Fractals & Chaos Final Day

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!

Fractals & Chaos Assignment for 5/26

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.

  • [2,10]
  • [1,45]
  • [1/3,60]
  • [1,30]
  • [5,180]
  • [1,7]
  • [1,120]
  • [1/2,36]
  • [2,90]
  • [1,10]

Fractals & Chaos Assignment for 5/22

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.

Fractals & Chaos Assignment for 5/3

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:

  1. Where between 3.0 << 3.1 does the first bifurcation occur? When is the 2-cycle born?
  2. Can a 3 cycle be found in the interval 3.4 << 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.
  3. Is there only chaos to be found for a > 3.6? Or is there something else hidden in there?

Fractals & Chaos Assignment for 3/10

Please sign up here to bring a small snack item to share with others at our FractaSketch Gallery Art Show.

For Monday, you should read Pollack’s Fractals, an article from Discover Magazine about the math underlying Jackson Pollack’s famous paintings.  For some interesting follow-up reading, check out this article from the New York Times about the use of fractal analysis to examine the authenticity of supposed Pollack paintings and this article from the Science Daily blog suggesting that such an analysis is not scientifically valid.

Fractals & Chaos Assignment for 3/9

For Friday, read this article from Nature about the connection between fractals and the famous Rorschach inkblot test.

You should also be finalizing your designs for the FractaSketch art show. I want at least one submission in at least three of the following categories (though more are certainly encouraged!)

  • Fern
  • Tree
  • Spiral
  • Realistic
  • Artistic

Save the templates through the Java application, and save fractals you like as screenshots.  See Printing from FractaSketch Online for instructions.

Fractals & Chaos Assignment for 3/7

Read Why Fractals Are So Soothing, a January, 2017 The Atlantic article by Florence Williams about some work done by physicist Richard Taylor (and his magnificent hair)

Continue working with FractaSketch (Safari or Internet Explorer only).  You should be close to having designs for at least three categories.

  • Fern
  • Tree
  • Spiral
  • Realistic
  • Artistic

Save the templates through the Java application, and save fractals you like as screenshots.  See Printing from FractaSketch Online for instructions.

Fractals & Chaos Assignment for 3/3

If you’re interested in reading a bit more about the central topics of the past two articles, check out this NPR article as a follow-up.  Also note the New York Times article linked there.  Both articles have really interesting information on Galileo’s Square Cube Law and something called Kleiber’s Law, which governs the relationship between an animal’s size and metabolic rate.

By Monday, try to come up with a fractal design (either as a template/line segments like how FractaSketch works or as a gasket/area like how we approached the Sierpinski Triangle and Carpet originally) that has a dimension of exactly 1.5.

Continue working with FractaSketch (Safari or Internet Explorer only).  You should hopefully be close to having designs for at least three categories.

  • Fern
  • Tree
  • Spiral
  • Realistic
  • Artistic

Save the templates through the Java application, and save fractals you like as screenshots.  See Printing from FractaSketch Online for instructions.