File:Relationship between Mandelbrot sets and Julia senshido.info - Wikimedia Commons
AuthorLink BlogLink Link The Mandelbrot Set and the Julia set are both famous sources of fractal images, partly because of the Points outside the Mandelbrot Set escape to infinity when we cycle the formula, points inside the set don't. .. The outline of the darker region in the centre of this table looks rather familiar. Julia sets are certain fractal sets in the complex plane that arise from the dynamics of complex polynomials. These notes give a brief introduction to Julia sets and explore some of their ing on the relationship of p1 to the unit circle: 1. If |p1| < 1. The Mandelbrot and the closely related Julia sets are both based on the idea of choosing two complex numbers z0 and c, and then repeatedly evaluating.
There are some deeper mathematical reasons for this, which we'll not go into here. Back to the Mandelbrot set But the shapes of the filled Julia set and the Mandelbrot set are connected in more ways than this. The appearance of the Julia set belonging to a c-value inside one of the decorations of M gives us a way of telling the period of that decoration, and vice versa. This filled Julia set is often called Douady's rabbit, after Adrien Douady who is one of the pioneers in this area of maths.
Note that the image looks like a "fractal rabbit. Everywhere you look you see other pairs of ears. Another way to say this is that the filled Julia set contains infinitely many "junction points" at which 3 distinct black regions in Jc are attached.
Fractals: The Julia and Mandelbrot Sets
In figure 14 we have magnified a portion of the fractal rabbit to illustrate this. The fact that each junction point in this filled Julia set has 3 pieces attached is no surprise to those in the know, since this c-value lies in a primary period 3 bulb in the Mandelbrot set. This is another fascinating fact about M.
If you choose a c-value from one of the primary decorations in M, then, first of all, Jc must be a connected set, and secondly, Jc contains infinitely many special junction points and each of these points has exactly n regions attached to it, where n is exactly the period of the bulb. Figure 15 illustrates this for periods 4 and 5. My website has a collection of movies showing how the Julia set changes and how the junction points arise as the c-value moves from the main cardioid of the Mandelbrot set into the primary bulbs.
Summary Thus, we've seen that the Mandelbrot set possesses an extraordinary amount of structure. Or we can take dynamical information and use it to understand the shape of M.
Unveiling the Mandelbrot set | senshido.info
This interplay between dynamics and geometry is on the one hand fascinating and, on the other, still not completely understood. Much of this interplay has been catalogued in recent years by mathematicians such as Adrien Douady, John H Hubbard, Jean-Christophe Yoccoz, Curt McMullen, and others, but much more remains to be discovered.
If you enjoyed this article, then maybe you can continue studying the Mandelbrot set and become one of the people making the discoveries. His main area of research is dynamical systems, primarily complex analytic dynamics, but also including more general ideas about chaotic dynamical systems.
He is the author of over one hundred research and pedagogical papers in the field of dynamical systems.
He is also the co -author or editor of thirteen books in this area of mathematics, including a series of four books collectively called A Tool Kit of Dynamics Activitiesall aimed at high school students and teachers. He has also been the "Chaos Consultant" for several theaters' presentations of Tom Stoppard's play Arcadia. Professor Devaney has delivered over 1, invited lectures on dynamical systems and related topics in all 50 states in the US and in over 30 countries on six continents worldwide.
He only needs Antartica to complete his goal of speaking on all continents — so if you teach at South Pole State and run some kind of seminar, give him a call! The Mandelbrot set and Julia sets The Mandelbrot set named after Benoit Mandelbrot is the most famous fractal of all, and the first one to be called a fractal. Just in case you haven't seen it before here it is shown on the right.
If zn does not tend to infinity the point z0,c is in the set colored blue or black in the pictures on this page. If we consider all possible z0 and c then we end up with a four dimensional set, whicht is rather hard to visualise. The mathematical definition is rather more complex, than this, but it turns out to be equivalent.
In fact this is what is called the "filled" Julia set. Officially speaking the Julia set is the boundary between points that tend to infinity, and those that don't.Interesting relationship between Mandelbrot and Julia fractals
These fractals can usually only be displayed approximately, because it is not always possible to tell how a particular point will behave. Spirofractal iterates each point up to times. Most points will either tend quickly to infinity, or converge into a repeating cycle of one or more values, but points near the boundary of the set, which is the interesting partcan take many iterations to do either.
The cycle may not be detectable for points near the boundary, because there is a limit to the accuracy of the calculations which may prevent convergence from occurring when it should. Spirofractal leaves points that are not definitely in or out of the set black, and varies the shade of blue according to the length of the cycle.
Most black points are probably in the set. See how intricate the Mandelbrot set is in "Seahorse Valley". A cardioid is a heart-shaped figure. It's studded with circles all around its boundary. Here's a magnification of the region of the circle on top of the cardioid.
Note that this circle as well as the other circles you can see are also studded with circles. There are infinitely many circles on the cardioid, each of those circles has infinitely many circles on them, and on and on ad infinitum.
That makes for a lot of circles! If you look close, you'll see the strands of dark blue above the circle under discussion. So, what's going on up there? Here's a blowup of that portion of the figure. There's something black up near the top of the picture.
It's another cardioid with associated circles! Not exactly the same, but close.