Kung begins by briefly mentioning a good "math fiction" tale -- Flatland, by the 19th century British teacher Edwin Abbott. In this lesson, the main character is called A. Square -- exactly because he is, in fact, a square -- who lives in a two-dimensional world. Kung points out that topologists often use Flatland because it's easy to visualize 2D examples before considering their 3D analogs.
Indeed, this is exactly what Kung does in much of the lecture. He considers a Mobius strip and points out how if A. Square lived on such a strip, he can travel all the way around the strip and see that all that was on his left is now on his right. This is why we say that a Mobius strip is non-orientable.
But after this, he moves on to examples of 3D non-orientable spaces. On the screen, Kung is standing inside a strange room. He walks out the right side of the door wearing a watch on his left arm, only to arrive on the left side of the door wearing a watch on his right arm. If he now takes a strip of paper and extend it from one door to the other, it becomes a sort of two-sided Mobius strip -- it's still not orientable since anyone walking to the right will end up on the left with directions reversed, yet it has two sides since nothing on the top of the strip ever ends up on the bottom of the strip.
In the Quick Conundrum, Kung has a student reach out his arms and interlock them (to form a torus), then Kung turns his sweater inside out without the student letting go of his arms.
What I found the most interesting about this lecture is a discussion of the Euler characteristic. A theorem states for any polyhedron, the vertices V, edges E. and faces F satisfy V - E + F = 2. This theorem doesn't appear in the U of Chicago text -- if it did, a logical place for it to appear would be Lesson 9-7 on polyhedra (just before Lesson 9-8 on the Four Color Theorem).
Back in July, I wrote about Euler's formula on the blog. That's because we were discussing spherical geometry, and I brought up how in another Geometry text, an old HRW text I own, topics like topology and spherical geometry appear in Chapter 11, in Lessons 11.4 and 11.5 respectively. Now Kung points out that Euler's formula can be proved using both topology and spherical geometry.
At this point you may ask, what does a polyhedron like a cube have to do with spherical geometry? As it turns out, that's easy to answer -- a cube is topologically equivalent to a sphere! Lesson 11.4 of the HRW text hints at the proof for a tetrahedron -- we topologically morph the tetrahedron into a sphere, and the four triangular faces now become spherical triangles. Kung generalizes this proof by taking any polyhedron, all of whose faces are triangles, and morph it into a sphere, where the triangular faces are now spherical triangles.
I won't post the full proof here, but both this proof and another proof that I alluded to last week (that if every Lambert quadrilateral is a rectangle, then Playfair holds) have something in common -- the relationship between the sum of the angles of a triangle and its area. I alluded to this fact in my last spherical geometry post in August -- the sum of the angles of a triangle is equal to 180 plus some constant times the area of the triangle (but we didn't get to the proof). If the units are chosen correctly (radian measure for the angles and 1 for the radius of the sphere), then the constant is 1. Then Kung uses this fact plus the known surface area of the unit sphere (4pi) to derive Euler's formula.
I've been thinking back to Kung's last lecture -- the one where he mentions the Four Color Theorem, and how it was first proved by a computer. Kung points out how this raises the issue of what exactly it means to "prove" or "discover" something in mathematics.
I bring this up because, as February begins, I reflect back to the month of January. The first month of 2016 was remarkable in the world of science, because scientists in three different fields made major discoveries during the month:
-- In early January, chemists discovered the 113th, 115th, 117th, and 118th _______________.
-- In late January, astronomers (may have) discovered the 9th _______________.
-- And in between, mathematicians discovered the 49th _______________.
Can you fill in the blanks?
The last discovery is probably the most well-known. Caltech astronomers may have discovered the ninth planet in our solar system (no, not Pluto):
The first discovery is that chemists around the world (including those in California, Japan, and Russia) have discovered four new elements on the periodic table:
But what did mathematicians discover the 49th of in mid-January? Well, they discovered the 49th Mersenne prime -- the largest known prime number to date:
As we know, Euclid proved that there exist infinitely many primes, so there is no largest prime. So what was discovered is the largest known prime.
Here's a thought experiment -- what's the largest prime you can think of? (As teachers, it may be interesting to ask this question in class -- especially an upper elementary or middle school class just learning about prime numbers.) I'm not sure what most students would answer, but maybe they might come up with 97, which has just two digits.
Even if you haven't clicked on the link above, you may have noticed the number 74207281 mentioned in the URL and might suspect that this is the prime. This number has eight digits, and just imagine trying to divide this number by 2, 3, 4, 5, and so on so see whether it's prime. (This, by the way, is known as trial division.) Well, yes, 74207281 is prime, but it's not the largest known prime -- indeed, it's not even close.
Some readers may be familiar with RSA encryption in computers. This involves primes that contain hundreds of digits -- so they're much larger than googol. But no, none of these are the largest known prime that was discovered last month.
