Lesson 5-7: Sums of Angle Measures in Polygons (Day 57)

This is what Theoni Pappas writes on page 310 of her Magic of Mathematics:

(nothing)

Oh yeah, that's right. We're covering Chapters 1 and 2 of the book in November and December. So let me restart that....

This is what Theoni Pappas writes on page 6 of her Magic of Mathematics:

-- Italian mathematician Giovanni Borelli proved that human muscles were too weak to support flight (1680).

Chapter 1 of the Pappas book is called "Mathematics in Everyday Things." And this is the second page of its first subsection, "The Mathematics of Flying." As we can see, math explains why humans can't fly unaided. And math (along with physics) also shows us how to build something that can fly.

Here are a few more excerpts from this page:

-- British inventor, Sir George Cayley, designed the airfoil (cross-section) of a wing, built and flew (1804) the first model glider, and founded the science of aerodynamics.
-- In 1903 Orville and Wilbur Wright made the first engine powered propeller driven airplane flights. They perfected their flying techniques and machines to the point that by 1905 their flights had reached 38 minutes in length covering a distance of 20 miles!

Here's how we get off the ground:
In order to fly, there are vertical and horizontal forces that must be balanced. To counteract the pull of gravity, lift (a vertical upward force) must be created. The study of nature's design of wings and of birds in flight holds the key. One does not always think of air as a substance, since it is invisible. The wing of an airplane, as well as the airplane itself, divides or slices the air as it passes through it. Bernoulli's principle explains how the shape of a wing creates the lift force. The top of the wing is curved.

OK, you may ask, so what does a curved wing top have to do with anything? The story continues on page 7, so you'll have to wait until tomorrow to find out.

Meanwhile, this is my first post after Daylight Saving Time and the clocks falling back one hour. It is a tradition for my blog to discuss the DST debate right around each time change, and as you can see, this post is no exception. (On the other hand, I missed the spring forward clock change, since this was during the Big March blog hiatus when I didn't make any blog posts.)

Let me start with something I wrote about last year -- the Kansen Chu bill that beings Year-Round Standard Time to my home state of California. That bill failed, yet Chu didn't give up. This year, he tried to introduce another bill, except this time it would be for Year-Round DST instead. Recall what I wrote last year about this:

But if I were to choose a single clock to use year-round, I'd personally prefer Year-Round DST. Part of this could be that I don't have young children. Parents of young children often prefer Year-Round Standard Time as sunrise and sunset on this clock are closer to the wake and bedtimes of children, while adults without young children prefer Year-Round DST, where sunrise and sunset are closer to the wake and bedtimes of adults.

Notice that legally, there is no such thing as Year-Round DST. Officially, Year-Round DST is actually the Year-Round Standard Time of the next time zone. So my favored Year-Round DST in California is formally Year-Round Mountain Standard Time. States have the power to adopt Year-Round Standard Time or have a biannual clock change, but they don't have the power to change time zones (which is what Year-Round DST actually is) -- instead it requires an act of Congress.

Here is a link to Kansen Chu's website:

https://a25.asmdc.org/press-release/assemblymember-kansen-chu-issues-statement-ahead-upcoming-end-daylight-saving-time

“On Sunday, November 5, for the second time this year, we will have to switch our clocks and move back one hour until next spring. It is a practice I aim to eliminate with AB 807, my bill that will implement critical first steps to move California to enjoy year round daylight saving time,” said Chu. “The practice of switching our clocks originated from World War I as a means to save energy. Studies today illustrate that there is little to no energy savings. In reality, switching our clocks twice a year directly impacts our health, as well as leads to increased crime and traffic. It is an outdated practice and I hope to work with the community at-large and my legislative colleagues to move to permanent daylight saving time all year.”

Again, Chu can't implement Year-Round DST with AB 807 alone -- Congress is required to act.

Meanwhile, the movement to implement Year-Round DST in New England is still active:

https://www.boston.com/news/local-news/2017/11/05/new-england-atlantic-time-zone

In this link, it's acknowledged that a single state shouldn't make the change alone, and so the New England states are trying to coordinate the conversion to Year-Round DST. But it's difficult to determine where the boundary between the status quo and Year-Round DST should be. For example, Connecticut is sometimes considered part of New England, but parts of it are also in the Tri-State New York metro area. In other words:

-- If the other New England states convert to Year-Round DST, then so should Connecticut (in order to avoid dividing the Boston metro area).
-- If Connecticut converts, then so should New York (to avoid dividing the NY metro area).

