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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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How is the Lorenz attractor a fractal?

Based on my intuition of what a fractal is, the Lorenz attractor doesn't fit that category for me. A fractal should have some self similarity, but the attractor seems just like two two-dimensional disks which meet in the middle.
grevel
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Have I discovered a new fractal / what is this fractal called?

So I was experimenting with fractals and created this. The equations are $$ A \leftarrow A^2 + x - B \\ B \leftarrow B^2 - y + A $$ and the set is defined as the points (x,y) on a graph which when the equations are iterated, $A$ AND $B$ remain…
George Russell
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Help understanding this 'Fractal' I've just made?

I was messing around in C++, making an image where the pixels change depending on the the rectangle's dimensions and whether or not the space bar is down, and I formed this image: Could anyone explain why such a drastic change exists between the…
Stumbleine75
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Square of $3/2$ dimensional fractal

It is known that the following fractal (Quadratic von Koch curve, or Minkowski's sausage) has a dimension $3/2$. It means that the square $X^{2} = X\times X$ may have dimension $2 \times 3/2 = 3$, which is a natural number. Is there a good way to…
Seewoo Lee
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Are the intriguing and lovely Mandelbrot Set hoops and curls the result of floating point computation inaccuracy?

Are the intriguing and lovely Mandelbrot Set hoops and curls the result of floating point computation inaccuracy? I have written various Mandelbot Set computer implementations, such as a dynamic zoom and playback. Some use fixed point arithmetic,…
Weather Vane
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What's the name of this fractal?

I created it with some simple chaos game restrictions: You can always move towards the point you moved to on the last step When on the top left corner, you can go to the right corners When on the top right corner, you can go to the left…
Francisco
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Critical points of a function

The literature on Mandelbrot and Julia sets mentions the phase "critical point" quite a lot, but usually doesn't bother to define what it means. As best as I can tell, a critical point is just any point where the function's derivative is zero. This…
MathematicalOrchid
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General Mandelbrot iteration formulas

Everybody loves the good old quadratic Mandelbrot set. As you probably know, both it and the corresponding quadratic Julia sets are defined by the iteration $f(z) = z^2 + c$. You might expect, however, that $f(z) = az^2 + bz + c$ would give you more…
MathematicalOrchid
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Why must fractals be self-referential?

Having an idea of what a fractal is done by example and heuristically, then seeking the actual definition is, at first, both obvious and imprecise. You'll see it defined as an object that is self-similar in some sense or other, but none of the…
j0equ1nn
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Is this plot of Ford circles actually a fractal?

Is this plot of Ford circles actually a fractal?
draks ...
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Area of filled Julia set

This is a vague question, and I know nothing about this area. We fix some $c\in\mathbb C$ and iterate the map $z\mapsto z^2+c$. This gives some filled Julia set, i.e. the set of points $z\in\mathbb C$ so that the orbit of $z$ is bounded. For…
Rob Silversmith
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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 Dwarves. Some of these tasks, however, quickly…
Raven Dreamer
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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?
AndyCap
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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 there some proof for that? Thank you in…
Slobodan
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What sort of pattern is this?

I'm not a mathematician, but a part time artist, and I enjoy drawing patterns. I came up with this drawing a couple years ago: Someone may have come up with this before me, I wouldn't be surprised. I don't know if it's a fractal, but as it is drawn…
sam matney
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