The Mandelbrot Set
Julia and Mandelbrot Sets. David E. Joyce August, Last updated May, Function Iteration and Julia Sets. Gaston Julia studied the iteration of. Question: What is the relationship between the Mandelbrot set and filled Julia sets? Answer: Remember that the Mandelbrot set is a picture of the complex c- plane. To choose other c-values, you may use the Mandelbrot/Julia Set Applet. The set of points that do not escape is called the filled Julia set, and the . both to explore Julia sets and the Mandelbrot set and also to explore the relationship.
For a julia set, for each pixel apply an iterated complex function. Z is initially the coordinates of the pixel, and will then constantly be updated through every iteration: If you keep iterating this function, depending on the initial condition the pixelz will either go to infinity, or remain in the circle with radius 2 around the origin of the complex plane forever.
- Lode's Computer Graphics Tutorial
- Julia and Mandelbrot Sets
- Julia and Mandelbrot Set Explorer
The points that remain in the circle forever, are the ones that belong to the Julia Set. So keep iterating the function until the distance of z to the origin 0,0 is greater than 2. Also give a maximum number of iterations, for exampleor the computer would be stuck in an endless loop.
The color value of the pixel will then become the number of times we had to iterate the function before the distance of z to the origin got larger than 2. The constant c in the formula can be anything really, as long as it's also inside the circle with radius 2.
Different values of c give different Julia Sets. Some Julia Sets are connected, others aren't. Here's an example of the calculations: First, you can choose a constant c for the function, which one you choose will determinate the shape of the fractal.Interesting relationship between Mandelbrot and Julia fractals
First, we transform the coordinates so it lies between -1 and 1 if you zoom or move around in the fractal a different transformation is required: Now we apply the function for the first time: What is the period of the "primary bulbs" and how can you see this period?
For that matter, what are the primary bulbs? First of all, the "primary bulbs" are the large, disk-like decorations hanging off the main cardioid in the Mandelbrot set. To the right is a picture. The largest black region in the Mandelbrot set is the "main cardioid. We'll have fun with the antennas later. For now, though, it is important to remember that a "primary bulb" is that large disk-like region directly attached to the main cardioid, not all the other junk that hangs from it.
Another thing that makes this map so interesting to study, apart from its apparent simplicity juxtaposed against the chaos it displays, is the fact that high iterates of most other complex functions begin to look locally like a wrapping around a point along with a translation.
So studying this map, the simplest nontrivial map of the complex numbers, can help us understand a wide variety of other functions on the complex plane. Dynamical Properties So what can happen to points as we iterate on them repeatedly?
This means that once points become large enough, then they always escape to infinity. The complex map has periodic orbits just like its real-valued counterpart, the logistic map, does.
Just as in the real case, it is possible for iterates of a point to converge to this periodic orbit. It is also possible for points to neither escape to infinity nor converge to a periodic orbit.
These points bounce around a bounded region of the plane close to the origin, but do so in a chaotic way, never repeating the same patterns. Julia Sets One way to visualize the dynamics of the map on the plane is to color points in the complex plane based on whether they escape to infinity or stay in a bounded region for all time. We can color points in the complex plane based on what their long-term behavior is.
We can color points that stay bounded one color, and color the points that escape based on how quickly they escape. Here, we will color the points that do no escape black, but other programs may color them differently. The set of points that do not escape is called the filled Julia set, and the boundary of the filled Julia set is called the Julia set.
Because of the fact that far enough away from the origin of the complex plane, points always escape, we know that everything of interest is happening in a small region around the origin, and this is all that is usually drawn. If we letthen our equation becomes.
From what we said earlier, each iteration of this function squares the radius of the point being iterated.
So points with a radius above 1 will always get larger in absolute value. Points with a radius less than will get smaller in absolute value and closer to 0, a fixed point of this map.
All future iterates of points with a radius of exactly 1 will also have a radius of exactlybecauseand they will stay on the unit circle around the origin. The following is a picture of the filled Julia set for: Here are some other pictures of filled Julia sets of maps with other values of. This one is called the Basilica, and occurs when: