almost never actually the same point except in this special case -- which is why I saved it for last.) Point D is the pole of the Prime Meridian BC, so it must lie on the Equator at longitude 90 degrees -- west, since it must lie on the same side of the Prime Meridian as point A. Point F is the pole of the meridian 118W, so it must lie on the Equator 90 degrees away from 118W in the direction of the Prime Meridian, so it must lie at longitude 28W.

Far from being congruent to ABC, DEF actually lies completely inside ABC. Now Legendre's 476 tells us that the angles of DEF are not necessarily congruent to the corresponding angles of ABC -- instead they are supplementary. Indeed, this is the case as angles D and F are still right angles, while the measure of angle E at the North Pole is now 62 degrees as it lies between 90W and 28W -- and 62 degrees is indeed supplementary to the original 118-degree angle.

Notice that the polar triangle of the polar triangle is the original triangle. We see that the polar triangle of DEF still has a vertex at the North Pole, and the other two vertices are on the Equator -- the side 90W moves 90 degrees east back to the Prime Meridian and the 28W moves 90 degrees west back to the 118W meridian.

Despite this, notice that polar triangles are not a transformation in the ordinary sense since we transform entire triangles, not individual points. Indeed, the location of point B tells us nothing about where point E is -- except that B and E must be in the same hemisphere defined by AC. We saw that point E would have been at the North Pole no matter where B was as long as it was in the Northern Hemisphere -- and if B were in the Southern Hemisphere, E would have been at the South Pole, all because A and C are on the Equator.

Let's move on to Legendre's Propositions 478 and 479. These two go together, since 479 is simply a "scholium" of 478:

478. The triangle ABC being given, if, from the pole A and with the distance AC, an arc of a small circle DEC be described, if, also, from the pole B, and with the distance BC, the arc DFC be described, and from the point D where the arcs DEC, DFC cut each other, the arcs of the great circles AD, DB be drawn; we say that, of the triangle ADB thus formed, the parts will be equal to those of the triangle ACB.

Demonstration. For, by the construction, the side AD = AC, DB = BC, and AB is common; therefore the two triangles will have the sides equal, each to each. We say, moreover, that the angles opposite to the equal sides are equal.

Indeed, if the center of the sphere be supposed in O, we can suppose a solid angle formed by the three plane angles AOB, AOC, BOC, we can suppose, likewise, a second solid angle formed by the three plane angles AOB, AOD, BOD. And since the sides of the triangle ABC are equal to those of the triangle ADB, it follows that the plane angles, which form one of the solid angles, are equal to the plane angles, are equal to the plane angles which form the other solid angle, each to each. But the planes of any two angles in the one solid have the same inclination to each other as the planes of the homologous angles in the other (359); consequently, the angles of the spherical triangle DAB are equal to those of the triangle CAB, namely DAB = BAC, DBA = CBA, and ADB = ACB; therefore the sides and angles of the triangle ADB are equal to the sides and angles of the triangle ACB.

479. Scholium. The equality of these triangles does not depend on an absolute equality, or equality by superposition, for it would be impossible to make them coincide by applying the one to the other, at least except they should happen to be isosceles. The equality, then, under consideration, is that which we have already called equality by symmetry, and, for this reason, we shall call the triangles ACB, ADB, symmetrical triangles.

Okay, Legendre is doing several things with these two propositions. Just as in 476, here we are starting with a triangle ABC and forming a second triangle. But unlike 476, the new triangle formed in 478 actually is congruent to ABC.

Notice that Legendre refers to poles of small circles, but we can easily imagine simply placing a large compass with the tip at A and the pencil C, drawing an arc on the other side of AB, and then placing the tip at B and the pencil at D, drawing another arc on the other side of AB. Then the point at which the two compass arcs intersect is the new point D.

In this proof, we have AD = AC, DB = BC, and AB = AB, so it would appear that we already have the two triangles ABC and ADC congruent by SSS. The problem is that we haven't actually proved that SSS works for spherical triangles yet! So we must show that the angles are congruent as well. We see this occur in traditionalist Euclidean geometry texts as well -- when students first learn about triangle congruence, about to learn about SSS and the other postulates in the next section. The students are given two triangles and they must show that all of the corresponding parts are congruent. The final step would be that the two triangles are congruent by "definition of congruent triangles" (i.e., that all the corresponding parts are congruent). Of course, as soon as students learn about SSS and the other postulates, they'll never use the sufficient condition part of "definition of congruent triangles" in a proof again.

But Legendre now uses the traditionalist "definition of congruent triangles" in his proof. We've already proved that the sides are congruent, so now we work on the angles. Notice that he goes back to solid angles, and uses our trick that since the sides (arcs of great circles) are congruent, so must the plane angles (the central angles of those arcs). Again, I point out that if the sphere has radius 1, the sides of the spherical triangles equals the radian measures of the plane angles.

