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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. Pick any two points that are members of the set, and we can’t draw a line that separates those two points without crossing the set somewhere. 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 (defined by the equation x² + y² = 25, for example) 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, for example defined by x² + y² < 25 — is simply connected, as anything not a part of 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.
Our last meeting will be on Wednesday, January 22nd in room G116. Our last topic will be a a brief history lesson of the discovery of the Julia Set that starts in Newton’s Method for Approximation.
By next Wednesday, 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
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
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:
And the 2-ball off of the 3-ball creates a Julia Set that is a Douady Rabbit of San Marcos:
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.
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!).
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 exceedsr = 2, the 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).
Assignments and Mathematical Musings from Mr. Kirk