And yes -- the Dialogue title mentions smoking tobacco. We saw back in Chapter 13 how some people prefer to avoid mentioning smoking on educational websites and texts (which was why the original "Who Owns the Zebra?" puzzle was changed). I'll keep Hofstadter's Dialogue just as the author wrote it in the early 1980's.
Anyway, here's how it begins:
"Achilles has been invited to the Crab's home."
Achilles: I see you have made a few additions since I was last here, Mr. Crab. Your new paintings are equally striking.
Crab: Thank you. I am quite fond of certain painters -- especially Rene Magritte. Most of the paintings I have are by him. He's my favorite artist.
At this point you might be asking, where's the Tortoise? Well, he's not at the Crab's house. Maybe Achilles is still upset at the Tortoise for the reptile's birthday trick.
Of course, Achilles compares the Crab's favorite artist Magritte to his own favorite artist -- Escher:
Achilles: I must say, Magritte's command of realism is astonishing. For instance, I was quite taken in by that painting over there of a tree with a giant pipe behind it.
Crab: You man, a normal pipe with a tiny tree in front of it!
The painting here is Magritte's The Shadows. It turns out that the crustacean is right -- because the pipe is real. The pair now start discussing another famous smoker -- Bach:
Crab: Old Bach was fond of versifying, philosophizing, pipe smoking, and music making (not necessarily in that order.) He combined all four into a droll poem which he set to music. It can be found in the famous musical notebook he kept for his wife, Anna Magdalena, and it is called:
Edifying Thoughts of a Tobacco Smoker
Whene'er I take my pipe and stuff it
And smoke to pass the time away,
My thoughts, as I sit there and puff it,
Dwell on a picture sad and gray:
It teaches me that very like
Am I myself unto my pipe.
... (I don't need to blog the entire poem here, really!)
Thus o'er my pipe, in contemplation
Of such things, I can constantly
Indulge in fruitful meditation,
And so, puffing contentedly,
On land, on sea, at home, abroad,
I smoke my pipe and worship God.
A charming philosophy, is it not?
Achilles: Indeed, Old Bach was a turner of phrases quite pleasin'.
I think it's neat that Bach, who wrote in German, managed to write a rhyming English poem. Anyway the Crab now plays a record on which his own voice rings out:
A turner of phrases quite pleasin'
Had a penchant for trick'ry and teasin'.
In his songs, the last line
Might seem sans design
What I mean is, without why or wherefore.
(That's instead of "without reason," as Achilles points out.) Anyway, the Crab's record reminds Achilles about the Tortoise's unplayable records.
Achilles: But when I last heard about your rivalry, it seem to me you had at last come into possession of an invincible phonograph -- one with a built-in TV camera, minicomputer and so on, which could take itself apart and rebuild itself in such a way that it would not be destroyed.
The Crab explains that his trick still doesn't work -- the Tortoise is able to destroy the subunit which controls the rebuilding process. Now the Crab tells Achilles about a book recommended to him by the Tortoise and written by one of the reptile's friends:
Crab: I wonder which friend it is ... Anyway, in one of the Dialogues, I encountered some Edifying Thoughts on the Tobacco Mosaic Virus, ribosomes, and other strange things I have never heard of.
Achilles: What is the Tobacco Mosaic Virus? What are ribosomes?
At this point, Hofstadter is presenting another paradox, known on the TV Tropes website as the "Celebrity Paradox":
https://tvtropes.org/pmwiki/pmwiki.php/Main/CelebrityParadox
A Celebrity Paradox describes the complications that arise from creating a fictional universe in which that fictional universe does not exist, and the actors playing roles within it do not exist either.
In this case, Achilles and the Crab are actually talking about the book Godel, Escher, Bach -- except that in their universe, the book is called Copper, Silver, Gold. And indeed, they are reading the same Dialogue that we are reading, except that it's about a Tobacco Mosaic Virus instead of a smoker. In fact, ribosomes copy messages just as a phonograph might.
Crab: Metaphorically, I suppose so. Now the thing which caught me eye was a line where this one exceedingly droll character mentions the fact that ribosomes -- as well as Tobacco Mosaic Virtues and certain other bizarre biological structures -- possess "the baffling ability to spontaneously self-assemble." Those were his exact words.
Achilles: That was one of his droller lines, I take it.