Legendre mentions that he uses Proposition 359 in his proof. This theorem essentially states that if two solid angles consist of three or corresponding congruent plane angles, then the two solid angles are themselves congruent, and so are the dihedral angles (the "inclinations") between the planes. This matters because Legendre defines angles of a spherical triangle in terms of these dihedral angles. And so the two triangles ACB and ADB must be congruent.

Now in the "scholium," Legendre states that these two triangles are equal "by symmetry." If we think about it, we notice that a simple reflection actually maps Triangle ACB to Triangle ADB -- the reflection over the (plane containing the) great circle AB. And so we see our first spherical isometry.

The concept of "superposition" goes back to Euclid. If we consider Euclid's proof of SAS, he uses superposition by translating and rotating one of the triangles until it coincides with the other. We see that Euclid omits reflections here -- so strictly speaking, he didn't actually prove SAS. Legendre corrects this error by counting only translations and rotations (isometries that preserve orientation) as "superposition" -- two figures that are mirror images of each other are equal "by symmetry" instead of equal "by superposition," except in the special case that the triangles are isosceles (since we could then rotate one triangle 180 degrees with the pole at the midpoint of AB to obtain the other).

I've mentioned earlier that in Euclidean geometry, there are four isometries (rotation, translation, reflection, and glide reflection), while in hyperbolic geometry, there are five isometries (rotation, translation, horolation, reflection, and glide reflection). So now we wonder, how many isometries are there in spherical geometry?

We recall the U of Chicago text and its definition of rotation -- it is the composite of two reflections in intersecting lines. Meanwhile, the definition of translation is that it is the composite of two reflections in parallel lines. But in spherical geometry, there are no parallel lines. Therefore, in spherical geometry there are no translations!

Notice that any transformation that appears to be a translation is actually a rotation, since any apparent "translation" is actually motion along either a great or small circle. The center of the rotation is simply the pole of this circle. So a "translation" east or west is a rotation centered at the North (or South) Poles, while a "translation" north or south is a rotation centered at a point on the Equator that is 90 degrees away from the current longitude. In either case, the magnitude (degree) of the rotation is equal to the change in longitude or latitude of the point.

This leaves us with the glide reflection. We notice that a glide reflection in Euclidean geometry is the composite of a reflection and a translation, but in spherical geometry there are no translations. Still, in spherical there exist transformations that are the composite of three reflections, yet are not themselves simple reflections. One may still call these "glide reflections" even though there are no translations -- these glide reflections are also the composite of a reflection and a rotation.

One key glide reflection is the antipodal map -- the function mapping each point to its antipodes. This map is the composite of a reflection in any line (great circle) and a rotation of 180 degrees centered at the pole of that great circle.

We can also think of the isometries of the spherical geometry in terms of the (three-dimensional) symmetries of the sphere itself. In this case, line (great circle) reflections correspond to plane reflections, point rotations correspond to axis rotations, and glide reflections correspond to what we call "roto-reflections," with the antipodal map corresponding to point inversion about the center of the original sphere.

If we think in terms of elliptic rather than spherical geometry, then all isometries are rotations! This is because we identity a point with its antipodes. So to perform a reflection on a figure, we simply take the (reversed) copy of that figure at the antipodes and rotate it all the way around the globe 180 degrees until it arrives at the place where we want the reflection to be. So in elliptic geometry, all isometries are "superpositions."

Let's get to our final proposition of the day -- but it's an important one:

480. Two triangles situated on the same sphere, or on equal spheres, are equal in all their parts, when two sides and the included angle of one are equal to two sides and the included angle of the other, each to each.

Demonstration. Let the side AB = EF, the side AC = EG, and the angle BAC = FEG, the triangle EFG can be placed upon the triangle ABC, or upon the triangle symmetrical with it ABD, in the same manner as two plane triangles are applied, when they have two sides and the included angle of one respectively equal to two sides and the included angle of the other (36). Therefore all parts of the triangle EFG will be equal to those of the triangle ABC; that is, besides the three parts which were supposed equal, we shall have the side BC = FG, the angle ABC = EFG, and the angle ACB = EGF.

Here, we see that Legendre is proving SAS Congruence for spherical triangles. He compares this to his Proposition 36, which is SAS Congruence for ordinary plane triangles.

Legendre proves SAS first by superposition -- EF can be rotated so that it lands on AB. Now there are only two possible places where G can end up -- C and D, where D is the vertex of ADB, the triangle that is "equal by symmetry" (i.e., the reflection image) to ACB.

The proof of SAS Congruence in the U of Chicago text is similar to this. The text transforms the original triangle ABC until its image lies on EF. Then the Isosceles Triangle Theorem is used to show that since CDG is isosceles, the reflection image of C must be G. Notice that as we traditionally use SAS to prove the Isosceles Triangle Theorem (and Legendre will do exactly this), we can avoid circularity by using the Ruler and Protractor Postulates to demonstrate why C reflected over AB must be the point G.