But notice that we can continue with this:

-- If New York converts, then so should New Jersey (to avoid dividing the NY metro area).
-- If New Jersey converts, then so should Pennsylvania (to avoid dividing the Philly metro area).

And by the time we reach Pennsylvania, it's no longer advantageous to have Year-Round DST due to extremely late sunrise. Sunrise in Philadelphia would be 8:22 AM at the winter solstice under DST (which is only eight minutes later than Boston), but it would be 8:43 in Pittsburgh. It's awkward, especially for young students, to have sunrise so close to 9 AM.

At some point, we must divide either a metro area or a state into different time zones. It's considered more important to keep metro areas intact, so most likely a state division would be required. But unfortunately, it appears that at least two (and possibly all three) of New York, New Jersey, and Pennsylvania would have to be divided. The situation in those states would be similar to the way it used to be in Indiana, with the west side in one time zone (with a biannual clock change) and the east side in another (with no clock change).

On my side of the country, it's logical for the states to coordinate a change as well. Perhaps Nevada should be the first to implement Year Round DST first. Then if the Chu bill passes, California, Nevada, and Arizona will have the same time all year. Again, late sunrise in the far north of California might make Year-Round DST undesirable there.

Last year, I also posted the Danzig plan, which proposes Year-Round Standard Time in some places and Year-Round DST in others, in order to reduce the mainland USA to two time zones, separated with a two-hour difference:

http://www.standardtime.com/

The Chu bill fits the Danzig plan, but the New England proposal does not. The Danzig plan actually recommends Year-Round Standard Time for the East Coast.

Oops -- I spent too much time in this post discussing DST. Let's get back to the U of Chicago text.

Lesson 5-7 of the U of Chicago text is called "Sums of Angle Measures in Polygons." This is one of the few lessons in this part of the book that is the same in both the Second and Third Editions.

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

Lesson 5-7 of the U of Chicago text discusses the sum of the angle measures in polygons, including triangles, quadrilaterals, and higher polygons. To me, this is the most arithmetic- and algebra-intensive lesson in all of the first semester.

The lesson begins with a discussion of Euclidean and non-Euclidean geometry. The 19th-century mathematician Karl Friedrich Gauss wanted to determine whether Euclidean geometry was true -- that is, that it accurately described the measure of the earth -- by experiment. The text shows a photo of three mountaintops that Gauss used as the vertices of a triangle, and the mathematician found that the sum of the angle measures of the triangle was, to within experimental error, 180 degrees.

Later on, the text states that if Gauss could have used a larger triangle -- say with one vertex at the North Pole and two vertices on the equator -- the angle-sum would have been greater than 180. The geometry of a sphere is not Euclidean, but is a special type of non-Euclidean geometry -- often called spherical geometry. As stated in the text:

"In a plane, two perpendiculars to the same line cannot intersect to form a triangle, but this can happen on a sphere. The surface of the earth can be approximated as a sphere. A triangle formed by two longitudes (north-south lines) an the equator is isosceles with two right base angles! Since there is a third angle at the North Pole, the measures add to more than 180 degrees. Thus neither the Two Perpendiculars Theorem nor the Triangle-Sum Theorem works on the surface of the earth."

But hold on a minute. It's obvious that the Triangle-Sum Theorem only holds in Euclidean geometry, as its proof uses the Alternate Interior Angles Consequence that depends on the Fifth Postulate. But we were able to prove the Two Perpendiculars Theorem on this blog, without using any sort of Parallel Postulate at all! So the Two Perpendiculars Theorem ought to hold for all types of geometry, both Euclidean and non-Euclidean -- yet it clearly doesn't hold for spherical geometry.

The truth is that spherical geometry differs from Euclidean geometry much more strongly than hyperbolic geometry differs from Euclidean.  We can obtain hyperbolic geometry from Euclidean simply by dropping the Fifth Postulate and replacing it with an axiom stating that there are many parallels through a point not on the line. But we can't obtain spherical geometry in a similar way.

