Dialogue 13 of Douglas Hofstadter's Godel, Escher, Bach is called "Aria with Diverse Variations." It begins as follows:
"Achilles has been unable to sleep these past few nights. His friend the Tortoise has come over tonight, to keep him company during these annoying hours."
Tortoise: I am so sorry to hear of the troubles that have been plaguing you, my dear Achilles. I hope my company will provide a welcome relief from all the unbearable stimulation which has kept you awake. Perhaps I will bore you sufficiently that you will at long last go to sleep. In that way, I will be of some service.
So the Tortoise tells Achilles the story of a famous Count:
Tortoise: Now, now, I wouldn't put it exactly that way, but ... Those days, Counts could get away with most anything. Anyway, it is clear that the Count was most pleased with the music, for he was constantly entreating his harpsichordist -- a mere lad of a fellow, name of Goldberg -- to play one or another of these thirty variations. Consequently (and somewhat ironically) the variations become attached to the name of young Goldberg, rather than to the distinguished Count's name.
The athlete asks more about these "Goldberg Variations," written by -- who else? -- Bach:
Achilles: That is a peculiar idea. Presumably, everybody thinks that the latest discovery was just a fluke, and that we now really do have all the Goldberg Variations. But just supposing that you are right, and some more turn up sometime, we shall start to expect this kind of thing. At that point, the name "Goldberg Variations" will start to shift slightly in meaning, to include not only the known ones, but also any others which might eventually turn up. Their number -- call it 'g' -- is certain to be finite, wouldn't you agree? -- but merely knowing that g is finite isn't the same as knowing how big g is. Consequently, this information won't tell us when the last Goldberg Variation has been located.
Bach wrote this Variations in 1742 -- which the Tortoise notices is the sum of two primes:
Tortoise: By thunder! What a curious fact! I wonder how often one runs across an even number with that property.
Achilles: But please don't let me distract you from your story.
In other words, the discussion has shifted from Goldberg to Goldbach and his famous conjecture:
Tortoise: Then in 1937, a sly fellow named Vinogradov -- a Russian too -- managed to establish something far closer to the desired result, namely, every sufficiently large ODD number can be represented as a sum of no more than THREE odd primes. For example, 1937 = 641 + 643 + 653. We could say that an odd number which is representable as a sum of three odd primes has "the Vinogradov property." Thus, all sufficiently large odd numbers have the Vinogradov property.
Achilles: Very well -- but what does "sufficiently large" mean?
The Tortoise explains it means that all odd numbers that are greater than v for some v. As it turns out, after this book was published, Harald Helfgott proved that v = 7 -- that is, all odd numbers greater than 7 have the Vinogradov property. (I've mentioned this result before on the blog.)
Tortoise: Indeed. But you know, there is a rather striking difference between the Goldbach Conjecture, and this Goldbach Variation, which I would like to tell you about. Let us say that an even number 2N has the "Goldbach property" if it is the SUM of two odd primes, and it has the "Tortoise property" if it is the DIFFERENCE of two odd primes.
Achilles: I think you should call it the "Achilles property." After all, I suggested the problem.
The reptile says that an even number that LACKS the Tortoise property has the Achilles property, but that no finite search can prove that a number has the Achilles property:
Achilles: I see what you mean. The Goldbach property is a detectable, or recognizable property of any even number, since I know how to test for its presence -- just embark on a search. It will automatically come to an end with a yes or now answer. The Tortoise property, however, is more elusive, since a brute force search just may never give an answer.
The pair now enter a discuss how infinity relates to this problem:
Achilles: Strictly speaking, I hope so. But you and I both know that you can't produce 29 by multiplying two numbers which are both bigger than 29. So in reality, saying "29 is prime" is only summarizing a FINITE number of facts about multiplication.
Tortoise: You can put it that way if you want, but think of this: the fact that two numbers which are bigger than 29 can't have a product equal to 29 involves the entire structure of the number system. In that sense, that fact in itself is a summary of an infinite number of facts. You can't get away from the fact, Achilles, that when you say "29 is prime," you are actually stating an infinite number of things.
The discussion now switches to order and chaos:
Tortoise: Perhaps they do; but have you ever considered that such chaos might be an integral part of the beauty and harmony?
Achilles: Chaos, part of perfection? Order and chaos make a pleasing unity? Heresy!
The Tortoise tells shows his friend the Escher picture Order and Chaos.
Achilles: Certainly.
Tortoise: The first type of search -- the non-chaotic type -- is exemplified by the test involved in checking for the Goldbach property. You just look at primes less than 2N, and if some pair adds up to 2N, then 2N has the Goldbach property; otherwise, it doesn't. This kind of test is not only sure to terminate, but you can predict BY WHEN it will terminate, as well.
Now the Tortoise gives an example of a test that is nonpredictably terminating:
Tortoise: An excellent choice. We begin with your number, and if it is ODD, we triple it and add 1. If it is EVEN, we take half of it. Then we repeat the process. Call a number which eventually reaches 1 this way a WONDROUS number, and a number which doesn't, an UNWONDROUS number.
