Several years ago, I wrote a bit about fractals. I am revisiting that post here.
I am so in favour of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author. Georg Cantor, 1845-1918
Fractals are awesome (too bad that adjective is so blunted by overuse and misuse). Sure, they are beautiful and, of course, the fact that math is involved is intriguing to me. But the real reason that I like fractals so much is because I can think of no better way to describe how I think about the universe and eternity. And as I was taking some time to just “contemplate,” I came to the conclusion that a fractal also best describes my understanding of scripture.
Most people have seen the lovely images of fractals, and many may even have some concept of the mathematical equations that generate the images. But some of you may still wonder, “What is a fractal, really?” OK, Fractals 101.
a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,” a property called self-similarity. Roots of mathematical interest on fractals can be traced back to the late 19th Century; however, the term “fractal” was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning “broken” or “fractured.” A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.”
[Note: If you have been reading along with my Gödel, Escher, Bach book discussion group (see here), this will really start to set off fireworks in your brain. Any time you see the word “recursion” again, you will definitely think of Hofstadter’s explanations.]
So what do fractals have to do with anything? For one thing, they are an easily accessible visual aid to the complex idea of infinity. One of the most amazing things to teach, I think, is astronomy. When you contemplate what you are really seeing when you look at the night sky it takes your breath away. With the naked eye you see the moon, constellations, planets. Zoom out… with binoculars or a telescope you can see more. And when you look at images from the Hubble telescope you are looking at stars and galaxies that are huge, old, distant, beautiful, and oh-so-numerous. A field of what looks like stars is really a field of entire galaxies. It kind of reminds me of the end of the movie Men in Black when the camera pans out from earth, to the solar system, to the galaxy, to a marble that contains the galaxy and is in the hands of some alien who is playing a game with it. I’m not interested in the alien aspect… I’ve just always been fascinated by the degrees of immensity.
Now go the other way… zoom in to the earth, to the rocks, the minerals, the elements, the molecules, the atoms, the sub-atomic particles. But it is so hard to see the “big” picture of the universe and the “small” picture of the details all at the same time… except in a fractal! Looking at that one image I can keep going deeper and deeper and pondering the immensity of the created universe.
[by the way, if you want to see the version of this video that explains the processes being animated, go here]
But just as amazing as that (or maybe even more so), is being able to look at scripture in the same way. For so long I looked at Scripture far too linearly. I saw the history and the stories and prophesies. I memorized the Gospel accounts and looked for the symbols and types in the Old Testament and fulfillment in the New. But now I see that there is so much more. Just as a fractal is one whole that beautifully communicates a single equation, so Scripture is one communication–a breaking into creation by God–with both a singleness and multiplicity. One story, unfolded in many iterations. Acted out over centuries, through many generations, cultures, languages, and individuals. Story upon story. But all part of the same one communication. God is breaking through to us to tell us about Himself. Using our own human context to speak to us, so that we can understand. It’s more beautiful and more amazing than a hundred thousand galaxies or the patterns repeated over and over in nature or the dancing movement of electrons and quarks and leptons.
Sometimes the beauty and majesty are hard to see in the black and white text, and so I keep a mental bookmark of a fractal image in my Bible to remind me of just how glorious and precious the book in my hands is.
A day late. Sorry folks. But I think I may be talking to myself by now. So, since we are at #7, I think I’ll let that be the perfect ending of our little discussion group (for now). I know this was a little ambitious to take on (schedule-wise), so I will re-think another book or topic to do next time.
Current Assignment: For Thursday, October 1 Read: Canon by Intervallic Augmentation and Chapter VI: The Location of Meaning Listen: Bach never multiplied the intervals of a theme by 3 1/3. He did multiply them by -1 in this canon by exact inversion, the Canon Perpetuus from the Musical Offering. An effect of the exact inversion is that the piece has to oscillate constantly between major and minor chords, and technically it can’t end.
