Questions on fractals, irregular-looking mathematical objects that display the property of self-similarity.

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20
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4answers
982 views

Mandelbrot fractal: How is it possible?

I'm a programmer and have recently played around a bit with rendering Mandelbrot fractals / zooming into them. What I can't grasp: How can such infinite, complex shapes come out of somewhat 10 lines ...
7
votes
2answers
426 views

Odd fractal-looking illusion with $x,y,z \in [0,1]$ such that $x+y+z=1$, what is wrong?

Thanks to comments, it should be a plane but why does it look a bit like a fractal? Does my code overlook something or some err in plotting tool? I used Python and GNUplot. Apparently an animated ...
3
votes
2answers
436 views

quasiconformal “automorphism” groups of julia sets

To motivate this question, let me begin with a picture: Each letter labels a "blob" of this quartic julia set. (is there a technical term for these parts?). Because of resolution limitations I ...
4
votes
0answers
114 views

The Hausdorff dimension of the set of solutions of a system of coupled differential equations

I am interested in the relationship between non-linear differential equations and the Hausdorff, or fractal, dimension of the set of solutions. For example, the Lorenz Attractor, which is a standard ...
14
votes
0answers
317 views

Visualizing the Partition numbers (suggestions for visualization techniques)

So Ken Ono says that the partition numbers behave like fractals, in which case I'd like to try to find an appropriately illuminating way of visualizing them. But I'm sort of stuck at the moment, so ...
8
votes
3answers
2k views

A way to determine the ideal number of maximum iterations for an arbitrary zoom level in a Mandelbrot fractal

I've created a JavaScript-based fractal drawer which you can see here: http://jsfiddle.net/xfF3f/12/ As you're probably all aware, a Mandelbrot Set is created by iterating over pixels as though they ...
7
votes
3answers
614 views

Need good material on multifractal analysis

I'm searching for some good reading material on multifractal analysis. Preferably something accessible that doesn't put the stress too much on mathematical proofs but rather on applications. As long ...
2
votes
1answer
342 views

Is Perlin Noise a “fractal”?

I have an old Spanish CG book that calls Perlin Noise a "fractal structure". After reading this I couldn't deny it or confirm it. Is it a fractal structure? What would it Hausdorff dimension be?
6
votes
2answers
334 views

Which features would be interesting for a mathematician in a fractal program?

Many years ago I wrote this fractal generator: http://uberto.fractovia.org/ It was shareware but then I put it as open source. It's written in Delphi, a language that I don't use anymore. So I'm ...
5
votes
2answers
393 views

Is the number of circles in the Apollonian gasket countable?

Is it correct to say that the number of circles in an Apollonian gasket is countable becuase we can form a correspondence with a Cantor set, as their methods of construction are similar? What about ...
11
votes
1answer
255 views

Reconstructing a Monthly problem: tree growth on the 2D integer lattice

I'm trying to reconstruct a problem I saw in the Monthly, years ago. Perhaps it'll look familiar to someone. In the integer lattice in the plane, we grow a tree in the following natural way: ...
4
votes
1answer
222 views

Sets of Constant Irrationality Measure

Let $\mu (r)>2$ be the irrationality measure of a transcendental number $r$, and consider the following set of points $P \in\mathbb{R}$: $P=\{r\in \mathbb{R}: \mu(r)=Constant\}$ Is this set a ...
10
votes
3answers
686 views

variant on Sierpinski carpet: rescue the tablecloth!

I was playing around with Sierpinski carpets (see pretty GPU-produced picture here), and came up with a variation that I have been unable to find mentioned elsewhere. I'm wondering if anyone can tell ...
4
votes
2answers
991 views

Magnet Mandelbrot Set

We know that the Mandelbrot set is derived from the iterations of z^2 + c. Do anyone know something about magnet Mandelbrot? I found it in the software UltraFractal, and it is much more beautiful ...
12
votes
1answer
439 views

Why are these two definitions of the Mandelbrot set equivalent?

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 ...
0
votes
2answers
287 views

How do I plot this red dot?

Here is the dot And here is how I arrived at it » Pretty simple IFS using a 2 x 2 grid as the base for the iterations. Is there a way to describe this point as well as its siblings, all of which ...
3
votes
4answers
515 views

Where have fractals gone since Mandelbrot?

What are some examples of cutting-edge research involving fractals or self-similar structures? Who's actively contributing high-quality research in this field?
6
votes
2answers
7k views

Continuous coloring of a Mandelbrot fractal

I've recently started making a small fractal app in Javascript using the famous Mandelbrot bulb $(z = z^2 + c)$. I've been trying to find the best method of coloring the points on the complex plane, ...
55
votes
4answers
2k views

Why does the Mandelbrot set contain (slightly deformed) copies of itself?

The Mandelbrot set is the set of points of the complex plane whos orbits do not diverge. An point $c$'s orbit is defined as the sequence $z_0 = c$, $z_{n+1} = z_n^2 + c$. The shape of this set is ...
13
votes
4answers
4k views

How to explain fractals to a layperson and to someone with more math training?

I have a Ph.D. in computational and theoretical chemistry with advanced but field-oriented knowledge of mathematics. I am fascinated by fractals, but I am unable to understand them from the formal ...
17
votes
11answers
3k views

Mandelbrot-like sets for functions other than $f(z)=z^2+c$?

Are there any well-studied analogs to the Mandelbrot set using functions other than $f(z)= z^2+c$ in $\mathbb{C}$?