In both the Euclidean and spherical cases, it may be instructive to find the actual transformation that maps AB onto EF. On the sphere, we notice that if AB and EF are on the same great circle, the correct rotation is obvious. If AB and EF aren't on the same great circle, then we notice that their great circles must intersect since all great circles intersect. The points of intersection can be taken as the poles of a rotation which maps the great circle of AB to that of EF. Now this reduces us to the first case, so a simple rotation maps the image of AB to EF.

Thus concludes this post.

Wednesday, July 22, 2015

How to Fix Common Core: Pi Approximation Day Version

Today is Pi Approximation Day, so-named because July 22nd -- written internationally as 22/7 -- is approximately equal to pi.

It also marks one full year since my very first post here on the blog. Yes, I've declared Pi Day of the Century -- March 14th, 2015, to be the ceremonial start date of this blog. But the chronological start date was July 22nd, 2014. I like celebrating both Pi Day and Pi Approximation Day -- including eating pie both days -- and so I celebrate both of them as launch dates for the blog.

Let me a little introspective here. Writing on this blog has taught me many things about myself that I have not realized.

For starters, I realize that I change my mind a lot. I say that I'm going to post something about a geometry topic, then I change my mind and post something else -- and then I say that I'll post a third thing when I revisit that topic next year. From the perspective of the blog readers -- you -- it must be infuriating to read something in the archives that I say that I'll post and then move forward in the archives only to find out that I never posted it.

Another thing I notice about myself is that too many posts contain too many topics. Let's just look at this very post, for starters. This post begins with this anniversary introspection, then it moves on to Eugenia Cheng's How to Bake Pi, a book about category theory, since I'm still in the middle of reading her book. Then this post is considered part of the "How to Fix Common Core" series -- and I wrote in the previous post of this series that I would discuss the Common Core tests. Finally, since it is after all Pi Approximation Day, I want to post some more links to videos about the number pi.

Of course, I want to say that this hodgepodge of topics is only because it's summer -- I could make several posts each with its own topic, but I don't want to make too many summer posts because this blog is all about the school year, not the summer. But too many of my posts during the school year contain several topics as well. For example, some of my posts which purport to be about the PARCC Practice Exam don't mention that test until deep within these posts.

I notice that some of the best blogs often use "tags" or "labels" -- these labels describe the content of each post. It's about time that I use these labels as well. If I used the labels correctly, today's post should have the following labels:

-- FAQ
-- Eugenia Cheng
-- How to Fix Common Core
-- Pi Day

And so let me proceed with the FAQ, since today's anniversary is an excellent day for me to remind or inform the readers about my intentions with this blog.

1. Who am I? Am I a math teacher?

I am David Walker. I have recently earned my clear credential in Single Subject Math here in my home state of California, but I have not been hired as a full-time teacher. Instead, I have worked as a part-time math tutor since October 2012 and a substitute teacher in two districts since October 2014.

This already explains some of why I change my mind so frequently. I changed the blog calendar (where my posts are labeled Day 1, Day 2, etc.) last October so that it follows the actual school calendar in one of the districts where I sub. This year, I plan on using the calendar for the other district instead. Day 1, the first day of school, will be on August 26th in this district. If I am hired to become a full-time teacher in a district, then of course I will change the calendar again to reflect the school year in that district. But as it is already Pi Approximation Day and I haven't been hired anywhere yet, so I am expecting to return to substitute teaching on August 26th.

2. What is Common Core Geometry?

Common Core Geometry refers to the new geometry standards included in the Common Core. In particular, I refer to the Core's new focus on translations, rotations, and reflections -- these transformations were not emphasized before the Core. Transformations appear in the standards for eighth grade and above.

The Common Core is neutral in that it supports either a traditionalist pathway (Algebra I, Geometry, Algebra II) or an integrated pathway. On the traditionalist pathway, of course it's mainly the Geometry class where the new geometric transformations appear. On an integrated pathway, the transformations appear in both Math I and Math II.

One of the districts where I work uses the traditionalist pathway, while the other is making the transition to the integrated pathway.

3. Am I for or against the Common Core Standards?

There are some things that I don't mind about the Common Core and some things I'd change. I know that some people oppose the Core for political reasons. I try to avoid politics here on the blog, but unfortunately the Core is inherently political. In particular, opponents of the political party that was in power when the Core was first adopted tend to dislike the Core as well.

Some people believe that the Tenth Amendment to the Constitution forbids Common Core -- only fifty separate sets of standards, one for each state, would be constitutional. I prefer national standards because I find that the top math nations -- the ones that the United States seeks to emulate -- have strong national standards. Some of the top math nations do have standards at their equivalent level of the state, and so I don't mind having separate sets of state standards, even though 50 is a lot. On the other hand, I oppose having completely localized standards -- one set for each city, town, or district -- because the top math nations do not have localized standards.

Some people oppose Common Core math because they do not like integrated math. I strongly disagree with this because the top math nations all use integrated math -- the United States is an outlier with its traditionalist Algebra I, Geometry, and Algebra II. In particular, the highly respected Singapore Math is based on the integrated pathway. Nonetheless, this website is currently based on the traditionalist Geometry class since I wish to focus on Geometry.