The Dialogue: Canon by Intervallic Augmentation
This Dialogue between Achilles and the Tortoise tries to resolve the question, ‘Which contains more information–a record, or the phonograph which plays it?”
(By the way, how many haikus can you find in this dialogue?)
Chapter VI: The Location of Meaning
This chapter discusses how meaning is divided among coded message, decoder, and receiver. Hofstadter gives examples of strands of DNA, ancient tablets containing undeciphered inscriptions, and some unusual phonographs. Continue reading →
What’s up with all these GEB posts? If you missed what we’re doing, read here. Remember, I’m no expert on all this, I’m just helping facilitate. I’m trying to read along with the rest of you!
Current Assignment: For Monday, September 28 Read: Little Harmonic Labyrinth and Chapter V: Recursive Structures and Processes Listen: The Little Harmonic Labyrinth by Johann David Heinichen. Waltz #2 by Billy Joel.
I think the Fall semester has hit most of us and our schedules are slipping away from our control (did we ever really have control??). But, in case someone is still interested in this, I’ll continue to post for a bit more.
When Justin Curry (MIT) taught this course, he quoted the following at this point in the book: This is your last chance. After this, there is no turning back. You take the blue pill -the story ends, you wake up in your bed and believe whatever you want to believe. You take the red pill -you stay in Wonderland and I show you how deep the rabbit-hole goes. Morpheus Continue reading →
I’ve been a bit under the weather the past few days, but I don’t want to get behind on this project. So… onward!
Current Assignment: Thursday, September 24 (where is the month of September disappearing to!!??) Read: Contracrostipunctus and Chapter IV: Consistency, Completeness, and Geometry Listen: Contrapunctus 19 from the Art of Fugue (BWV 1050). This performance abruptly ends in the same place that the score ended due to Bachs death. Bach left his name in the music, as the German notes B-A-C-H, a few measures before the end.
Summary of Contrastipunctus
This dialogue is central to the book because it contains a set of paraphrases of Gödels self-referential construction and of his Incompleteness Theorem. One of the paraphrases of the Theorem says, For each record player there is a record which cannot play. The Dialogues title is a cross between the word acrostic and the word contrapunctus, a Latin word which Bach used to denote the many fugues and canons making up his Art of the Fugue. Some explicit references to the Art of the Fugue are made. The Dialogue itself conceals some acrostic tricks.
(Brief) Summary of Chapter IV: Consistency, Completeness, and Geometry
The preceding Dialogue is explicated to the extent it is possible at this stage. This leads back to the question of how and when symbols in a formal system acquire meaning. The history of Euclidean and non-Euclidean geometry is given, as an illustration of the elusive notion of undefined terms. This leads to ideas about the consistency of different and possibly rival geometries. Through this discussion the notion of undefined terms is clarified, and the relation of undefined terms to perception and thought processes is considered.
(for the Dialogue) GEB pp. 81 For instance, Lewis Carroll often hid words and names in the first letters (or characters) of the successive lines in poems he wrote. Poems which conceal messages that way are called acrostics. Might this quote apply to this dialogue?
Why does DRH keep apologizing about his use of the term isomorphism?
Whats the problem with interpreting mathematical objects? In the case of the modified pq-system? In the case of Euclidean geometry? Whats wrong with our interpretation of a straight line?
(that’s enough to get you started)
Click on the photo to view how this image was created.
Up Next: For Monday, September 28
Read: Little Harmonic Labyrinth and Chapter V: Recursive Structures and Processes
Listen: The Little Harmonic Labyrinth turns out not to be by Bach at all! It was written instead by his much lesser-known contemporary, Johann David Heinichen. Disappointingly, it doesnt even have a fake resolution near the end, as the dialogue implies. Also, its boring. A completely unrelated piece, however, does have a clear pushing and popping structure to it, and a fake resolution: Waltz #2 by Billy Joel. Yes, that Billy Joel, retired from pop and writing classical music. Allow Achilles and the Tortoise one more anachronism and pretend this is what theyre listening to.
