Please watch the video posted here for the final part of the course. Play around with the application featured in the video here (though watch the video, first!). Thank you to Mrinal Thomas for developing this application!
Please also complete the course evaluation posted here by June 14th. Your feedback is very helpful for me to improve the course. Feel free to be honest!
If you’d like to play around with the piecewise complex linear function fractal generator, you can find a link to download the program here.
If you weren’t in class, we debriefed/formed some conclusions about linear functions then moved on to quadratics. For now, we’re keeping it simple and just iterating z -> z^2. We iterated several test seeds to start to see what kind of behavior would be expected in this iteration process, so pick a few below and iterate them (remembering that [r,theta]x[r,theta] = [r^2,2theta]) to see what happens for yourself. Should the value of theta ever exceed 360 degrees, reset it to a value within the 0 < theta < 360 domain.
We discussed in class that a fractional value of T will (eventually) result in a number of spokes equal to the denominator of T. If T = 1/3, for example, then every third iteration ends up exactly where you started, only closer (if you’re attracting with a value of R<1) or farther (if R > 1). But this spoke-like behavior doesn’t always appear immediately; sometimes you’ll see spirals before you see spokes. Our next question is to explore why, but first, please finish the front of the Complex Paint Worksheet. For each of these, start with R of .9, then gradually slow it down (R = .95, R = .99, R = .995, etc) so you can watch the different spiral/spoke patterns emerge. The last two can be entered as PI and PHI directly into Complex Paint.
Watch this video for a debrief on part I of the Complex Paint Worksheet, then get started on Part II as instructed.
Please complete the front of Complex Number Transformations, converting between polar and rectangular forms and using them to identify translations and dilation/rotations.
Now that you’ve had some opportunity to explore Devaney’s Orbit Diagram, please read the Feigenbaum Plot Article you got in class today. It will expand a bit on the work we’ve been doing in class, demonstrate a very interesting phenomenon in the pattern of bifurcations and, most importantly, answer the question of what order to cycles appear in?
Also, check out this Numberphile video about the concept of the Feigenbaum Constant, mentioned in the article.
We’ve spent some time exploring the logistic function analyzers linked at right, and have found some patterns. For example, when a = 3.0, we have a fixed point attractor, but at a = 3.1, there’s a 2 cycle. There’s a 4 cycle at a = 3.5, and while there appear to be separate bands of chaos at a = 3.6, they’ve merged by the time we get to a = 3.7. Please investigate the logistic function some more, with the following questions in mind:
- Where between 3.0 < a < 3.1 does the first bifurcation occur? When is the 2-cycle born?
- Can a 3 cycle be found in the interval 3.4 < a < 3.5? If so, this would suggest that one part of a cycle can bifurcate before the other. If not, perhaps bifurcation is an all or nothing thing.
- Is there only chaos to be found for a > 3.6? Or is there something else hidden in there?
Use the cobweb diagram apps on MathInsight.org to finish working on Part 4 of the Iterated Functions sheet (use this Supplement to record your answers).
Use the cobweb diagram apps on MathInsight.org to analyze the non-linear functions of Part 4 of the Iterated Functions sheet (use this Supplement to record your answers). Work up to 4g.
We’re working on some more iterating functions in class, but tonight I want you to read this article from Scientific American: Chaos and Fractals in Human Physiology. It talks about many of the same ideas discussed in the NOVA documentary we watched, but there’s some very important clues on some upcoming phenomena that I want you to pay attention to also.
Continue working on Iterated Functions, finishing part 1 and starting with 3 a, b, c. Make both a time diagram and a cobweb diagram for each part of 3, and start to look for patterns in the cobweb diagram that might correspond to patterns on the time diagram.