Crab: That's just what the other character in the Dialogue thought. But that's a preposterous interpretation of the statement.
The conversation returns to the Tortoise's records -- that even though the records will always break the phonographs, there is a silver lining:
Crab: Oh yes -- the silver lining. Well, eventually, I abandoned my quest after "Perfection" in phonographs, and decided that I might do better to tighten up my defenses against the Tortoise's records. I concluded that a more modest aim than a record player which can play anything is simply a record player that can SURVIVE: one that will avoid getting destroyed -- even if that means that it can only play a few particular records.
The idea is that the phonograph can only play records with the Crab's own label. The label must not be obvious, or else the Tortoise would figure it out and copy it. Thus the label must be integrated somehow with the music itself.
Crab: Precisely. But there is a way to disentangle the two. It requires sucking the data off the record visually, and then --
Achilles: Is that what that bright green flash was for?
Crab: That's right. That was the TV camera scanning the grooves.
Achilles now asks to learn more about the camera part of the phonograph:
Achilles: Terrific! May I try it out?
Crab: Certainly.
Achilles (pointing the camera at the Crab): There YOU are, Mr. Crab, on the screen.
Crab: So I am.
Achilles: Suppose I point the camera at the painting with the burning tuba. Now it is on the screen, too!
The painting, by the way, is Magritte's The Fair Captive.
Achilles: So I see ... But I don't understand what's on the screen now -- not at all! It seems to be a strange corridor. Yet I'm certainly not pointing the camera down any corridor. I'm merely pointing it at an ordinary TV screen.
Crab: Look more carefully, Achilles. Do you really see a corridor?
It turns out to be a set of nested copies of the TV screen itself, getting smaller and smaller. Indeed, Hofstadter includes other images formed when the camera takes a picture of the same screen that is displaying the camera's image. Some of these involve Common Core transformations such as rotation and dilation.
Achilles (turns away from the camera): I'll say! What a wealth of images this simple idea can produce! (He glances back at the screen, and a look of astonishment crosses his face.) Good grief, Mr. Crab! There's a pulsating petal-pattern on the screen! Where do the pulsations come from? The TV is still, and so is the camera.
Crab: You can occasionally set up patterns which change in time. This is because there is a slight delay in the circuitry between the moment the camera "sees" something, and the moment it appears on the screen -- around a hundredth of a second.
The Crab puts his pipe down in front of another famous Magritte painting:
Achilles: I'm still a little woozy. (Points at the Magritte.) That's an interesting painting. I like the way it's framed, especially the shiny inlay inside the wooden frame.
Crab: Thank you. I had it specially done -- it's a good lining.
The painting is The Air and the Song, and it reads "Ceci n'est pas une pipe" -- French for "This is not a pipe," even thought the Crab just put a pipe there.
Crab: Oh, you misunderstood the phrase, I believe. The word "ceci" refers to the painting, not to the pipe. Of course the pipe is a pipe. But a painting is not a pipe.
Achilles: I wonder if that "ceci" inside the painting refers to the WHOLE painting, or just to the pipe inside the painting. Oh, my gracious! That would be ANOTHER self-engulfing. I'm not feeling at all well, Mr. Crab. I think I'm going to be sick ...
Chapter 16 of Douglas Hofstadter's Godel, Escher, Bach is called "Self-Ref and Self-Rep." Here's how it begins:
"In this chapter, we will look at some of the mechanisms which create self-reference in various contexts, and compare them to the mechanisms which allow some kinds of systems to reproduce themselves. Some remarkable and beautiful parallels between these mechanisms will come to light."
Oh, so "Self-Ref and Self-Rep" means "Self-Reference and Self-Replication." I already wrote that I won't be able to describe the entire chapter, so let me just write some of the key ideas.
Let's start with some self-referential sentences:
(1) This sentence contains five words.
(2) This sentence is meaningless because it is self-referential.
(3) This sentence no verb.
(4) This sentence is false. (Epimenides paradox)
(5) The sentence I am now writing is the sentence you are now reading.
Hofstadter moves on to actual programs called quines -- programs that reproduce themselves. He attempts to write quines in his language BlooP, but I've read about quines in real computer languages:
https://cs.lmu.edu/~ray/notes/quineprograms/
He eventually discusses something that reproduces itself in biology -- DNA. He begins with a simulation of DNA reproduction which he calls "Typogenetics," for "Typographical Genetics." He starts with a string, or strand, of bases A, C, G, and T. Then there are enzymes, or rules applied to the strands, to obtain new strands. The author tells us that this is similar to his MIU-system.