So, what is this huge prime, then? Well, it's not 74207281, but 2 ^ 74207281 - 1 -- that is, it's one less than the 74207281st power of two. You see, that's what a "Mersenne prime" is -- a prime one less than a power of two. They are named for a 17th century French mathematician, Marin Mersenne.
The Mersenne prime 2 ^ 74207281 - 1 is gigantic -- while the RSA primes have mere hundreds of digits, this Mersenne prime has over 22 million digits! It seems as if it would take forever just to divide it by three, much less all the numbers with 2, 3, 4, 5, ... digits -- and yet we know for a fact that none of these smaller numbers will divide this number evenly! How can this be?
One of my favorite YouTube channels is Numberphile -- and yes, I've posted several links to Numberphile earlier on the blog. Well, here is a series of Numberphile videos about the new prime (including how it was discovered):
Notice that Numberphile printed out the number in full. He needed three full notebooks -- with each notebook approximately a ream (500 pages) in size, printed front and back, with the digits in small print (possibly 10 pt or even 8 pt -- I couldn't tell for sure).
Three years ago, when the previous (48th) Mersenne prime was discovered, I told one of the students I was tutoring about the huge prime, and it astounded him. I didn't print out the prime in full like Numberphile, of course, but I showed him the first and last page, including the page number count (which I believe was something over 1000, if not 2000). I'd found the digits at this site:
This site belongs to Landon Noll, the discover of some of the previous (much smaller) primes. Here are two links referring to the newest Mersenne prime:
The other link writes out the number in words. According to the link, the new prime is approximately:
three hundred septenmilliamilliaquadringensexquadraginmilliaduocenquattuortillion
Notice that Mersenne primes -- numbers of the form 2 ^ p - 1 -- has this form for a reason that goes back to Algebra II factoring. We can prove that numbers of the form 3 ^ p - 1, 4 ^ p - 1, 6 ^ p - 1, and so on, can all be factored using a generalization of the difference of squares/cubes formula. And for the reason, the p in 2 ^ p - 1 must be prime, otherwise it can also be factored algebraically.
By now you're wondering, what does any of this have to do with the Four Color Theorem? Well, like the proof of the 4CT, these Mersenne primes were discovered on computer. A professor named Dr. Curtis Cooper proved it using dozens of computers at the University of Central Missouri -- and he had so many computers running at the same time that one of them had calculated the new prime in September, yet Cooper didn't notice it for months! So this is another example where we wonder, what does it mean for something to be discovered?
The official answer is that a Mersenne prime has been discovered as soon as some human is aware of the exponent. So even though Cooper's computer calculated the prime in September, the official discovery date is not until January.
Another issue that comes up with Mersenne prime discovery is the order. Sometimes larger primes are discovered before smaller primes. In some ways, this is a bit like chemistry, where the 112th and 114th elements were discovered before the 111th and 113th elements. Then again, we knew exactly which element is the 114th simply by counting its protons -- we can't tell as easily whether the new prime is actually the 49th Mersenne.
In that sense, this is more like planet discovery. The newly discovered Planet Nine is so far away that it's possible there could be yet another undiscovered planet closer to the sun, in which case "Planet Nine" would actually be Planet Ten. As of now, we know that there are no undiscovered primes less than the 44th, so the names "45th" through "49th" Mersenne primes are actually misnomers until all smaller exponents have been checked.
Meanwhile, here's the link if you want to volunteer your computer for the Great Internet Mersenne Prime Search:
Lesson 12-5 of the U of Chicago text is about similar figures. There is not much for us to change about this lesson from last year, except for the definition of similar itself. Recall the two definitions:
-- Two polygons are similar if corresponding angles are congruent and sides are proportional.
-- Two figures are similar if there exists a similarity transformation mapping one to the other.
The first definition is pre-Core, while the second is Common Core. The U of Chicago text, of course, uses the second definition. But that PARCC question I mentioned last week must be using the first definition, since it requires that we know what similar means before we can define dilations and ultimately similarity transformations.
In the U of Chicago text, the Similar Figures Theorem is essentially the statement that the second definition implies the first definition. We would actually need to prove the converse -- that the first definition (at least for polygons) implies the second. But the proof isn't that much different -- suppose we have two figures F and G satisfying the first definition of similar -- that is, corresponding angles are congruent and sides are proportional, say with scale factor k. Then use any dilation with scale factor k to map F to its image F'. Now F' and G have all corresponding parts congruent, so there must exist some isometry mapping F' to G. Therefore the composite of a dilation and an isometry -- that is, a similarity transformation -- maps F to G. QED
So I don't change the worksheet from last year. Read this worksheet and get ready for Kung's next lecture, as the geometrical paradoxes continue.