I once heartily endorsed the Singapore Math standards as a great example of an integrated math program for the United States to adopt. But since then I fear that Singapore Math, especially its Secondary Year Two program, is too advanced for most American eighth graders and would only set them up for failure. So instead, I prefer the other highly respected integrated program, Saxon Math, since I feel that its Algebra 1/2 (half) text is more realistic for American eighth graders -- and this text covers much of the same material as the Common Core.

There are some things that I would change about the Common Core. The least popular aspect of the Common Core is, of course, the testing associated with it, especially PARCC and SBAC. In today's post, I want to discuss what I would change about the Common Core tests.

4. What is the "U of Chicago text"?

In 1991, the University of Chicago School Mathematics Project published a series of secondary mathematics texts. I was able to purchase both the Algebra I and Geometry texts for two dollars each at a local public library. One thing I discovered was that the U of Chicago Geometry text is based on the same transformations that appear in the Common Core Standards -- even though the U of Chicago published the text nearly two decades before the advent of the Core.

The common denominator is NCTM, the National Council of Teachers of Mathematics. This group was behind both the U of Chicago text and the Common Core Standards. Still, this means that one can satisfy most of the Common Core Standards by teaching out of the U of Chicago text.

There are newer versions of the Geometry text than the one that I own. I've read that while my old version introduces reflections in Chapter 4 and the other isometries in Chapter 6, in the new version all isometries appear in Chapter 4.

The U of Chicago text is not the only book that I mention here on the blog, not by a long shot. Many posts here refer to the Glencoe Geometry text. This is mainly because I am a math tutor, and I've been tutoring an eighth grader who is using the Glencoe text. His seventh grade teacher fell behind in Algebra I, so this year the eighth grade teacher finished the Algebra I text during the first quarter and started Geometry during the second quarter. He ended up reaching Chapter 9 of the Glencoe text by the end of the year -- which just happened to be the chapter on transformations.

Sometimes when I sub at various schools, I comment on the various texts that I see in the classrooms where I am working. I especially like listing the various Geometry texts as well as Integrated Math I and II texts so that I can point out the Geometry chapters.

Unfortunately, I've never seen a copy of the Saxon Algebra 1/2 text that I mentioned earlier. The only Saxon text that I own is the Saxon 65 text for fifth and sixth graders.

To learn more about some of these texts, here are some links to a webpage by Cathy Duffy. Duffy's goal is to help homeschooling parents choose a textbook for their children. Here is her review of the U of Chicago text:

and the Saxon series:

5. Who are David Joyce, Dr. M, and Hung-Hsi Wu?

These are mathematicians who have commented on how geometry should be taught. Dr. David Joyce actually discusses pre-Common Core Geometry standards. At the following website, he strongly criticizes a Prentice-Hall Geometry text:

Dr. Joyce strongly believes that theorems should be proved before they can be used -- that is the overarching theme of his criticism. I don't agree with everything he writes -- for example, he criticizes Chapter 3, the chapter on isometries. Pre-Common Core, his criticism would have made sense, but now with the advent of the Core, things have changed. We now actually use isometries to prove SSS, SAS, and ASA all as theorems. Still, I attempt to use the Joyce ideal of using nothing before it is proved -- especially considering what Joyce writes about coordinate geometry:

In a return to coordinate geometry it is implicitly assumed that a linear equation is the equation of a straight line. A proof would depend on the theory of similar triangles in chapter 10.

The Prentice-Hall text doesn't prove this, nor do most other texts (not even the U of Chicago text) -- but the proof is actually required in the Common Core eighth grade standards!

Oh, and as a bonus, Dr. Joyce links to a copy of Euclid's Elements -- the world's first geometry text:

Dr. Hung-Hsi Wu is a Berkeley mathematician. He has written extensively on how to teach Geometry according to the new Common Core Standards:

I have used some of Dr. Wu's proofs, including the use of rotations to prove some of the properties of parallel lines. I also once tried to use Dr. Wu's proof of the Fundamental Theorem of Similarity -- but this proof is extremely complex and inappropriate for high school students. After posting the long Dr. Wu proof, I changed my mind and suggested that a Dilation Postulate be used instead. This is what Dr. Joyce means when he writes:

Chapter 10 is on similarity and similar figures. (Appropriately for this level, the difficulties of proportions are buried in the implicit assumptions of real numbers.)

Dr. Wu calls this the "Fundamental Assumption of School Mathematics." It is Dr. Wu who defines pi as the area of the unit disk -- this definition favors pi above the numbers tau and lambda. (Yes, today is Pi Approximation Day, so here's a pi reference.) Meanwhile, Wu has also created an extensive document on how to teach fractions under the Common Core:

Dr. Franklin Mason is a Geometry teacher at an Indiana high school. He has created an online text:

He has changed his text several times since first posting it. I've surmised that most likely, Dr. M wrote the original version based on what the Common Core Standards say, but the new version is based on what actually appears on the PARCC and SBAC tests. The original text followed the David Joyce ideal of using nothing before it is proved, but the new text replaces theorems with postulates if the proofs are more difficult than anything that appears on the PARCC or SBAC.

