Articles of fractals

Taylor series of mandelbrot bulb boundaries

What I am looking for is a way to find an approximation to the boundaries of hyperbolic components of the Mandelbrot set. I would like to be able to write a program to find the equations which describe this approximation. For example, I know that the period 2 region boundary is C = -1 + […]

Hausdorff Dimension Calculation

This is a homework question that I don’t know where to start with it. Can somebody help me please? I am trying to work out the Hausdorff dimension of the set $\{0,1,\frac{1}{4},\frac{1}{9},…\}$. I have worked this sort of thing out for a couple of fractal examples, but never for a set of numbers and I […]

Why does the mandelbrot set and its different variants follow similar patterns to epi/hypo trochodis and circular multiplication tables?

So the $z^2 + c$ variant has a cardioid shape at the center. This shape is made by an epitrochoid with a ratio of the radi being one, or from the two times table when we display it in a circle (as seen in this video https://www.youtube.com/watch?v=qhbuKbxJsk8). The next variation $z^3 +c$ has a nephroid […]

How to draw a Mandelbrot Set with the connecting filaments visible?

The M-Set is connected. But the M-Set viewers I’ve found create cool pictures that don’t really show the connecting filaments. This mini-Mandel beetle should be connected to a larger min-Mandel by a black filament going into its “butt crack”, but you can’t see it here: Monochrome pictures that show just the M-Set itself (not the […]

Fractal derivative of complex order and beyond

Is there some precise definition of “complex (fractal) order derivative” for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would like to know if some mathematician has defined a complex order derivative valid without restrictions for all complex number z. I mean: Is a well-defined definition […]

What is known about nice automorphisms of the Mandelbrot set?

It is often stated that fractals, such as the Mandelbrot set M, are self-similar, although I’ve never heard of any functions to formally model this perspective. I’m curious to learn about any functions f that map M to itself in nontrivial ways. By “nice” in the title, I mean to exclude trivial or very artificial […]

Can monsters of real analysis be tamed in this way?

Consider the Weierstrass Function (somewhat generalized for arbitrary wavelengths $\,\lambda > 0$ ): $$ W(x) = \sum_{n=1}^\infty \frac{\sin\left(n^2\,2\pi/\lambda\,x\right)}{n^2} $$ $W(x)$ is an example of a function that is everywhere continuous but nowhere differentiable (Or, for the nitpickers among us: differentiable only on a set of points of measure zero). We seek to remove the pathological […]

How to prove Mandelbrot set is simply connected?

In this lecture note of Harvard, it is proved that Mandelbrot set is connected, a result due to Douady and Hubbard. However, I lack necessary knowledge to comprehend it. Then in the same note it is said that the simply connectedness of Mandelbrot set is easier to prove. However, I cannot find a proof through […]

Mandelbrot set: periodicity of secondary and subsequent bulbs as multiples of their parent bulbs

In the Mandelbrot set, all points of the main carodioid are asymptotic (that is, the iterations of c^2 + c approach a constant). In contrast, it seems that all bulbs have a periodicity greater than 1, that is, the iterations settle into a cycle with a certain period. There are several questions to be asked […]

Are mini-Mandelbrots known to be found in any fractals other than the Mandelbrot set itself?

This is a generalization of the question Are there mini-mandelbrots inside the julia set? @Hagen raises an issue I was afraid of, which is that even the mini-Mandelbrots in the Mandelbrot set are not exact copies. I don’t know enough to give a rigorous answer to this question, but I’m thinking of whatever (implicit) standard […]