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

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Is the maximal temperature of the curlicue fractal acheived by $e\times\gamma$?

The Curlicue Fractal is defined as follows: Choose an irrational number $s$ and a horizontal unit segment with angle $\phi_0 = 0$. Define $\theta_{n+1} = \theta_{n} + 2 \pi s \pmod{2 \pi}$, with ...
8
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2answers
557 views

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 ...
3
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1answer
214 views

$\dim_\text{topology}(\text{Cantor Sets}) \leq \dim_\text{hausdorff}(\text{Cantor Sets})$?

Please, explain: the Cantor set (a zero-dimensional topological space) is a union of two copies of itself, each copy shrunk by a factor 1/3; this fact can be used to prove that its Hausdorff ...
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0answers
236 views

Julia Sets in Mathematica

stackexchange geniuses! I'm a high school student doing engineering research and am in need of some technical assistance. I'm working on a paper on using fractals in civil engineering and need to ...
0
votes
1answer
59 views

Why this Mandelbrot program add current point in the Mandelbrot

I read some posts about the Mandelbrot. I read that the Mandelbrot should be defined by $f(z)=z^2+C$. In my understanding, I think, the $C$ should be a constant, like $0.27$ or $2.1+4.5i$. However, in ...
3
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0answers
582 views

How can I generate grid-based Fractals?

Please let me know if there's a better site to ask a question like this. I play a little indie game called Dwarf Fortress and a major part of the game involves building the titular Fortress for your ...
6
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0answers
242 views

Do fractals contain solutions for problems?

I notice that fractals resemble natural shapes such as leaves or rivers. Leaves and rivers are solutions to problems in themselves. A leaf is trying to distribute the water to the leaf while the leaf ...
4
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2answers
333 views

Independent High School Research

StackExchange Gurus! I'll keep this question short and to the point. I'm going into my senior year of high school, will have an independent research period, and have few ides of what to do during ...
9
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3answers
348 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 ...
4
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1answer
322 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 ...
13
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2answers
388 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
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1answer
170 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 ...
28
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2answers
661 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
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3answers
519 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 ...
7
votes
1answer
228 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
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1answer
246 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
296 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
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1answer
143 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 ...
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1answer
169 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 ...
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4answers
837 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 ...
6
votes
2answers
400 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
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2answers
382 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 ...
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0answers
104 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 ...
9
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0answers
208 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 ...
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3answers
1k 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 ...
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3answers
494 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
290 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
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2answers
318 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 ...
4
votes
2answers
337 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
240 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
217 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 ...
7
votes
3answers
590 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 ...
3
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2answers
803 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 ...
10
votes
1answer
367 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
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2answers
282 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 ...
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4answers
475 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?
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2answers
5k 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, ...
51
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
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4answers
3k 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 ...
15
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11answers
2k 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 $\mathbb{C}$?