Actually, this already has a name -- the Collatz conjecture -- but Hofstadter never mentions the name "Collatz" in this dialogue. Here are some examples:
3-10-5-16-8-4-2-1
7-22-11-34-17-52-26-13-40-20-10-5-16-8-4-2-1
15-46-23-70-35-106-53-160-80-40-20-10-5-16-8-4-2-1
Thus 3, 7, and 15 are all wondrous. So far, it's known that every number up to 10^20 is wondrous.
Tortoise: I understand what you mean, but it's not that different from saying "29 is prime," or "gold is valuable" -- both statements attribute to a single entity a property which is has only by virtue of being embedded in a particular context.
Now our pair writes about making a story more suspenseful by adding padding so that one can't tell how many pages are left:
Achilles: If this were standard practice, it might be effective. But there is a problem. Suppose your padding were very obvious -- such as a lot of blanks, or pages covered with X's or random letters. Then it would be as good as absent.
Just in case you've forgotten that Achilles is an insomniac:
Achilles: I am most flattered that you have stayed up for so long, and at such an odd hour of the night, just for my benefit. I assure you that your number-theoretic entertainment has been a perfect antidote to my usual tossing and turning. And who knows -- perhaps I may even be able to go to sleep tonight. As a token of my gratitude, Mr. T, I would like to present you with a special gift.
The special gift is an Asian box -- but watch out, Tortoise, it's a trap!
Achilles: I haven't the foggiest idea. It seems suspicious to me. Why don't you go hide behind the dresser, in case there's any funny business.
Tortoise: Good idea. (Scrambles in behind the dresser.)
Achilles: Who's there?
Voice: Open up -- it's the cops.
Achilles: Come in, it's open.
It turns out that the gold box -- the "gift" the Tortoise receives -- is stolen property. Achilles has just set his friend up!
Cop: So there it is! And so Mr. Tortoise is the varmint, eh? I never would have suspected HIM. But he's caught, red-handed.
Achilles: Haul the villain away, kind sirs. Thank goodness, that's the last I'll have to hear of him, and the Very Asian Gold Box!
Chapter 13 of Douglas Hofstadter's Godel, Escher, Bach is called "BlooP and FlooP and GlooP." It begins as follows:
"BlooP, FlooP, and GlooP are not trolls, talking ducks, or the sounds made by a sinking ship -- they are three computer languages, each one with its own special purpose. These languages were invented specially for this Chapter."
This chapters sounds as if it would be fun to write about -- but unfortunately, Hofstadter's Law strikes again, and I don't have much time to write about it!
I will write Hofstadter's BlooP example:
DEFINE PROCEDURE "TWO-TO-THE-THREE-TO-THE" [N]:
BLOCK 0: BEGIN
CELL(0) <= 1;
LOOP N TIMES:
BLOCK 1: BEGIN
CELL(0) <= 3 X CELL(0);
BLOCK 1: END;
CELL(1) <= 1;
LOOP CELL(0) TIMES:
BLOCK 2: BEGIN
CELL(1) <= 2 X CELL(1);
BLOCK 2: END;
OUTPUT <= CELL(1);
BLOCK 0: END.
FlooP and GlooP are supposed to be more powerful versions of BlooP. Actually, FlooP is more powerful than BlooP, but GlooP is no more powerful than FlooP. The reasons are quite complex -- the Chapter discusses both Cantor's Diagonal and Turing's Halting Problem. We skip to the final line, which is about how all of these relates to the power of number theory, TNT:
"If TNT could not represent primitive of general recursive predicates, then it would be incomplete in an uninteresting way -- so we might as well assume it can, and then show that it is incomplete in an interesting way."
This is what I wrote last year about today's lesson:
Lesson 15-4 of the U of Chicago text is "Locating the Center of a Circle." According to the text, if we are given a circle, there are two ways to locate its center. The first is the perpendicular bisector method, which first appears in Lesson 3-6. (Recall that the perpendicular bisectors of a triangle are a concurrency required by Common Core.) This section gives the right angle method:
1. Draw a right angle at P (a point on the circle).
2. Draw a right angle at Q (another point on the circle).
3. The diameters
This method is based on the fact that a chord subtending a right angle is a diameter -- a fact learned in the previous lesson. Indeed, "an angle inscribed in a semicircle is a right angle" is a corollary of the Inscribed Angle Theorem.
Notice that unlike the perpendicular bisector method, this is not a classical construction. That's because the easiest way to construct a right angle is to construct -- a perpendicular bisector, which means that if we have a straightedge and compass, we might as well use the first method. The text writes that drafters might use a T-square or ell to produce the right angles, while students can just use the corner of a sheet of paper.
Today is an activity day. I've decided to expand the Exploration Question from this lesson into a full activity page. (It's one of the few second semester activities that falls on Friday both last year and this year.)
"Each of the three circles below intersects the other two. The three chords common to each pair of circles are drawn. They seem to have a point in common. Experiment to decide whether this is always true."
As it turns out, these three chords are indeed concurrent, except for a few degenerate cases such as if the circles have the same center or if the centers are collinear. (The concurrency of perpendicular bisectors has the same exceptions.) The students are asked to experiment rather than attempt to prove the theorem that these three lines (called radical lines) intersect at a common point (radical center, or power center). The name "power center" refers to "power of a point" -- a dead giveaway that we must wait until Lesson 15-7 before we can attempt to prove the theorem.
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