First of all, what exactly is a line in spherical geometry? (Recall that line is one of the undefined terms, so we can't rely on its definition.) Any figure that we think is a "line" on earth goes all the way around the world, and so is actually a circle. What we want is for a "line" to be the shortest distance between two points. Notice that smaller circles on the globe clearly look curved, but larger circles that go around the world look like straight lines to a traveler. Therefore the most "linear" circle is the largest possible circle -- one that shares a center with the earth. This is called a great circle -- and this is why the example in the text mentions two longitudes and the equator -- these are great circles. But the so-called "parallels of latitude" are not great circles and so are not "lines" (geodesics).

Now what postulates does this spherical geometry violate? Notice that there are no parallel lines on the sphere, because any two great circles intersect. (Once again, note that "parallels of latitude" are not great circles.) Any two longitudes meet at the poles, and so the Unique Line Assumption part of the Point-Line-Plane postulate is violated -- through the poles there are infinitely many lines rather than just one.

But any two great circles that intersect at the North Pole must intersect at the South Pole. And any two great circles that intersect at one point intersect at the point directly opposite that point -- often called the antipodes, or antipodal point. So one way to avoid this problem is to declare that two antipodal points are actually one point. The resulting geometry is called elliptic geometry.

Yet elliptic geometry still violates the postulates. Here I link to David Joyce's website for more discussion of elliptic geometry:

http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI16.html

Notice that this is a link to the first theorem of Euclid that fails in elliptic geometry. It is the Triangle Exterior Angle Inequality Theorem, or TEAI. Dr. Franklin Mason follows Euclid and uses the TEAI to derive the Parallel Tests. The Parallel Tests do not hold in elliptic geometry (of course not, since they prove lines parallel and there are no parallel lines).

In discussing which of Euclid's five postulates that fail in elliptic geometry, the link above writes:

Elliptic geometry satisfies some of the postulates of Euclidean geometry, but not all of them under all interpretations. Usually, I.Post.1, to draw a straight line from any point to any point, is interpreted to include the uniqueness of that line. But in elliptic geometry a completed “straight line” is topologically a circle so that any pair of points on it divide it into two arcs. Therefore, in elliptic geometry exactly two “straight lines” join any two given “points.”
Also, I.Post.2, to produce a finite straight line continuously in a straight line, is sometimes interpreted to include the condition that its ends don’t meet when extended. Under that interpretation, elliptic geometry fails Postulate 2.

Both of these are essentially part of our Point-Line-Plane Postulate. So this is the postulate that we'd have to rewrite if we want elliptic or spherical geometry. Our other postulates still hold -- we can still measure angles, we can still perform reflections, and we still have plane separation (of course, we'd call these halves "hemispheres").

Notice that ironically, our Fifth Postulate still holds in spherical geometry. Of course, it's vacuously  true -- there are no parallel lines, so any statement of the form "if lines are parallel, then..." or something about a line intersecting two parallel lines, is vacuously true. The Parallel Consequences are also vacuously true in spherical geometry. Playfair also holds, provided that we write it the way that Dr. M writes it on his site:

"Through a point not on a given line, there’s at most one line parallel to the given line."
(emphasis mine)

"At most one" allows for the possibility of zero parallel lines. Technically, this is the form of Playfair that we proved earlier this week -- we only showed that at most one parallel line exists. The proof that at least one parallel line exists uses rotations and is not valid in spherical geometry.

Some teachers believe that we should briefly introduce high school students to non-Euclidean geometry -- and usually spherical geometry is suggested as it describes the earth. This is opposite what a college non-Euclidean geometry class would do -- in college, the emphasis is usually on hyperbolic geometry because its theorems are more similar to those of Euclidean geometry.

[2017 update: Actually, some high school teachers do. In her weekly Monday Must Reads series, Sarah Carter links to a teacher, Chris Bolognese, who combines a lesson on spherical geometry with a Halloween activity by using a pumpkin as the sphere. As usual, there's only a Twitter link to the activity, but fortunately the tweet is only a week old, so I do post it below:]

https://twitter.com/EulersNephew/status/925087175527591936

But it's often interesting to discuss with students how spherical geometry affects the earth. A classic brainteaser often goes as follows:

http://www.murderousmaths.co.uk/books/bearpuz.htm

• A bear hunter sets out from camp and walks one mile south.
• He sees a bear and is about to shoot it.
• The bear grabs his gun and eats it.
• The hunter runs away one mile east.
• He then walks one mile north and gets back to his camp and changes his underwear.
• What colour was the bear?

