Fractals & Chaos Recap for 1/16

We discussed an open problem related to the connectedness of the Mandelbrot Set today. It seems silly, but “connected” is actually a tough idea to define mathematically. We discussed the following ideas:

  • Not disconnected: That is, the set is not divided into pieces. We can’t draw a line “between” pieces of the set without crossing the set. The Mandelbrot set is not disconnected
  • “Simply” connected: Both the set and its complement (everything not a member of the set) are not disconnected. A circle is not simply connected, because it divides the plane into regions inside and outside of the circle, which are disconnected from each other by the circle. A disk — a circle that includes its interior — is simply connected, as anything not in the disk is outside the disk. The Mandelbrot Set is simply connected
  • “Path” connected: Any two points of the set can be joined by a path that stay in the set. We used the Topologist’s Sine Curve to define an example that is not “path” connected. The Mandelbrot Set, though, is path connected.
  • “Locally” connected: Around any point of the set, we can draw a circle small enough such that the set is not connected inside of it. We used the idea of a comb defined by the line segments y = 1/2^x from x: [0,1] as an example of a space not locally connected. Mathematicians are still unsure whether the Mandelbrot is Locally Connected or not, an open problem known as the MLC.

The last topic, which we will get into tomorrow, is a brief history lesson of the Julia Set that starts in Newton’s Method for Approximation.

For tomorrow, please read this article from the November, 1991 issue of Science News, published just after an important discovery was made about the border of the Mandelbrot Set, as well as this follow-up of sorts from 2017 in Scientific American

Fractals & Chaos Recap for 1/15

We wrapped up yesterday’s game of “Find the Julia Set” and reviewed some other interesting properties of the Mandelbrot Set:

  • “Area” Julia Sets come from “areas” in the Mandelbrot Set, and “string” Julia Sets come from “strings” in the Mandelbrot Set
  • There are “veins” through the main cardioid of the Mandelbrot Set where we can watch the orbit of a single fixed point die towards a 2-, 3-, 4-, or any magnitude cycle. Along these veins, the fixed point forms spokes. Offset slightly from these veins and they become spirals
  • Relatedly, there are balls off of the main cardioid for every possible cycle, and every variant of every cycle (1/5-cycles, 2/5-cycles, etc.). Imagine a circle centered on the cusp of the cardioid. Any angle rotation around that circle points straight at a ball on the edge of the cardioid, whose orbit-cycle magnitude exactly matches the fraction used to find that angle. A 120 degree rotation, representing 1/3 of a full rotation, points straight at the 3-cycle ball on the top of the Mandelbrot set, and a 240 degree rotation, 2/3 a full rotation, points straight at the 3-cycle ball on the bottom. A 72-degree rotation, 1/5 of a full, points straight at the first 5-ball. Rotations of 144, 216, or 288 degrees point straight at the second, third, and fourth 5-balls.

Our last two topics of the course: an open problem related to the connectedness of the Mandelbrot Set, and a discussion of Newton’s Method of Approximation, and its surprising connection to the Mandelbrot Set

Fractals & Chaos Recap for 1/11

We spent the period playing around 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)
  • 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.
  • 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…

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 Monday).

Fractals & Chaos Recap for 1/10

Download Fractal Zoomer.exe from here (don’t click the green button, that downloads something else). Use the navigation guidelines posted yesterday to help you use the program.

Play around with the program, but do so with intent. I have a few things I want you to explore:

  • Take unusual values of C from the Complex Paint worksheet and verify the connected/disconnectness of the corresponding Julia Set
  • Explore: Where in the Mandelbrot Set can we find values of C that correspond to area versus string Julia Sets?
  • Explore: What is the relationship between the location of C in the Mandelbrot Set and the orbit pattern within the corresponding Julia Set?
  • Challenge: We saw earlier that orbits of quadratics can contain up to Nine 6-cycles. Some are attracting, some are repelling, but most are complex. Can you find all nine of them in the Mandelbrot Set?

Also, if you’re interested in learning a bit more about three dimensional fractals, check out the game or information linked here.

Fractals & Chaos Lesson Recap for 1/8 & 1/9

We’ve discussed and proven a number of properties of the Julia Sets that we’ve observed, including that all Julia Sets have 180-rotational symmetry, and only Julia Sets with real-number parameters will have x- and y-axis symmetry. We also discussed an argument that all Julia sets are either completely connected (any two points can be connected by a path along the Julia Set) or completely disconnected (every point is disconnected from every other point). This difference gives us a convenient way to categorize Julia Sets, allowing us to create a catalog view of them, much like we did for the Feigenbaum Plot in the real numbers.

What we need then is a convenient way of identifying whether or not a Julia Set is connected without having to actually draw it. Fortunately, iterations of points contained within the Julia Set give us a way to do that: if we can find seeds that do not diverge to infinity, then the Julia Set is connected. More specifically, we have argued that the origin, z = 0, is a convenient starting seed to iterate. If the orbit of z = 0 eventually tends to infinity, we know the Julia Set is disconnected. But how do we know the path of an orbit will actually tend to infinity and never return?

Fortunately, there’s a radius of no return. We further proved in class that if the path of an orbit of the origin exceeds r = 2the orbit will never come back. This means that any value of |c| > 2 results in a Julia Set that is automatically disconnected (since if z_0 = 0, then z_1 = 0^2 + c = c, and we’re already past 2), and for any value of |c| < 2, we just need to iterate until the orbit exceeds 2.

If we imagine the complex plane as a computer image, where every pixel corresponds to a single value of c, then we can colorize those pixels based on whether or not their orbit has “escaped”. As we increase the number of iterations, more and more pixels will become colorized (we can use the same colors for pixels that escape at the same number of iterations). Eventually, we will see a shape form. That shape is the Mandelbrot Set.

Download and investigate the Mandelbrot Set using the program Fractal Zoomer. Make sure you download Fractal Zoomer.exe (unfortunately, this software only functions on Windows-based computers). There are three modes in the basic view screen: Zoom Mode, Julia Mode, and Orbit Mode

  • In Zoom Mode:
    • Left Click = Zoom in
    • Right click = Zoom out
    • Ctrl+F3 = Set new center
  • Press J to activate Julia Mode
    • Left Click anywhere in the Mandelbrot Set to create the Julia Set at that value of C
    • Ctrl+F3 will allow you to spawn a Julia Set of a specific value of C (use this to investigate Julia Sets from the Complex Paint Worksheet)
  • Press O to activate Orbit Mode
    • Left click anywhere in the Mandelbrot Set and it will superimpose the orbit of seeds within the Julia Set that uses that value of C
    • Ctrl+F3 will allow you to specify the orbit of a particular value of C (or if you are in a Julia Set, of a particular seed).

Fractals & Chaos Lesson Recap for 1/7

We spent some time reviewing the results from Complex Paint Worksheet 2. We have observed a few things, the most obvious of which are that all Julia Sets have 180-degree rotational symmetry and the only Julia Sets that have x- and y-axis symmetry have a parameter that is a real number. It turns out that both of these observations are provable facts about Julia Sets, which we will investigate tomorrow!

Secondly, we observed that many Julia Sets are connected (areas and strings) but many are not connected. This distinction – between connected and unconnected Julia sets – will form the basis for the formation of the Mandelbrot Set. Be sure you are in class tomorrow, as it will be an important day!