Before we leave Typogenetics, I found the following link, which not only provides more examples of quines but also purports to solve many of Hofstadter's Typogenetics problems as well:
https://www.bamsoftware.com/hacks/geb/index.html
Afterward the author moves on to real DNA and RNA. He mentions a couple of analogies -- one between molecular biology and mathematical logic, and the other between molecular bio and the Contracrostipunctus Dialogue (broken records and whatnot).
EDIT: I did finally return to this Chapter in my June 14th post. Here's a link to it:
https://commoncoregeometry.blogspot.com/2019/06/flag-day-post-self-ref-and-self-rep.html
Lesson 15-7 of the U of Chicago text is called "Lengths of Chords, Secants, and Tangents." In the modern Third Edition of the text, lengths of chords, secants, and tangents appear in Lesson 14-7. As for the new Lesson 14-6, this is a lesson that doesn't appear in the old text. It's all about three circles associated with a triangle -- the circumcircle, incircle, and nine-point circle. The first two of these circles are directly related to the concurrency theorems that appear in the Common Core Standards, so it's good that the authors include this lesson in the third edition.
This is what I wrote last year about today's lesson:
Here are the two big theorems of this lesson:
Secant Length Theorem:
Suppose one secant intersects a circle at A and B, and a second secant intersects the circle at C and D. If the secants intersect at P, then AP * BP = CP * DP.
Given: Circle O, secants AB and CD intersect at P.
Prove: AP * BP = CP * DP.
There are two figures, depending on whether P is inside or outside the circle, but proofs are the same.
Proof:
Statements Reasons
1. Draw
2. Angle BAD = BCD, 2. In a circle, inscribed angles intercepting
Angle ADC = ABC the same arc are congruent.
3. Triangle DPA ~ BPC 3. AA~ Theorem (steps 2 and 3)
4. AP / CP = DP / BP 4. Corresponding sides of similar
figures are proportional.
5. AP * BP = CP * DP 5. Means-Extremes Property
This leads, of course, to the definition of power of a point.
Tangent Square Theorem:
The power of point P for Circle O is the square of the length of a segment tangent to Circle O from P.
Given: Point P outside Circle O and Line PX tangent to Circle O at T.
Prove: The power of point P for Circle O is PT^2.
Proof:
Statements Reasons
1. Draw Ray TO which 1. Two points determine a line.
intersects Circle O at B.
2. Let
Circle O at A and B. two points.
3.
and TAB in semicircle
4. PTB right triangle 4. Definitions of right angle, right triangle,
with altitude
5. PT^2 = PA * PB 5. Right Triangle Altitude Theorem
6. The power of point P 6. Definition of power of a point
for Circle O is PT^2
By the way, we can now finally prove the Bonus Question from Lesson 15-4. I think I'll dispense with two-column proofs here and give a paragraph proof. We begin with a lemma:
Lemma:
Suppose two circles intersect in two points. Then for each point on their common secant line, the power of that point for first circle equals the power of that point for the second circle.
Given: Circles A and B intersect at C and D, Point P on secant CD
Prove: The power of point P for Circle A equals the power of point P for Circle B
Proof:
For both circles, the power of P is CP * DP, no matter whether P is inside or outside the circle. This common secant has a name -- the radical (or power) axis of the two circles. QED
Theorem:
Suppose each of three circles, with noncollinear centers, overlaps the other two. Then the three chords common to each pair of circles are concurrent.
Proof:
The proof of this is similar to that of the concurrency of perpendicular bisectors of a triangle (which I'll compare to this proof in parentheses). Let A, B, and C be the three circle centers. Every point on the radical axis of A and B has the same power for both circles. (Compare how every point on the perpendicular bisector of
Notice that if three centers are collinear, then the three radical axes are parallel (just as if three points are collinear, then the perpendicular bisectors are parallel).
I decided to make this the activity day for this week, since the Exploration section includes two questions rather than one. This is more like a puzzle, since the key to both questions isn't Geometry but arithmetic (or Algebra) and the properties of multiplication!
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