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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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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 well known, why is it that if you zoom into parts of…
anon
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Does this Fractal Have a Name?

I was curious whether this fractal(?) is named/famous, or is it just another fractal? I was playing with the idea of randomness with constraints and the fractal was generated as follows: Draw a point at the center of a square. Randomly choose any…
Silver
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What exactly are fractals

I have always been amazed by things like the Mandelbrot set. I share the view of most that it and the Koch snowflake are absolutely beautiful. I decided to get a deeper more mathematical knowledge of this, but sadly Wikipedia hasn't been of much…
Guy
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Perturbation of Mandelbrot set fractal

I recently discovered very clever technique how co compute deep zooms of the Mandelbrot set using Perturbation and I understand the idea very well but when I try to do the math by myself I never got the right answer. I am referring to original PDF…
NightElfik
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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 point of view. To my level of understanding, they…
Stefano Borini
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Is The *Mona Lisa* in the complement of the Mandelbrot set.

Here is a description of how to color pictures of the Mandelbrot set, more accurately the complement of the Mandelbrot set. Suppose we have a rectangular array of points. Say the array is $m$ by $n$. Suppose also we have a number of color names. Now…
Jay
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Fractal identification

I was trying different algorithms out, and after a while, I found this fractal: The generation has similarities to Koch's curve, but instead of putting triangles on triangles, I put circles on top of circles. The algorithm is the following: I go…
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Can we prove the Mandelbrot set is a fractal? Which maps/processes produce fractals?

So, as you probably noticed, I have two questions. The second leads on from the first. Can we prove the Mandelbrot set is a fractal? It is very easy to see that something like the Sierpinski triangle is fractal by design. Yet it's not obvious to me…
ShakesBeer
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What real numbers are in the Mandelbrot set?

The Mandelbrot set is defined over the complex numbers and is quite complicated. It's defined by the complex numbers $c$ that remain bounded under the recursion: $$ z_{n+1} = z_n^2 + c,$$ where $z_1 = 0$. If $c$ is real, then above recursion will…
Carl Brannen
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Mandelbrot boundary

Is there a sequence of parameterized expressions for the border of all the major bulbs of the mandelbrot set? By major meaning all bulbs with diameter greater than 0.01 for example. I am interested in generating the major features of the set to…
PMay
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How to draw a fractal from $z \mapsto z^2 + c$ explained for a mere mortal?

I am interested in: 1) Understanding in detail how fractals are draw. 2) Coding a computer program to draw a simple fractal. Can someone with good explaining skills take care of 1) for me? I don't think this link does a good job. I step-by-step…
chrisapotek
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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 functions f that map M to itself in nontrivial ways. By…
Tyler
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Is the Fractal Dimension of a Space-Filling Curve in a Plane Always 2?

I have been playing around with space-filling curves that completely fill the unit square. All of them that I have seen have a fractal dimensional of 2. Makes sense that it would be 2, but a Google search hasn't turned up any comments on this at…
bob.sacamento
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Prove that the box dimension of $\{0,1,\frac{1}{2},\frac{1}{3},...\} $is$ \frac{1}{2}$

I'm supposed to consider the difference $\frac{1}{n+1}-\frac{1}{n}$ and let it equal to $\epsilon$. Hence $\epsilon=\frac{1}{n(n+1)}$. But how do I show that the number of boxes of size $\epsilon$ to cover the set is $N(\epsilon)=2n$? After that,…
Chuck Franklin
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We know the dimension of the Koch snowflake's perimeter, but does it have a measure?

I start with an equilateral triangle with side perimeter three meters. I can define a Koch snowflake by the following sequence of figures. Starting with that triangle, produce the next figure by replacing the middle third of each line segment with…
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