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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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What is the condensation set of a fractal?

Is there a definition of the condensation set of a fractal that is both clear and rigorous? I've been searching around to get a sense of what exactly the condensation set of a fractal is - I've encountered papers like…
David Faux
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Is there a general metod to construct a fractal?

I would like to construct a fractal (traditional, self-affine, and fat fractal) with a given embedding and fractal dimension, but I don't know how to do it programmatically. The shape of the fractal is negligible.
Alíz
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Box dimension of $\{\frac{ 1}{5^n} : n \in \mathbb{N}\}$

I am working through a first course in Fractal Geometry, and have encountered a problem that has asked me to calculate the box-counting dimension of $F=\{ \frac{1}{5^n} : n \in \mathbb{N}\}$ However, I am stuck straight away. Thus far, I've only…
Mystery_Jay
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Are fractal image generators one-way functions?

Is it hard to calculate the coordinates and zoom factor that was used to generate a fractal image of, say, the Mandelbrot set? If you know the rest of the parameters, like how many iterations where used, in generating the image.
Lars Holm Jensen
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Representing a 3D Hilbert Curve as an L-system

A 2D Hilbert curve can be represented as the following L-system: A → -BF+AFA+FB- B → +AF-BFB-FA+ where F denotes a step forward, - denotes a 90 degree turn left, and + a 90 degree turn right. (source:…
Yuval Adam
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test for membership in mandelbrot bulb of period n

Is there a efficient test (formula or inequality) of whether a given point is in a bulb of period n? In other words, something other than running the iteration a lot of times to see if it converges to a period n cycle. I would like to apply such a…
PMay
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Negative Fractal dimension values in plants images

After calculating lengths and angles from a plant i represented it with the help of L-system fractals (see image below). I made that process for many plants and then i went to matlab to calculate their fractal dimension. I used the boxcount…
F.N
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Do fractals really happen in nature?

We live in a 3 dimensional world. So, line and plane as 1 and 2 dimensional objects do not exist in reality although using these concepts are useful for modeling some problems such as motion in one direction. I have more difficulty about fractals.…
MOON
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Defining strict self-similarity

I have been reading through John Hutchinson's paper "Fractals and Self-Similarity" and some other things, and I haven't really found a definition of strict self-similarity to work with that makes much sense to me. Heuristically we want (any?) part…
Dom
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Two questions on fractal

Suppose we have two given fractals $K_1$, $K_2$ of dimension $d_1$,$d_2$ respectively. What can we say about the dimension of $K_1 \cap K_2$, and $K_1 \cup K_2$? Is there any technique to describe a fractal with given dimension $d$? What if the…
Arsenaler
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Box counting fractal dimension

S: 1,0.3,0.22,0.12,0.06 N: 7,201,478,2595,17950 (no idea how to put this in a tally) I've got a question here where S is not shrunk by the same fraction throughout, I know how to work out the dimension Fractal when the Shrink is continuous E.G 1/3…
Unknownstarz
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"Necks" of the Mandelbrot set

I'm looking at some "neck" points of the Mandelbrot set where there is just one point connecting a part of the set to another, such as the point (-0.75,0). I'm currently interested in finding out the precise values of the 2 necks connecting bulbs…
Suzumisc
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How do I algebriacally analyze these results from the mandlebrot set?

The equation for the mandelbrot set is z = z^2 + c, in my exploration, I changed the values of c and analyzed the iteration in the table below What algebriac patterns emerge in this table?
Ram Tewari
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How to locally distinguish the interior of a fractal curve from the exterior?

After staring at a small portion of a Koch snowflake, it seems to me that it is not possible to tell the inside of the snowflake from the outside if you can only see a small portion of the boundary. Small view of the Koch snowflake: Expanded…
Yly
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Is the intuition that the fractal dimension measures the extent of space filling not valid for box fractals?

Box fractal refers to various iterated fractals created by a square or rectangular grid with various boxes removed or absent and, at each iteration, those present and/or those absent have the previous image scaled down and drawn within them. from…
HAL
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