Here's an example of the evolution of the Dr. M text: originally, Dr. M used several theorems, including SAS, to prove a Triangle Exterior Angle Inequality (TEAI), and then used the TEAI to prove his Parallel Line Tests. Later on, he dropped the proof of TEAI and made it a postulate. Still later on, he dropped the proof of the Corresponding Angles Test and made it a postulate.

But Dr. M still includes a proof that all circles are similar. Dr. Joyce writes:

So the content of the theorem is that all circles have the same ratio of circumference to diameter. This theorem is not proven. It would require the basic geometry that won't come for a couple of chapters yet, and it would require a definition of length of a curve and limiting processes.

Dr. M actually includes these "limiting processes" by introducing a least upper bound. (Yes, here's another pi reference for Pi Approximation Day.)

There are a few more resources for Common Core Geometry. One is the EngageNY website:

The State of New York has developed a strong Common Core curriculum. Much of this is based on the recommendations of Dr. Wu. For example, Lesson 18 uses 180-degree rotations to develop the Alternate Interior Angles Test. This idea goes back to Dr. Wu:

Here is one last resource that I wish to mention, the Geometry Common Core website:

The website creator, Mike Patterson, teaches Geometry at a Nevada high school. I sort of like his idea of using translations to prove the Corresponding Angles Test, rather than rotations to prove the Alternate Interior Angles Test, since I consider corresponding angles to be more basic:

But unfortunately, I'm not quite sure whether one can actually write this proof without introducing some circularity, as the properties of parallel lines are uses to prove the properties of translations (at least in the U of Chicago text).

6. Who are "the traditionalists"?

I use the phrase "traditionalists" to refer to those who oppose Common Core for mathematical -- not political -- reasons. Such mathematicians include:

Dr. Katharine Beals
Dr. Barry Garelick
Dr. James Milgram
Dr. Ze'ev Wurman
SteveH (often comments at the websites of the other traditionalists)

The traditionalists tend to believe that many of the topics in the Common Core Math Standards are taught too late. In particular, they believe that the standards in Grades 4-6 that require the use of the standard algorithm for arithmetic should move down at least one grade level.

I tend to agree with the traditionalists on this point here -- most of the reasons for opposing direct instruction, rote memorization, and drill don't apply to the youngest students. I remember once learning about two nonstandard algorithms for addition -- the Left-Right Method and the Plus-Minus Method -- on an old 15-minute math show for sixth graders, "Solve It," that used to air on our local PBS station back when I was that age. For mental math, these nonstandard algorithms are great, but for a young elementary student first learning to add two-digit numbers I prefer the standard algorithm.

The traditionalists also prefer that high school seniors take AP Calculus in order to open the door to college STEM majors. In particular, some STEM colleges here in Southern California, such as Cal Tech and Harvey Mudd, require calculus for admission. Even colleges that don't formally require calculus are likely to send students rejection letters if they apply without having calculus on the schedule, especially if they apply to STEM majors. Therefore they want the classes leading up to senior year to prepare students to take AP Calculus -- including Algebra I in the eighth grade. They oppose Common Core because its eighth grade math standards do not comprise a full Algebra I class.

I only partially agree because I don't want seniors who can't handle AP Calculus or eighth graders who can't handle Algebra I to be labeled failures. Just as AP Calculus can open the door to STEM, it can close the door to a non-STEM career if students are forced to spend too much effort trying to pass their high school math classes. Then again, my preferred eighth grade text, Saxon Algebra 1/2, does provide a path to a Calculus class: Saxon Algebra 1 (third edition) in freshman year, Saxon Algebra 2 (third edition) in sophomore year, Saxon Advanced Math in junior year, and then Calculus.

Most traditionalists dislike the U of Chicago math texts. However, when a traditionalist refers to U of Chicago math, he is nearly always referring to the elementary school texts -- he rarely, if ever, considers the U of Chicago texts for sixth grade and above. Most likely, traditionalists would dislike the high school texts as much as they do the elementary texts -- especially since they already oppose Common Core and I emphasize how well the U of Chicago texts are Core-aligned.

Here are links to some of the traditionalist websites:
(The last is an article on which SteveH has commented heavily.)

7. How would I fix the Common Core tests?

Well, I said that I would post this today, so let me do exactly that. I repeat that the testing is the least popular aspect by far about the Common Core. Myriads of parents around the country chose to opt their students out of the Common Core tests.

What I'd like to do in this section is design a test that parents would want their students to take, so that there wouldn't be so many opt-outs. Of course, PARCC and SBAC may have already poisoned the well so much that parents may opt their children out of any national standardized test, no matter how good the test may be. To such parents, nothing that I can write in this post would produce a test that they'd allow their children to take.

