Fractals & Chaos Assignment for 12/14

Keep playing with IFS Construction Kit and try to recreate the Koch Curve, the Dragon Curve, and a fern. See Wednesday’s post for instructions about installation and usage.

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Fractals & Chaos Assignment for 12/13

Most computer-created fractals utilize an Iterated Function System (or IFS) to construct the images. One such software is the IFS Construction Kit, developed by Larry Riddle at Agnes Scott College. With this software, you can code a series of equations that the program will iterate to create an image.

Click the “Download Now” button at the bottom of the page and the software and relevant files will download as a zip folder (like the other student-created programs we have used have done). Open your Downloads folder and Right Click+Drag the zip folder to wherever you want to extract it do (e.g., the desktop) and click “Extract…” When the extracted folder opens, you can read the IFS Help file, or just double click the Sierpinksi Triangle icon labelled IFS Construction Kit to open the program.

This program has a lot of pretty neat capabilities, but the essential steps are as follows:

  • Under the Window dropdown, you should activate IFS and Fractal at a minimum. This will give you the IFS window, where you can enter the parameters of an equation you want to iterate, and the Fractal window, where the Fractal will appear.
    • The Design window will give you an ability to design your fractal by moving small boxes around, the number of boxes determined by what you’ve entered in the Fractal window. The parameters of the equations in the Fractal window will change as you move the boxes. Click the “Show/Hide Initial Polygon” button (a little blue dotted line box) in the window to see the bigger picture. The smaller boxes represent the scaled down versions of this bigger picture.
  • I recommend you use the “Scale/Rotation Form” to enter your parameters in the Fractal window. Either choose “Scale/Rotation Form” from the “Code” dropdown or just click the R on the toolbar.
    • The Scale portion represents R of the polar form for A in the Az + B format of complex linear equations
    • The Rotation portion represents Theta (note that you can give differing values for each of these, though I recommend you use consistent values for now!).
    • The Translation portion represents B in the Az + B format of complex linear equations.
  • Once you’v entered the parameters of an equation, you can just hit the “Draw” button in the Fractal window to see your design.

Your challenges are to try and use this software to create the Koch Curve, the Dragon Curve, and a Fern. Think carefully about how you constructed these objects using FractaSketch and calculate how the direction, position, and placement of each individual segment of the sketch compared to the original dotted baseline that formed the 0th iteration. Use the default Sierpinski Triangle as a template of sorts, or use the Design window to try and click/drag a design that matches the FractaSketch template.

We’ll share what you’ve achieved on Friday!

Fractals & Chaos Assignment for 11/30

Read the Feigenbaum Plot Article article that you got at the end of class today. In particular, pay attention to the converging ratio of gaps between bifurcation points, and the second page that discussing the ordering of cycles. You may also want to check out this Numberphile video about this same topic.

Also, keep playing with the Feigenbaum Plot applet! Remember that you’ll need to use a Java-permitting internet browser in order to view it.

Fractals & Chaos Assignment for 11/20

Finish question 5 from Iterated Functions, using the apps on MathInsight.org. Note that all of these functions have fixed points at zero, where the instantaneous slope at zero is equal to 1. This creates some interesting behavior for some fixed points, so make sure you check the behavior of seeds on either side of the fixed point.

Fractals and Chaos Assignment for 11/3

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 Tuesday.

Importantly, on Tuesday we will also discuss the Chaos Hits Wall Street article you got in class on Thursday. 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 Tuesday.

UPDATE: The Chaos Hits Wall Street discussion will happen on MONDAY, not Tuesday.