The answer is that the "colour" (sorry -- this is obviously from a British website) of the bear is white, since the puzzle describes a polar bear at the North Pole. Technically, this is not a spherical triangle, since the "one mile east" is along a parallel of latitude, not a great circle. It's not even close to being a great circle -- if the hunter ran approximately six miles east he would have walked in a complete circle around the pole.

Here's another puzzle related to spherical geometry. I've tutored students who've taken a long transoceanic flight, from California to Seoul, South Korea. Along the way, the plane ends up flying very close to Alaska. The question is, why does it fly so close to Alaska, rather than take a more sensible route closer to, say, Hawaii? The answer is that the flight near Alaska is actually shorter -- the flight follows a great circle, and the great circle through California and Korea passes near Alaska.

One final related question -- any two great circles meet at two antipodal points. Where exactly is the point on the globe that is antipodal to where we are standing now? Despite all the talk about "digging a hole to China," that country is not antipodal to the United States. As it turns out, most of the Lower 48 United States are not antipodal to land at all. If one dug a straight hole through the center of the earth starting anywhere in California, we'd end up in the Indian Ocean. But Hawaii is antipodal to parts of Botswana and Namibia in Africa, and of course Alaska is antipodal to Antarctica.

Here is a link to a map that calculates antipodes:

http://www.findlatitudeandlongitude.com/antipode-map/

Returning to Euclidean geometry, here's the proof of the Triangle-Sum Theorem given in the U of Chicago text. Since the book gives a two-column proof, I'll convert it to a paragraph proof:

Triangle-Sum Theorem:
The sum of the measures of the angles of a triangle is 180 degrees.

Given: Triangle ABC
Prove: angle A + angle B + angle C = 180

Proof:
Draw line BD with the measure of angle 1 (ABD) equal to angle A. By the Alternate Interior Angles Test, lines BD and AC are parallel. Then angle 3 (the angle on the other side of BC -- the text doesn't name it, but we can call it CBE if E is a point such that BE and BD are opposite rays) has the same measure as angle C, by the Alternate Interior Angles Consequence. By the Angle Addition Postulate, angles 1, 2 (ABC), and 3 add up to 180 degrees. Substituting, we get that angles A, ABC, and C add up to 180 degrees. QED

Right now, I am a substitute teacher, but last year I interviewed for a position as a regular teacher, and one of the things I was asked to prove was the Triangle-Sum Theorem. (I also had to derive the Quadratic Formula.) I gave a two-column proof similar to the one given in the text, and the principal told me that it was satisfactory, but that he might have preferred something like this:

Statements                                              Reasons
1. Draw line BD parallel to line AC        1. Uniqueness of Parallels (Playfair)
2. angle 1 = angle A, angle 3 = angle C    2. Alternate Interior Angles Consequence
3. angle 2 = angle ABC                            3. Reflexive Property of Equality
4. angle 1 + angle 2 + angle 3 = 180        4. Angle Addition Postulate
5. angle A + angle ABC + angle C = 180  5. Substitution (steps 2 and 3 into step 4)

So we include step 3, to show students that we are making three substitutions. Calling the same angle by two different names -- angle 2 and angle ABC -- emphasizes the need for a Reflexive Property to show that this angle equals itself. The U of Chicago just changes angle 2 to angle ABC without any explanation whatsoever. On the other hand, the U of Chicago distinguishes between the Angle Addition Postulate and the Linear Pair Theorem (which is the just the Angle Addition Postulate in the case that the angles add up to 180). The hope is that this form of the proof is the best for students to understand, which is our goal.

The Quadrilateral- and Polygon-Sum Theorems are just corollaries of the Triangle-Sum Theorem, as we expect. As I mentioned earlier, calculating (n - 2)180 (and dividing by n to find each angle of a regular polygon) is the most complicated algebra that I want students to have to do in first semester of the geometry class.