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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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Formal definition of "inifinite detail" regarding factals such as the Mandelbrot Set

The Mandelbrot Set is typically described as having infinite detail, e.g.: Images of the Mandelbrot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing…
jon hanson
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What is the fractal dimension of this tetramino shape?

This tetramino is used to form a fractal by copying a smaller copy of the shape into each of the four square parts that form the original shape. The picture only has the first step, but assume the final shape contains additional steps that do the…
HAL
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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 creating a highlighted point representing the…
KTys
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Spherical Fractals

Alright so for my work i am in the need of a Fractal with a spherical shape. Now, i have implimented a way to generate Fractals using a set of Equations. But so far i couldnt find a comprehensive source looking into Fractals which are to some extend…
ErikHall
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Mandelbrot set incorrect picture

I'm writing an algorithm to generate the Mandelbrot set in Java. However, the final picture is incorrect. It looks like this I was wondering if the algorithm was incorrect. public void Mandlebrot() { float reMax=2; float imMax=2; …
simplicity
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Mandelbrot artwork: how do I generate "way-out" lines?

I found this cool rendering somewhere on the net. Apparently the author generated nice tendrils you'd have to follow if you want to escape. I wonder about the math behind this. I know how to calculate the derivatives (generating a single thread),…
Matthias Urlichs
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How is golden ratio related to fractals (if it really is)?

I came across an information that the golden ratio is related to fractals in way a that: "Fractal patterns created using the golden ratio, however, are optimized in a way that does not occur with any other number" or sth like that. But such…
pw1822
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Fractal dimension of landscape generated with Fournier's midpoint algorithm (recursive subdivision) is 2.0 no matter what. Why?

I am using Fournier's midpoint algorithm (recursive subdivision) to construct a landscape of 1025 x 1025 gridpoints, then I am using an advanced version of box counting to find fractal dimension, D. No matter what Hurst exponent I use in Fournier's…
user993563
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(fractals) What is the link between my bifurcation diagram of a population and the mandelbrot set?

This is my first time here as I'm investigating fractals. I started of by writing about the similarity between a bifrucation diagram of a population, using formula $z_{n}=a*z_{n-1}*(1-z_{n-1})$. I ended up getting the following image:bifurcation…
AruLeF.
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The Coastline Paradox and The Mandelbrot Set

I've been doing some research on the Mandelbrot Set and have discovered that it's boundary is a fractal. I was wondering if the Coastline Paradox can be applied to the boundary of the Mandelbrot Set? Surely the perimeter of the set would approach…
M.Hatton
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Is there a way in finding the area of an island with its coastline's Hausdorff Dimension?

I have applied the Hausdorff method to find the fractal dimension of the coastline of one of my country's islands. Is there any way for me to find the area of the islands with the fractal dimension I have calculated? Perhaps, there is a way reverse…
Draveenn
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Proof for range of mandelbrot set?

Really simple question, I'm aware that if the value of c used for the mandelbrot set ever gets more than 2 units away from the origin, the value will tend to infinity and it doesn't belong in the set. Is there any proof for this? How do we know that…
Tomas Premoli Muniagurria
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Image of Hilbert's Space filling Curve

I am trying to find the image of $[0,1/2)$ in Hilbert's Space Filling Curve. What I thought is $[0,1/2]\times$ $[0,1]$ \ $[1/4,1/2]\times[1/2,3/4]$. Is my understanding correct? The sets considered are as usual. From $[0,1]$ to $[0,1]^2$.
Grazel
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are graphs with fractal dimensions possible?

As I am new to understanding fractals, I appreciate their beauty when graphed in cartesian coordinates. My intuitive understanding of the definition of a 'fractal dimension' is the relation of the length of a fractal segment as a proportion of the…
aquagremlin
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"Border" versus "perimeter" for Mandelbrot sets

Why do most writers used "border"--not "perimeter"--for Mandelbrot sets?
Daruis soli
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