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Questions tagged [fractals]

For questions on fractals, which are irregular, rough, or "fractured" sets that often possess self-similar structure.

The term fractal, derived from the Latin fractus meaning "broken" or "fractured," was coined by Benoît Mandelbrot in 1975 in order to describe mathematical objects (shapes, sets, processes, etc.) which possess irregular or rough structure at all scales. While there is little consensus on the precise definition of the term, fractals are typically characterized by self-similarity. The Cantor set, Sierpinski carpet, Koch Snowflake, and Mandlebrot set are examples of fractal sets.

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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 transformed surface f(X(s))? More clarification…
Frederik Larsen
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Can we check whether a Cantor set is self-similar or not?

Given a Cantor set $C$ on the real line, do we have some ways to determine whether it is self-similar or not? In particular, how can we check that $C$ is not self-similar? Edited: Definition: Let $\{f_i\}_i$ be a family of contraction maps, i.e.…
MichaelNgelo
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Can a "Julia set" fractal be described in a "closed form"?

What I mean by that is, consider, say, the "Koch snowflake" curve. It is formed by repeatedly applying a substitution to the lines of a triangle to get the final curve in the limit. What I am after is whether or not you can find a substitution and…
The_Sympathizer
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Choosing an appropriate sequence of $\{1,2,3\}$

Let $f_1,f_2,f_3$ be the contracting maps $f_i:x\mapsto \frac{1}{2}(x+p_i)$ from $\mathbb{R^2}$ to itself and $p_i\in \mathbb{R}^2$. Denoted by $S$ the attractor Sierpinki gastek of the iterated function system $(f_1,f_2,f_3)$. I want to prove the…
EQJ
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Is there a koch circle?

Is there some fractal like the koch snowflake, but only with many circles around a bigger initial circle, each of them surrounded by smaller circles and so on (but all of them kissing one bigger circle)? So circles instead of the triangles in a koch…
user2103480
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Name of this fractal

I am writing my final paper in the field ob computer enginering my work are on fractals. Some time ago, I found this fractal. Now I need to refer to it in my work but i have no clue what is it called. I searched the internet but found nothing really…
Shawn
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Fractal dimension of the boundary of a fractal

Sorry if this is a stupid question, but I'm a physicist, not a mathematician, and fractals are pretty new to me. Is there a simple relationship between the fractal dimension of a set and the fractal dimension of that set's boundary? For…
Zane Beckwith
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Did I just invent a new fractal?

I am not a mathematician but I am an artist and fractal enthusiast who spends lots of time creating art using this geometry. I recently discovered online you can make fractals in MS Paint (there's plenty of info online about this) and one of the…
katzenklavier
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Box counting method for fractal dimension

First I saw here that the box counting fractal dimension defined by $D = \lim_{\epsilon \rightarrow 0}{ {\log N( \epsilon)} \over {\log { {1}\over{ \epsilon }}}}$ which makes sense for me. Then I saw here a mathematica code for calculating it.…
MBolin
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What algorithm can be used to reconstruct a self similar time series from a portion of it?

I am working on a process which produces time series similar to the one shown in the graph below: I calculated Fractal dimension $D$ and Generalised Hurst Exponent $H$, to confirm that equality $D=n+1-H$ holds for $n=1$. I followed the line of…
Arash
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Categories of fractals

I have a question about classifying a few fractals I've been programming. I understand that there are types of fractals like L-systems (Barnsley's Fern, Fractal plant, ...), IFS systems (Sierpinski's gasket, Fractal flames, ...). I'm a bit confused…
xtrinch
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interior distance estimate for Julia sets - getting rid of spots

From wikibooks colouring the Julia set, the distance estimate $\delta(z)$ can be calculated by: $$\begin{aligned} \delta(z) &= \lim_{n \to \infty} \frac{|z_n| \log |z_n|}{\left|\frac{\partial}{\partial z} z_n \right|} &\text{ for } z_n \to \infty…
Claude
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Collage theorem to generate a spiral

I need to answer a question on fractals from the book Fractals Everywhere by M. Barsley and I have been struggling with it for a while: Use collage theorem to help you find an IFS consisting of two affine maps in $\mathbb{R}^2$ whose attractor is…
Lei Gioia
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How should I assign RGB colors to points in the Mandelbrot Set?

I decided to learn about the Canvas object in javascript by implementing a display of the Mandelbrot Set. I am mimicking the Mandelbrot psuedocode found on wikipedia. The thrust of it is that the number of iterations it takes for a point to diverge…
Darth Egregious
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Let $A$ and $B$ be fractals with box dimension of $x$ and $y$ respectively. Then prove:

Let $A$ and $B$ be fractals with box dimension of $x$ and $y$ respectively. Then prove that the Cartesian product $A \times B$ has box dimension $x+y$. Any hints to start out? (note that box dimension is sometimes called Minkowski dimension)
Chuck Franklin
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