We spent the weekend at Bald Head Island with family… what a great way to say goodbye to summer. In the BHI Conservancy shop we found a tempting book/kit: M. C. Escher Kaleidocycles: An Illustrated Book and 17 Fun-to-Assemble Three-Dimensional Models.
According to the publisher:
A Kaliedocycle is a three-dimensional ring made from a chain of solid figures enclosed or bonded [sic] by four triangles. These kaleidocycles are adaptations of Escher’s two-dimensional images of fish, angels, flowers, people, etc., transformed into uniform, interlocking, three-dimensional objects whose patters [sic] wrap endlessly. Kaleidocycles contains a 48-page book with over 80 reproductions and diagrams, assembly instructions, and a fascinating discussion of the geometric principles and artistic challenges underlying Escher’s designs and their transformation to three-dimensional models; and seventeen die-cut, scored, three-dimensional models (11 kaleidocycles and 6 geometric solids).
I was so tempted to buy the set, but I resisted. You might enjoy finding some similar projects online for free (let me know if you find any). Anyway, on to the assignment at hand.
Current Assignment: Monday, September 21 Read: Sonata for Unaccompanied Achilles and Chapter III: Figure and Ground Listen: Sonata No. 1 for solo violin: Adagio (BWV 1001). If an accompanied version of this exists somewhere, I dont know where to find it.
I hope the MU puzzle didn’t discourage too many of you from continuing to read this book (see here for explanation of what we are doing and here for a schedule). I’m going to assume it was a busy weekend and people just didn’t get around to posting anything about the last section. That’s ok. Let’s continue on.
Current Assignment: Thursday, September 17 Read: Two-Part Invention and Chapter II: Meaning and Form in Mathematics Listen: Two-Part Invention in C major (BWV 772)
Summary of Chapter II:
A new formal system (the pq-system) is presented, even simpler than the MIU-system of Chapter I. Apparently meaningless at first, its symbols are suddenly revealed to possess meaning by virtue of the form of the theorems they appear in. This revelation is the first important insight into meaning: its deep connection to isomorphism. Various issues related to meaning are then discussed, such as truth, proof, symbol manipulation, and the elusive concept, form.
Discussion Questions: What is the difference between meaning in a formal system and meaning in a human language?
What is the difference between active and passive meaning?
For those of you silently reading/skimming along, don’t get discouraged about all this mathematics stuff. We’re just laying a foundation for the meatier discussions to come. I’m hoping you will start to see connections to other areas of study and life.
Up Next: For Monday, September 21
Read: Sonata for Unaccompanied Achilles and Chapter III: Figure and Ground
Listen: Sonata No. 1 for solo violin: Adagio (BWV 1001). Anyone know where to find an accompanied version of this?
Well, we’re off and running (pun intended). I hope you are beginning to see that this book is about much more than the intersection of mathematics, art, and music.
Current Assignment: Monday, September 14 Read: Three-Part Invention and Chapter I: The MU-puzzle Listen: The Three-Part Ricercar, from the Musical Offering (BWV 1079), introduces the Kings theme (which appears in nearly every piece of the Musical Offering) and the fugue style in general.
What story is recreated in this dialogue?
In what ways is this dialogue self-referential?
Is there any significance in positioning the Tortoise upwind of Achilles?
What are some more differences between people and machines? Hofstadter talks a lot about observing patterns, but who is doing the observing and from where?
Do you think that being able to jump out of a task and look for patterns is an inherent property of intelligence? What do you think of the following?
Of course, there are cases where only a rare individual will have the vision to perceive a system which governs many people lives, a system which had never before even been recognized as a system; then such people often devote their lives to convincing other people that the system really is there, and that it ought to be exited from! (pp. 37)
What, or who, does this make you think of?
Hofstadter calls the U-Mode a Zen way of approaching things. (pp. 39) What does this mean?
