# Questions tagged [fractals]

For questions on fractals, which are irregular, rough, or "fractured" sets that often possess self-similar structure.

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### How to find a fractal with a predetermined Hausdorff dimension?

For many patterns that display self-similarity, the Hausdorff dimension can be found. Sometimes the dimension is calculated and approximate - as is the case with the Feigenbaum attractor - but often ...
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### Experimenting with code when generating the Sierpinski triangles resulted in a beautiful picture of two birds kissing each other

So I recently found out about the Sierpinski triangles and decided to code it up myself because it seemed like a fun thing to do. The original algorithm is as familiar: Take three points in a plane ...
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### Proving that ${\mathscr{V}}^s( E \cup \widetilde{E})< 2 \leq {\mathscr{V}}^s( E ) + {\mathscr{V}}^s( \widetilde{E} )\quad$

For $E$ a (totally) bounded subset of $X$ and $s>0$, the Hewitt-Stromberg pre-measure is defined as follows, $$\overline{\mathscr{V}}^s(E)=\limsup_{\delta\to0} M_\delta(E) \;(2\delta)^s,$$ where ...
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My question is simple: how do you add a shadow effect (like the one in Kalles Fraktaler 2)? I have tried distance estimation, but have failed at creating it reliably. I would like a relatively fast ...
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### Is the Mandelbrot set path-connected if and only if it is locally connected?

This question mentions that it's an open question whether the Mandelbrot set is path-connected and the answer conflated it with the more famous open question of whether the Mandelbrot set is locally ...
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### Does an approximation to $f^n(x)=x^2+c$ exist?

The question half iterate of $x^2+c$ and answer describe how to approximate $f^n(x)=x^2+c$, when $x$ is large. However, using values of $c$ other than $0.25$ (or $0.5 < c < 1$) yields invalid ...
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### The continuity of a curve created through a geometric progression

Can I create a continuous curve using numerical discrete data? In this case, I'm looking at the area of Sierpinski triangles which decrease at a rate of $\frac{3}{4}$ per reiteration of the fractal. ...
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### Fractal Dimension of Lag Plots

I have created a number of lag plots from a single time series, namely the EUR/USD exchange rate, with lag = 1,2,...,20. E.g. Lag Plot I want to get the Fractal Dimension between the original time ...
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### Proving that the n-dimensional Wallis sieve has the same hypervolume as the n-ball

The Wallis sieve is a variation on Sierpinski's carpet, where you start with a square, and in the $i$th step you divide each square into $(2i+1)^2$ smaller squares and remove the middle one. The total ...
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### Sample points on the boundary of the Mandelbrot-set

I'd like to know if theres a simple way of sampling points on the boundary of the Mandelbrot-set. The last few hours I've been coding a bot to visualize a specific region of the set. I now want to ...
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### Do the Zermelo-Fraenkel axioms prevent the existence of fractals?

It is well known that Zermelo-Fraenkel axioms solve the Russell's paradox - i.e. A set can not contain itself (singular set) without leading to logical contradictions- just saying that this kind of ...
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### Is a random walk in 1.585-dimensional space transient or recurrent? [closed]

A drunk man will eventually find his way home, but a drunk bird may get lost forever. Will a drunk squirrel eventually climb down the 3-branch tree?
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### Newton fractals with nonzero measure?

The german wikipedia article about Newton fractals currently contains the following claim: Überraschenderweise kann die Julia-Menge (das Newtonfraktal) auch positives Maß in der Ebene haben: das ...
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