Tag Archives: Julia Set

Fractals & Chaos Recap for 1/16

We wrapped up yesterday’s game of “Find the Julia Set” and reviewed some other interesting properties of the Mandelbrot Set (many of which were hinted at by Mr. Drix yesterday):

  • “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° 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° rotation, 2/3 a full rotation, points straight at the 3-cycle ball on the bottom. A 72° rotation, 1/5 of a full, points straight at the first 5-ball. Rotations of 144°, 216°, or 288° point straight at the second, third, and fourth 5-balls.
    • This note also explains why we can only find two 6 balls along the edge of the cardioid. A 1/6 (60°) rotation points to a six ball, but a 2/6 rotation points to the 3-ball (2/6 = 1/3), a 3/6 rotation points to the 2-ball (3/6 = 1/2), and a 4/6 rotation points to the other 3-ball (4/6 = 2/3). The only two 6-balls are at 1/6 and 5/6 rotations around the cusp.

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 1/15

The central question we have remaining is: In what way are the shape of the Julia Set, the size of the cycle with it, and its location in the Mandelbrot Set related to each other? We’ve discussed parts of this question, but today with Mr. Drix we answered it in earnest.

First, we observed that orbits along the horizontal x-axis of the Mandelbrot Set bifurcate in the same way that they do in the Feigenbaum plot, and for good reason. Furthermore, we noticed that elsewhere along the edge of the central cardioid of the Mandelbrot Set, we can find balls that yield cycles of any magnitude, and in general the size of the ball is inversely related to the magnitude of its cycle (bigger ball = smaller cycle).

When we look at the Julia Sets we get in each of these balls, we realize that each of them have a central area, centered at the origin, and each with an infinity of “pinch points” found along its edge. The number of areas pinched together at this point exactly matches the size of the orbit contained! Given a mystery Julia Set, then, we can determine the size of its contained orbit just by its shape!

Finally, we observed that every ball connected to the central cardioid has an infinity of balls attached to it, also of decreasing size. The fractal nature of the Mandelbrot Set manifests itself here: the largest sub-ball corresponds to a bifurcation, the next largest to trifurcations, and so on. Thus, the 2-cycle found in the San Marco Julia Set can bifurcate into a 4-cycle, trifurcate into a 6-cycle, or any other multiple of two. The 3-cycle found in the Douady Rabbit can bifurcate to a 6-cycle, trifurcate to a 9-cycle, and so on for any multiple of 3. And this pattern extends to the Julia Sets themselves: the 3-ball off of the 2-ball creates a Julia Set that is a San Marco of Douady Rabbits:

3 off 2
San Marco made up of Douady Rabbits — The 2-cycle of the San Marco trifurcates to create a 6-cycle

And the 2-ball off of the 3-ball creates a Julia Set that is a Douady Rabbit of San Marcos:

2 off 3
Douady Rabbit made up of San Marcos — The 3-cycle of the Douady Rabbit bifurcates to create a different 6-cycle

With this in mind, we can play the game: Find the Julia Set. For each Julia Set in the attached file, use the shape/structure of the Julia Set to predict where in the Mandelbrot Set it can be found, and as a result the magnitude of the cycle found within it. Then verify your predictions using Fractal Zoomer.

 

Fractals & Chaos Recap for 1/14

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!).

Fractals & Chaos Recap for 1/13

Download Fractal Zoomer.exe from here. 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. All nine can be found as attracting cycles in the Mandelbrot Set. Can you find all of them?

 

Fractals & Chaos Recap for 1/10

We finished proving some of the observations made yesterday, and 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 Recap for 1/9

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 proceeded to start, and will finish tomorrow.

Fractals & Chaos Recap for 1/7

We discussed the results from iterating yesterday’s seeds in the function z², finding that some seeds will attract to zero, some will spiral off to infinity, but others seem trapped in a unit circle around the origin, falling into a cycle or landing on (1,0). This unit circle forms a boundary between seeds that diverge and seeds that do not, and that boundary is called a Julia Set, named after French mathematician Gaston Julia, and developed by Julia and Pierre Fatou (Fatou names the complement of the Julia Set; in the case of C = 0, the Fatou set is the entirety of the rest of the complex plane except for the unit circle).

Not all Julia Sets (in fact pretty much none of them) look as simple as this, and Complex Paint is a great tool to help us see them. Watch this video for instructions on the derivation of Julia Sets, plus instructions on how to use Complex Paint to create and understand this new class of mathematical object. Feel free to explore whatever parameter values you would like, but in particular we will be using the ones found on Complex Paint Worksheet 2. We started this today and will finish it tomorrow. For each, note the type of Julia Set you get (area, string, or dust), any symmetry you observe, and the destination of orbits inside any area Julia Sets.