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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 to generalize a Moore Curve to 3 dimensions?

I understand the concept of a Moore curve in 2D: However I find it a bit tough to conceptualize and generalize it to 3D or higher dimensions. Can someone kindly help me out by providing some resources and maybe a computer code to do so…
Ujjwal Aryan
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How to map coords (x,y) to a H-tree?

For a mini game I want to have a map resembling the H-Tree Fractal. The line would be road and you can drive around but only on the road. The map is infinite and needs to be generated as you drive around. So given a coordinate (x,y) how do I decide…
Goswin von Brederlow
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Distance Estimation Rational Julia Set

I have the following julia set: $$ f(z)=z\cdot\Omega=z\cdot\left(\sum_{j=1}^A\omega_j(z)^{-1}\right)^{-1} $$ where $$ \omega_j(z)=C\prod_{i=0}^{N_j}(z-r_{j,i}) $$ Where $C$ and all the $r_{j,i}$ are complex constants. Depending on what I choose for…
catmousedog
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Mandelbrot set approximation

Is there a function $f:\mathbb N\to \mathbb R$ such that $\lim_{n\to\infty} f(n) = 0$ and for every $c\in\mathbb C$: If $z_0=0$, $z_{n+1}=z_n^2+c$ and $|z_k|<2$, then there exists a point $c'$ in the Mandelbrot set satisfying $|c-c'|
janek37
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Is there a Weierstrass function with asymptotic line?

I haven't seen a Weierstrass function with asymptotes. They are all fractals and the sum of trigonometric functions. If you know a function that is nowhere-differentiable but is continuous and has asymptotes, please let me know, and if not, please…
Galois
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What is the box dimension of the image of a continuously differentiable function on $[0,1]$?

Im going through some notes for fractal geometry and the following exercise is stated: "Let $f:[0,1]\rightarrow{}\mathbb{R}$ be a continuously differentiable function with $f(0)\neq{}f(1)$. Show that the box dimension of the image of $f$, namely…
kam
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Set with Hausdorff dimension $s = \log 2 / \log 3$ but $H^s = \infty$?

I am trying to solve the following problem: Find a set $X \subset \Bbb R$ s t.$\dim_H (X)= s$ where $s = \frac{\log 2}{\log 3}$, but $H^s (X) = \infty$. Here I am using the notations from Fractal Geometry by Kenneth Falconer. From exercise 4.9…
Wolfgangg
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what is the fractal dimension of the henon map?

I have some questions about the Henon map that are not clear for me. I have seen that the correlation dimension of the Henon map is approximately 1,21, is that measure similar to its fractal dimension? Also, can the Henon map could be considered…
Lila
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Can we learn anything about international boundaries if we model them as fractals?

Many country borders are determined by rivers, which are fractal in nature. Other country borders are determined by other natural phenomena such as mountain ranges and coastlines. (edited) Main question Because they are often defined by natural…
rajah9
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Are Fractals always hollow? If so, how can they have volume or area?

When calculating the dimension of a fractal shape, using the intersecting boxes method (where the number of intersecting boxes at different scales is compared; this is described at the starting at around the 10:30 mark in this video), boxes within…
Obada_alhumsi
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Isn't everything in real life fractal?

I've learned that an object that its Hausdorff dimension strictly exceeds its topological dimension is a fractal - which implies that an object that has roughness everywhere is a fractal. However I think that all the objects in the real life has…
Dimen
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3D Mandelbrot - Multibrot

The basic Mandelbrot equation is well known $f(z) = z^2 + c$. The formula for the 'MultiBrot' or 'MandelShape' varies the value '$2$' in the formula $f(z) = z^d + c$. There are other descriptives for 3D versions of the Mandelbrot fractal but sadly…
J King
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Coloring the Mandelbrot set using iterated points

I'm working on rendering the Mandelbrot set using K. I. Martin's method (http://www.superfractalthing.co.nf/sft_maths.pdf), and I am able to successfully use it to approximate the values of points in the set under iteration. However, now I have…
Johnny Laptop
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What does the Mandelbrot set look like with holes?

We are interested in the boundary of the Mandelbrot set. It is closed, so does have a boundary. It seems a good idea to increase the boundary by making holes in the set. Which is easy to do. We then have not lost anything and have gained additional…
Stephen Wynn
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