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

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9
votes
3answers
496 views

Hilbert curve, understanding the original article

I'm trying to read and understand the article in which Hilbert gave an illustration of a space filling curve, namely "Ueber die stetige Abbildung einer Linie auf ein Flächenstück". It's only a short 2 ...
6
votes
1answer
457 views

Hausdorff dimension of graphs of one-dimensional Brownian motion

First question here, my apologies if it is a duplicate or inappropriate. There is a page on Wikipedia listing fractals by Hausdorff dimension and it includes the graph of a "regular Brownian ...
14
votes
2answers
469 views

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 ...
6
votes
1answer
188 views

Fractal dimension after nonlinear transformation

Let's assume X(s) is a fractal surface with Hausdorff dimension D. Now we take a nonlinear transformation f which transforms X(s) to f(X(s)). In this case, what will be the Hausdorff dimension of the ...
33
votes
2answers
1k views

Why does this distribution of polynomial roots resemble a collection of affine IFS fractals?

Consider the following spectacular image, created by Sam Derbyshire and described in John Baez's article "The Beauty of Roots": In this image are plotted all the complex roots of all polynomials of ...
7
votes
3answers
783 views

Given a Pattern, find the fractal

Is it possible, given a pattern or image, to calculate the equation of the fractal for that given pattern? For example, many plants express definite fractal patterns in their growth. Is there a ...
8
votes
1answer
257 views

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 ...
5
votes
1answer
296 views

What is the mathematical principle that describes a series of dots on concentric circles that form a spiral pattern?

Apologies for the vagueness of the question, I'll clean it up once an answer helps me describe it better. I'm fascinated by the pattern demonstrated in this image. It's made up of dots on a series ...
3
votes
2answers
344 views

Dimension of fractals

I would like to know is it possible to generate a fractal in the plane with dimension higher than 2? If that is possible, please could you explain the intuition behind that? If it is not possible, is ...
0
votes
1answer
166 views

What is the tangent point of any given co-prime on the Mandelbrot Set in pseudocode?

Given a computer program generating the Mandelbrot Set - using this one for example, which uses a module called mandel.js - what would be the pseudocode to find the complex coordinates, capable of ...
0
votes
1answer
178 views

Interesting non-stem questions about Koch/Sierpinski fractals

Exam time and I am having a hard time finding any inspiring questions about fractals for our "contemporary math" course. We found the perimeter and area of various Koch snowflakes and Sierpinski ...
20
votes
4answers
991 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 ...
8
votes
2answers
432 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
443 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
115 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 ...
17
votes
0answers
389 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
619 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
348 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
337 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
394 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
256 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
689 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
1k 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
452 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
519 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 $\mathbb{C}$?