# Tagged Questions

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### Simplest way to determine if a number is a member of the Mandelbrot set?

I'm writing JavaScript code to plot the Mandelbrot set on an HTML5 Canvas element. (That's probably not relevant to the answer to this question). A core part of the problem is to write a simple ...
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### Why is the bailout value of the Mandelbrot set 2?

For the past few days I've been studying the Mandelbrot set, and many say that if the iterations of a point stay within a magnitude of 2, the point converges. A very natural question of "why is the ...
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### 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 ...
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### 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 ...
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### Is $g(z)=\frac{1}{z}+\frac{1}{z^2+1}+\frac{1}{(z^2+1)^2 +1}+…$ analytic for $|z|>2$?

Let $z$ be a complex number. Let |.| denote be the absolute value. Let $n$ be a positive integer. Let $f_1(z)=z^2+1$. Let $f_n(z)=f_1(f_{n-1}(z)).$ Is ...
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### Do there exist periodic fractals $A_f$ of this type?

Let $z$ be a complex number. Meromorphic here means meromorphic on all of the complex plane $C$. Lets define a fractal $A_f$ on the complex plane as the result of iterating a meromorphic function ...
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### Hausdorff Dimension of Arbitrary Julia Set

I am looking to find an exact solution to the Hausdorff dimension of a Julia set $J(f)$ for a polynomial $f: z \mapsto z^2 +c$ given an arbitrary $c$. I know this question is known for a number of ...
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### Buddhabrot Sewing machine [closed]

The Buddhabrot fractal traces the orbits of the points outside the Mandelbrot set. What design considerations need to be taken into account to create a computerised sewing machine that traces out ...
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### Properties of the Mandelbrot set

Are there any properties of the Mandelbrot set that can be analysed without a knowledge of complicated topology? Considering the fact that the set is based on a quadratic function, are there any ...
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### any idea what fractal algorithm might generate this shape?

I Found this image around, and i'm curious what algorithm generates this kind of shape In particular, i'm curious how the flow lines are generated, since usually the Mandelbrot iteration just ...
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### Simple examples of entire functions that have fractal properties.

Im looking for simple examples of entire functions that have "fractal properties". With "fractal properties" I mean that $|f(z)| < 1$ has a "fractal structure" in the complex plane. With "fractal ...
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### How to prove a property regarding periodicities of points in the Mandelbrot set?

While studying a visual representation the Mandelbrot set, I have come across a very interesting property: For any point inside the same primary bulb (a circular-like 'decoration' attached to the ...
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### Julia Set of polynomials

If $f$ is a polynomial and $z\in\mathbb{C}$, show that either $f^n(z)\rightarrow\infty$ or $\{f^n(z) : n\geq 1\}$ is a bounded set. Here, $f^2(z)=f(f(z))$ and $f^n(z)=f(f^{n-1}(z))$ for $n\geq 2$ ...
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### Number of limit points of a continued exponential

Inspired by the work of C. Bender, I recently played with continued exponentials (like continued fractions but with exponential functions ;) ). Given all prefactors are equal to 1, the continued ...
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### coloring the inside point for Julia Fractal

I am trying to continuous coloring the inside point for a fractal image,such as $z \to z^2+C$. For those outside point, we can use the escape iteration to determine the color, just as the description ...
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### Fractal behavior along the boundary of convergence?

The complex power series $$\sum_{n=1}^{\infty}\frac{z^{n^2}}{n^2}$$ has radius $1$ (Ratio Test) and is absolutely convergent along $|z|=1$. Recalling something that my calculus professor (Ray Mayer, ...
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### Interesting but elementary properties of the Mandelbrot Set

I suppose everyone is familiar with the Mandelbrot set. I'm teaching a course right now in which I am trying to convey the beauty of some mathematical ideas to first year students. They basically know ...
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### Mini Mandelbrots, are they exact copies?

(This one was found by magnifying 280,000,000 times.) In popular "zoom movies" of the Mandelbrot set the last image is often what appears to be an exact copy of the original set. This is always ...
The definition of the Mandelbrot set that most enthusiasts first encounter is that of the set of all complex numbers $c$ for which the sequence $z_{n+1} = z_n^2 + c$ starting from $z_0 = 0$ does not ...