Throughout this "How to Fix Common Core" series, I would often go back to the traditionalists to see what their specific objects are and how I would address them. I link above to the comment section of an article where the traditionalist SteveH made several posts. Let's find out what SteveH had to say about the Common Core testing:

"The ACT and College Board are now in the process of developing and improving their own college readiness assessments, and the College Board, in particular, offers assessments geared towards AP classes and STEM careers. Some states, like Alabama, have dropped PARCC and are now going with ACT. We need more than CCSS, and my hope is that states will use tools from ACT and the College Board to define a high-level curriculum path back to the lowest grades. The words in the CCSS document are too vague and open to vast differences in interpretation and level of expectation. States need to pick the assessment tests and let those tests drive the curriculum."

OK, I agree with what SteveH writes here. One way to improve the testing is simply to reduce the number of tests that students must take. Since high school students already take the ACT, that can be already replace the PARCC and SBAC tests that high school students must take. Like SteveH, I also look forward to seeing what the ACT will do to design a test for the lower grades.

Unfortunately, I don't completely agree with what SteveH writes in the next paragraph:

"Life will go on for many students and parents who know better than to look to a one-size-fits-all standard for how their kids are progressing. Most middle schools will continue to offer proper pre-algebra and algebra courses as lead-ins to a rigorous geomptry, algebra II, pre-calc, and calc sequence in high school - NOT integrated math curricula.. These math classes will continue to be driven by teacher-centered direct instruction, and these courses will continue to produce the STEM-prepared students that colleges are looking for. These students have not, and will continue to not be created by standardized state tests. For the most part, these STEM kids will be created by parents who reteach at home and with private tutors. STEM students will continue to be NOT be created by hands-on, group work in class. They will be created by a steady diet of individual homework sets, and parents, tutors, and teachers who ensure that students master those skills. I find it ironic that K-8 educators talk about having little Johnnie or Suzie think like a real mathematician or engineer while those professionals are telling them that they got it completely wrong."

Here are the parts of this paragraph with which I disagree:

-- "NOT integrated math curricula." Many traditionalists are convinced that integrated math is inferior to the usual Algebra I, Geometry, and Algebra II. In reality, the majority of the world's students who succeed in Calculus learned Integrated Math to prepare them for Calculus. Ironically, many of the parents and tutors mentioned later in the paragraph probably used the Singapore and Saxon Integrated Math texts.

-- "driven by teacher-centered direct instruction." Like most traditionalists, SteveH strongly prefers the direct "sage on the stage" to the progressive "guide on the side." The only problem with this is that it presumes that students would willingly listen to the teacher and accept him or her as the "sage on the stage." This may be true for younger students -- and this is one reason that I agree with traditionalism in the younger grades -- but it isn't true for the teenagers who would be taking the classes discussed in this paragraph.

Teenagers are at an age where they begin to question authority. Therefore, in math classes, they ask questions like "Why do we have to learn this?" or "When will we use this?" I can only assume that if SteveH taught such a class, he'd answer something like "It will prepare you for Calculus, which is necessary for a STEM career." But such an answer might not necessarily convince teens to get up and do the work -- especially those who have no intention of going into a STEM career.

SteveH mentions "individual homework sets" later in the paragraph. He appears under the impression that all if he were the teacher, all he'd to do is give lectures everyday in class and long homework sets every night, and teens will just do it because he is the "sage on the stage." But merely assigning individual homework sets won't prepare students for Calculus if they don't do the assignments -- or if they are no longer individual because someone else is doing the homework.

I have a problem with forcing everyone onto the STEM path in high school, in particularly in the name of "keeping doors open." Actually, eighth grade Algebra I isn't as much of a problem because colleges mainly look at the high school grades, so a D or F in eighth grade Algebra I doesn't really close doors. (I do note that some middle schools don't allow eighth graders to participate in the promotion ceremony if they are failing English or math during the last semester/trimester. I'd hate to see an eighth grader miss promotion after passing every math class except the last term just because he can't factor a quadratic polynomial.)

The real problem with the SteveH plan is in high school -- especially sophomore Algebra II, as there is a huge leap in difficulty between Algebra II and any math class that precedes it. If a student can't participate in sports or get a part-time job -- both of which might lead to a great non-STEM career -- because he's too busy trying to finish SteveH's "individual homework sets" for an Algebra II class that's unnecessary for his career path, then he's likely to see Algebra II and Calculus as closing doors, not keeping them open.

SteveH writes that he doesn't want his standards to be "one-size-fits-all." He points out that while there are five scoring levels on the PARCC and four on the SBAC, there are actually only two levels that matter. I'll call these levels "proficient" and "deficient." Schools are judged by how many students score as "proficient." There is no incentive to raise students from "low deficient" to "high deficient," or, much to SteveH's dismay, from "low proficient" to "high proficient." Therefore students end up spending all of their resources trying to raise students from "deficient" to "proficient" and hardly any trying to raise them from "proficient" to "Calculus-ready." And so let's try to design a test that would break out of this "one-size-fits-all" mold.