Is the notion of truth different for a theorem than an axiom?
What other rabbit trails we can pursue?
Up Next: For Thursday, September 17
Read: Two-Part Invention and Chapter II: Meaning and Form in Mathematics
Listen: Two-Part Invention in C major (BWV 772)
Watch this video of the enigmatic Canon 1 à 2 from J. S. Bachs Musical Offering (1747). The manuscript depicts a single musical sequence that is to be played front to back and back to front. A nice tie-in to our reading of Gödel, Escher, Bach. We’ll be reading about and listening to Crab Canon at the beginning of October.
HT: Patrick Wynne
On Thursday of this week we will begin our group discussion of Hofstadter’s book, Gödel, Escher, Bach (see here for more details). I’m posting this a little early since it is the first assignment/discussion. The plan is to post on Thursdays and Mondays so that participants can start commenting (this means you will have to do the reading BEFOREHAND, so check and follow the schedule!). We’re still working out the best day/format for a live group chat. Stay tuned.
You still have time to order the book from Amazon.com (or pick one up at a used book store, mine was 25¢). This is a no guilt discussion group. If you miss a week, try to jump back in. You can also read/listen ahead by looking at the schedule posted here.
Douglas Hofstadter wrote a very helpful preface to the 20th anniversary edition of the book. The actual content of the book remains unchanged. However, if you read the preface, you will get a good idea of what Hofstadter was hoping readers would “get” in his book. He wrote it because so many people over the years have not understood what he was really trying to say. I guess you could say there was a real chasm between reader response and authorial intention! You can read a fair bit of it by creatively using the Amazon “Look Inside” feature. Actually, you can read all of it if you try hard enough (first puzzle of the course to solve).
Feel free to start posting your thoughts and/or questions on the Introduction!
Up next (For September 14):
Read: Three-Part Invention and Chapter I: The MU-puzzle
Listen: The Three-Part Ricercar, from the Musical Offering (BWV 1079), introduces the Kings theme (which appears in nearly every piece of the Musical Offering) and the fugue style in general.
The Penrose triangle, also known as the tribar, is an impossible object. It appears to be a solid triangle made of three straight beams of square cross-section which meet at right angles. It is featured prominently in the works of artist M.C. Escher, whose earlier depictions of impossible objects partly inspired it. (Image by MIT OCW.)
Here’s the course description:
How are math, art, music, and language intertwined? How does intelligent behavior arise from its component parts? Can computers think? Can brains compute? Douglas Hofstadter probes very cleverly at these questions and more in his Pulitzer Prize winning book, “Gödel, Escher, Bach”. In this seminar, we will read and discuss the book in depth, taking the time to solve its puzzles, appreciate the Bach pieces that inspired its dialogues, and discover its hidden tricks along the way.
In the bricks-and-mortar version of the course, they met twice a week for an hour. In our virtual version, we’ll discuss two sections in a week starting after Labor Day. We’ll follow this reading/listening schedule which follows the same order of the original MIT course. I’ll create a post for each of the reading and listening assignments (one on Mondays and the second on Thursdays). You can join in the discussion at any time during that week but make sure you have done the reading/listening first! It will be most productive if we move through the material together as much as possible. I reserve the option to close comments after a week, so that we keep moving forward and focus on the discussion for the most current reading/listening. However, if you get behind, you should feel free to jump back in later in the semester.
There will be an optional chat discussion once a week. We’ll figure out the format (iChat, AIM, Google chat, etc) and the day and time once I know who is interested.
UPDATE: There is now a page (see tab in the blue bar above) for the “Gödel, Escher, Bach” Course Schedule. This page has the reading and listening schedule and links to MP3 files for the music referenced. The dates listed are for when the discussion on the reading/listening will commence (so be prepared ahead of time). If you cannot keep up with the full schedule, you are welcome to participate in whatever chapters you are able to prepare for.