It’s spring break. We just finished the long walk from February break to April. When we get back, we’ll only have 7 short weeks before the end of the school year. But you have a week’s rest before we get there. So here’s your assignment for this week:
Stay up until 3. Sleep in until 2. Sleep, rest. Read a new book. Read a book you read when you were 8. Read a book to a younger sibling, or to kids in your neighborhood. Play a new video game. Go to watch a movie. Go make a movie. Start a YouTube Channel. Write a song, play a song, or sing a song. Write a poem. Draw something, anything. Go to the Commons and play music for the people there, then eat at Waffle Frolic. Order something from a restaurant you’ve never tried before. Play a board game with your family. Go for a hike in Cornell’s Plantations or in a gorge. If you’re in AP Stats, work a bit out of your review book (Practice Exam 1 would be a good choice!). If you’re in Fractals, keep playing around with mySolarSystem and complete 1d/e and all of 2 of the Iterated Functions sheet. Go for a run before the sun rises. Go for a swim. Bingewatch an entire season of The Great British Baking Show on Netflix. Look at the stars at night. Make dinner for your family. Do your chores. Do someone else’s chores. Visit a relative you haven’t seen in a year. Talk to someone who has significantly different opinions than you. Go to the farmer’s market. Play Pokemon Go (there’s an egg event happening with double XP!). Work on some IXL practice problems. Go to the SPCA and pet the cats (or the dogs). Take the road less traveled. Whatever you do to rest, do it. Just make sure that when you come back on the 24th, you’re ready to work. We’re almost done.
Keep playing with mySolarSystem and try to create a stable system with the following features:
- One sun, one planet
- One sun, and 2, 3, or 4 planets
- Binary system (2 suns)
- Binary system with planet
- One sun, one planet with moon
Also, read pages 49-54 out of Fractals: The Patterns of Chaos about space and planetary motion
If you missed class, please start watching the NOVA documentary on Chaos found here (we got to around the 27:00 mark today).
For Wednesday, please read the following three sections from your Fractals: The Patterns of Chaos book:
- Pages 55-60 – Weather
- Pages 93-97 – Symmetry
- Pages 99-106 – Landscapes
For this weekend, create a diagram showing the relationship between the following variables for a “damped” pendulum (a pendulum that slows down over time)
- Y-position as a function of x-position
- Y-position as a function of time
- Velocity as a function of time
- Velocity as a function of x-position
For a more modern take on Chaos Theory in the financial investment industry, check out this 2010 blog post from MoneyMorning.com: What We Can Learn from the Stock Market Genius That Wall Street Loves to Ignore.
Keep in mind, however, that this “technical analysis” approach is not without its detractors. For a counterpoint, see this bluntly-titled post from The Motley Fool, also from 2010: Technical Analysis is Stupid
Keep playing with Iterative Canvas and Robert Devaney’s Chaos Game. Make a note of any interesting patterns you find with Iterative Canvas; we’ll do a brief “show and tell” in class on Friday.
Importantly, on Friday we will also discuss the Chaos Hits Wall Street article you got in class on Wednesday. I know that some of you don’t always read every article I ask you to take a look at, but this is a must read. I expect everyone to contribute meaningfully to the discussion tomorrow.
By Friday, please read the Discover Magazine article Chaos Hits Wall Street. While this is an old article (1993!) it addresses a lot of the topics we’ll be discussing in the second part of the course. We’ll see some updated takes on the theories presented in the next articles.
For tonight, employ some random procedure to sketch a zigzag pattern across the piece of graph paper you got in class. Suggestions include using a coin and moving up a square on a heads and down a square on a tails, or using your calculator to generate random integers and moving up on an even number and down on an odd. Bring your completed zigzag in tomorrow for a quick explanation on the beginnings of chaos!
For some more interesting Tales From the Fourth Dimension, check out the following:
- The Adventures of Fred, Bob, and Emily – a detailed look at how the lives of a 2D (Fred), 3D (Bob), and 4D (Emily) creature interact with each other. See especially the “World” section, where the author, Garrett Jones, imagines how wheels, water, and war would work in these universes.
- And He Built a Crooked House – a short story by “Big Three” science fiction writer Robert A. Heinlein about an eccentric architect who designs a house in the shape of an unfolded hypercube. An earthquake hits, and the house folds back up on itself to concerning results (see also this student film version of the story)
- Some Notes on the Fourth Dimension – some animations and movies showing the geometry of the fourth dimension, including the ones we saw in the Flatland special features and the “folding” hypercube we looked at in class
You’ll need to report back your data about your Box Counts. Do that here.
We started the fourth and final method of finding dimension of fractals, the Box Count Method. In class, I gave you a Great Britain BoxCount Method sheet. Please count the boxes (on just the large island of Great Britain only!) for each scale of boxes. I want everyone to go all the way to 1/24, but please give 1/32 a shot. It’s tedious, but having more data is definitely worth it.