Suggestion #5 (the first four suggestions appear in earlier June posts): Here is a detailed design for how I would create a new computer-based Common Core test.

We begin with the third grade test, as this is typically the lowest grade that the tests are given. The third grade test will begin with simple one-digit multiplication problems.

Now the most important part of the test will be the scoring system. I want the average third-grader at the end of the year to receive a score of 400. In fact, the scoring should go like this:

100: Average first-grader at start of year
200: Average second-grader at start of year
300: Average third-grader at start of year
400: Average fourth-grader at start of year
500: Average fifth-grader at start of year
600: Average sixth-grader at start of year
700: Average seventh-grader at start of year
800: Average eighth-grader at start of year

So our average third-grader, taking the test at the end of the year, should score 400, since he or she is getting ready to start the fourth grade.

The test begins with our third-grader having a baseline score of 300 points. The goal is for the student to reach 400 points in the 30 minutes allotted for the test -- since I've already stated that I want to reduce the test time to 30 minutes. We could set each one-digit multiplication problem to be worth one point each. So if the student gets one question correct every 18 seconds, then this will get them the 100 points that they need in 30 minutes.

Notice that this would obviously be a computer-adaptive test, just like the current SBAC. But I don't believe that the SBAC takes full advantage of its computer adaptivity. Our test here keeps a running score and asks questions based on the current student score. Each time a student answers a question correctly, he or she gains a point, and each time a student answers a question incorrectly, he or she loses a point.

Traditionalists say that the most important math for a third-grader to learn is multiplication -- but it need not be the only math that they learn. We could set it up so that once a student reaches a score of 350, the questions switch so cover the rest of the third grade Common Core Standards. This will mean that 50% of the third grade score is devoted to multiplication -- which is probably more than the current PARCC and SBAC third grade tests. Questions can be worth more than one point, especially if they take more time to answer, so as to maintain the average score of 400.

Notice this means that, at least during the 300-350 multiplication section, a student who gets only half of the questions right will gain and lose equal numbers of points, so the score remains at 300. This is consistent with the fact that 50% in class is a failing grade -- and we certainly want third graders to get more than half of the multiplication questions correct. I'd argue that for more complicated questions, students can gain several points for correct answers but lose only one point if the answers are incorrect. Otherwise, a student who gets all of the easy answers right and all of the hard answers wrong would have a low score (since they'd gain points one at a time and lose them two at a time).

If the score drops below 300, then the test would switch to second grade questions, with more of a focus on addition and subtraction. Once again, the types of questions a student receives is based on the running score.

If the score rises above 400, then the test would switch to fourth grade questions. Notice that an exceptionally bright third grader might make it all the way to a score of 500 before the 30 minutes are up -- if this happens, the test would switch to fifth grade questions. I don't want the running score to appear on the screen while the student is taking the test, but when the thirty minutes are up, the score is reported immediately, on both the screen and a computer printout. This means that every question must have either a multiple-choice or numeric answer. Performance tasks where the student has to answer with sentences will not appear on this test.

When a fourth-grader returns to take the test the following year, the questions with which the test begins are based on the score the student received the previous year. So a student with a score of 400 starts with fourth grade questions, a student with a score of 273 starts with second grade questions, a student with a score of 546 starts with fifth grade questions, and so on.

The maximum score of this test should be 800, just like the SAT. But this time, a score of 800 represents the beginning of eighth grade -- which should be Algebra I, according to SteveH. The problem is that we don't want the test to favor either the traditionalist or integrated pathway -- it is just as unfair for an Algebra I student to answer geometry questions as it is for an Integrated Math I student to answer polynomial questions. Recall that this was the problem when California tried to implement eighth grade Algebra I on top of the Common Core Standards -- the problem was that eighth-graders in Algebra I would be unable to answer geometry questions on the SBAC eighth grade math test.

So I want this test to end with seventh grade math. We can have separate end-of-course exams for Algebra I, Geometry, and Integrated Math I/II, or roll them into other tests for students. Otherwise, a student who reaches a score of 800 but doesn't take the Algebra I or Integrated Math I tests should not take any math test at all.

We notice that this scoring system fits very well with the path system described in other posts. Of course, no national test can force schools to use paths, but it can certainly encourage it by means of the scoring system described in this post. We can also subtly encourage eighth grade Algebra I via the scoring system, simply by reporting scores on the Algebra I test on a scale of 800 to 900.

Actually, I've changed my mind slightly about the path system. Recall that in my proposal, students are divided into paths based on their reading levels -- students reading at a first or second grade level would be on the Primary Path, those at a third or fourth grade level on the Transition Path, and so on.

This makes sense at elementary schools, since reading is such a critical skill. But an argument can be made that at the middle school level, the most critical skill is math -- this is, after all, the crux of what SteveH is saying. Back when California still reported state scores for ELA and math, I can just look at a school's score and determine whether it is elementary or secondary. If the math score is higher, then it's probably an elementary school (2+2 is easy, but reading is hard). If the ELA score is higher, then it's probably a secondary school (they know how to read, but now algebra is hard). This trend persisted regardless of the student demographics.

