We spent some more time investigating the Mandelbrot Set with with Fractal Zoomer, and have made a few observations:
- The central “bulb” of the Mandelbrot Set is a cardioid, and each other bulb off of that central body is a perfect circle.
- Each bulb has its own cycle of orbits. Some bulbs have the same size orbit as others, but with a different “order” (1,2,3,4,5,6 vs 1,3,5,2,4,6 for example)
- Along the central “spike” on the left side of the Mandelbrot Set, we can find baby Mandelbrot Sets strung along that string. The main body of these baby Mandelbrots show cycles instead of fixed points. Realizing that the horizontal axis on which the Mandelbrot set is centered is the real number axis, this suggests that there may be a meaningful connection between the Mandelbrot Set and Feigenbaum Plot…
- If you tile the plane with Julia Sets of a sufficient density, the collection of Julia Sets make a photo-mosaic of the Mandelbrot Set. This further reinforces that there is something important between the location of C within the Mandelbrot Set and the shape of the corresponding Julia Set.
This last point suggests there is a significant, meaningful connection between the location and shape of a Julia Set in the Mandelbrot Set. This will be what we’ll explore tomorrow.
For now, keep working with Fractal Zoomer and considering the questions posted yesterday. Please also read the section on Mandelbrot Set from pages 74-81 in your copy of Fractals: The Patterns of Chaos (and bring that book back on Thursday!).
From Chapter 14, read pages 363-371 (up to “Tables, Venn Diagrams, and Probability”)
From page 359, do exercises 17, 23, 25, 27
From pages 384-385, do 15, 21
Check your answers to these problems in the back of the book!
Your next Personal Progress Check has been assigned. Log into AP Classroom and complete the Unit 4 MCQ Part A PPC by Friday, January 10
We spent the first portion of class today exploring the Feigenbaum Plot today by asking a few supplemental questions to question 3 (is there anything to be found after chaos):
- Which comes first, a 3 cycle (so far found at a = 3.84) or the 5 cycle (found at a = 3.74 but also a = 3.906)?
- Why do some cycles happen more than once (see the 5-cycle above, or the two 6-cycles at a = 3.63 and a = 3.845 or two 4-cycles at a = 3.5 and a = 3.961)?
- In general:
- Where do cycles come from? How are they “born”?
- Is there some order to cycles?
Instructions on using Paul Fischer-York’s Bifurcation Diagram
- Use the Darkness slider to make the image darker and easier to see.
- Left-click and drag a rectangle around any portion of the diagram to zoom in on that portion.
- Right-click anywhere in the diagram to run a series of iterations at that value of a along the x-axis. The software will identify the magnitude of the cycle you’re looking at (though sometimes it makes a mistake, so look at the list as well!)
After some time to explore, we made a few observations about the general questions. The first is that cycles are born in two ways:
- Bifurcations of “lower” cycles, or
- Spontaneously arising from chaos
This suggests a certain ordering of cycles: the 2-cycle comes first, then the 4-cycle, the 8-cycle, the 16-cycle, and so on for powers of 2. Eventually, these cycles become so large that the system becomes chaotic. But this only explains cycles that are powers of 2. What about any other? For this, please read this article about the Feigenbaum plot, its constant, and the ordering of cycles.
We reviewed results from 4b through 4h on the Iterated Functions sheet. Every example had a single attracting fixed point (and possibly an additional repelling one) until 4f, which had two repellers. But what did these repellers repel to? It wasn’t infinity, it was a cycle of values between 0 and -1. This is our first example of a limit cycle, a surprising result in algebraic iterations where a pattern of steps is pushed into a repetitive cycle.
Then we pushed the function a little further, from x^2 – 1 to x^2 – 2. Still with two repellers, this time the system of iterations dissolved into complete chaos, careening randomly within the interval -2 to 2, never settling into one spot. Changing the value of the seed by only one ten-thousandth produced a completely different pattern of iterations, demonstrating the sensitivity to initial conditions that are characteristic of chaos.
We will see plenty more examples of weird behavior in our iterations in the days to come. For tomorrow, be sure to read the article posted yesterday.
Read pages 313-319 (up to the section on Confounding), then from the exercises do 7, 17, 30
For some additional reading about the ongoing Replication Crisis in science, see this article from The Atlantic, or this Crash Course video. Check out this New York Times article for more about Brain Wansink, the Cornell food science researcher who resigned amid this scandal.
We discussed the last of the exercises from #3 of the Iterated Functions sheet, then debriefed all of the observations of Ax+B again, deciding once and for all that Ax+B is solved. A fixed point can be found by the expression x = B/(1-A), but only the value of A impacts the behavior of the fixed point:
- If |A| < 1, the fixed point is attracting
- If |A| > 1, the fixed point is repelling
- If A > 0, the pattern is direct
- If A < 0, the pattern is alternating
- If A = 1, there is no fixed point
- If A = -1, the fixed point is neutral
- If A = 0, the fixed point is “super-attracting”
We also observed that as |A| gets nearer to 1, the attracting/repelling behavior becomes slower.
We looked at another hypothetical non-linear system, then set to work on exploring more of #4 from the IF sheet, using the Iterated Functions Supplement as a guide. Your homework is to finish up through #4h (by tomorrow), and read Chaos and Fractals in Human Physiology, from the February, 1990 edition of Scientific American
We discussed the reading from Fractals: The Patterns of Chaos and looked at some references to the “Butterfly Effect” in popular culture. We spent most of the rest of the period playing with the Solar System simulator online.
For Monday, please read pages 49-54 from your book, the section on “The Fractals and Chaos of Outer Space.”