And so one might introduce a sort of path system where students are divided onto paths based on their reading levels in elementary school but their math scores in middle school. Of course, by the time the students reach middle school they are already having several teachers throughout the day, so a path system may be irrelevant at all except the smallest middle schools.

Let's get back to Eugenia Cheng's book How to Bake Pi. I've now reached Part II which is on Category Theory -- the main topic of the book. In particular, I've completed Chapter 13, "Sameness."

As I've said earlier, category theory is just as new for me as it is for most of you. Here is what Cheng says about category theory: "Category theory is the mathematics of mathematics. Category theory, then, is the process of working out exactly which parts of math is easy, and the process of making as many parts of math easy as possible.

In Chapter 11, Cheng draws several examples of categories -- many of these look just like the networks found in Section 1-4 of the U of Chicago text. Here the nodes are objects and the vertices are actually directed line segments, or arrows, that represent "morphisms," which are sort of like functions, but Cheng describes them better as relationships.

I'm still trying to make sense of what I'm reading in Cheng's book. But just as I'm struggling to learn category theory from her book, many students in our high school algebra and geometry classes are struggling to learn from us. Algebra is as difficult to them as category theory is to me. This is why I am not as eager to put students into higher math as SteveH is, but my hope is that with a test similar to the one that I propose here, we'll be able to get our best math students -- those who are able to make that great leap of abstraction -- into the top math classes necessarily for success in STEM.

I remember when I was learning about groups and rings at UCLA, the professor often drew what he called a "commutative diagram." Although he never used the term "category," it is now obvious to me after reading Cheng's book that my professor was using category theory. In the same way, sometimes it takes a higher class, like Calculus, to see why something in Algebra or Geometry makes sense -- and this goes right back to the SteveH door to Calculus. Here is a simple example of such a commutative diagram, except this time I used sets of geometry figures as the objects. Here the morphism is the inclusion map:

Parallelogram --> Rectangle
          |                    |
          |                    |
         V                  V
Rhombus       --> Square

Cheng begins the last chapter that I've read so far, Chapter 13, with a recipe for "Raw Chocolate Cookies" -- another recipe that Cheng created herself. She points out that "raw chocolate cookies" may be so different from "cookies" that it's unclear whether they should still be called "cookies" -- just like her "olive oil plum cake" from earlier in the book. The chapter describes how it's not always clear what it means for two objects to be the same. She gives triangle similarity as an example of sameness (same shape). Of course, we think of congruence as being a stronger example of sameness, while topological equivalence is a weaker example of sameness.

But today is Pi Approximation Day, and I like to eat apple pie -- which really is the same as a pie -- since I associate apple pie with this time of year ("as American as apple pie," and this is the month that we celebrated Independence Day").

In honor of Pi Approximation Day, let me post some more videos about pi. The first few of these mention several ways to approximate pi.

The History of Pi. This video mentions that even Legendre -- the author of the geometry text we've been discussing in some of my recent posts -- contributed to the history of pi. In particular, Legendre proved that not only is pi irrational, but so is pi^2 (unlike sqrt(2), which is irrational yet has a rational square, namely 2).

This video also mentions a quote from the mathematician John von Neumann: “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” We've seen Eugenia Cheng, in her How to Bake Pi, say almost exactly the same thing.

The Infinite Life of Pi:

Here is one of the oldest pi videos on YouTube. It appears to have been posted close to Pi Approximation Day in 2006:

No Pi (Approximation) Day is complete without a Numberphile video. In this video, we learn that pi is not just irrational, but is also transcendental -- that is, there doesn't even exist a polynomial f, with integer or rational coefficients, such that f (pi) = 0 (unlike sqrt(2), which is algebraic). This is even stronger than Legendre's result that pi^2 is irrational.

Here is a longer video, but it contains some more series approximations of pi:

This one's in German, but it does contain some interesting information. The video credits the Greek mathematician Archimedes as the first to use today's approximation 22/7 for pi, then it moves on to the question of whether pi is a normal number -- i.e., with random digits. (Oh, and BTW, an episode of Futurama, in one of its background jokes, refers to 22/7 as "Fool's Pi.")

Here is a short Pi song:

No, I can't avoid another one of my favorite math YouTube posters, Vi Hart. Of course, in this video she argues that there's nothing special about pi at all:

Let's wrap up with one more Pi song, a longer one this time:

Finally, this link is not a video, but it's one of my favorite links for Pi Approximation Day. We know that 22/7 is approximately equal to pi. As it turns out, 22/7 is actually more than pi -- and we can find out why 22/7 > pi using calculus. It's possible that an AP Calculus student -- a senior on the SteveH plan -- might be able to calculate this integral:

A few other Pi Approximation Day (aka "Casual Pi Day") links:

And so I wish everyone a  Happy Pi Approximation Day.

Tuesday, July 14, 2015

Spherical Geometry (Legendre 469-476)

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.