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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How to add shadow effect to 2D fractals?

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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Which objects can be Minkowski halved?

The Minkowski Sum of two subsets $A,B \subset \mathbb{R}^n$ is $$A \oplus B = \{a + b | a \in A, b \in B\}$$ For a given $A$, is there some condition that tells me when I can find a $B$ such that $A = ...
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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 ...
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Is the lower Hausdorff dimension of measures a Borel measurable function on measure space?

Let $X\subset\mathbb{R}^n$ be a compact subset, $\mathcal{B}(X)$ be the Borel $\sigma$-algebra on $X$. Denote $\mathcal{M}(X)$ the collection of all Borel probability measures on $X$. It is clear that ...
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Banach Circle Fractal: Find factor scale $s$ such that between small circle is not overlap in $n$-th step as $n\to\infty$

Source: from stackoverflow I want to draw a Banach Fractal as picture above. First step, draw a circle with radius $r$. Second step, draw 9 smaller circles with radius $s\cdot r$ ($s$ is factor scale,...
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Exponential Interpolation for Fractal Zoom with Perspective Camera in 3D Space

I'm creating an endless looping 3D animation of a zoom into a fractal, but I'm having an issue with the interpolation of the camera movement. Essentially it is a heart made out of other hearts, which ...
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How to determine the fractal dimension of a random walk in 2D?

I have to determine the fractal dimension of a random walk in 2D. How can I do that? I am first supposed to picture the random walk for 2D as in the following site. However, I am also asked to ...
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Implementation of rotation in chaos games

I am trying to implement rotation in chaos games. I have the following implementation for determining the next point: ...
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Inequality on the uniform norm of a conformal differentiable transformation

I'm currently working on a question on chaotic dynamics on fractals. Particularly if we have an iterated function system of conformal differentiable contraction mappings {$f_1$,...,$f_n$} we denote ...
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Making the Mandelbrot Fractal in Desmos Online Graphing Calculator

I would like to make the following "animation" of the Mandelbrot Set using the Online Desmos Graphing Calculator app - as seen over here: https://www.youtube.com/watch?v=naqgsOOEHJs I ...
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How are Mandelbrot Animations Made?

I have always been interested in learning about how computers are able to animate the "Mandelbrot Set" (https://en.wikipedia.org/wiki/Mandelbrot_set). I tried to learn about how pictures of ...
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What shapes can be composed of scaled copies of themselves, with no intersection?

A square can be subdivided into 4 squares, each scaled by a factor of $\frac{1}{2}$. A right-angled triangle can generally be subdivided into 2 similar triangles of different size. What other shapes, ...
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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 ...
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Official Definition of a Fractal? [duplicate]

Informally, I have heard for a function to be considered as a fractal - the function can never be "smooth". This means that if you keep "zooming in", you will never have a "...
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Is it possible to prove an irrational to be normal? [closed]

We don't know if pi is normal but it seems like it is. is there any irrational number known to be normal through a formal proof? I ask this question because I have a fractal structure that I want to ...
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How would I go about generating this sort of fractal?

Don't really know if this question belongs here or on stack overflow, so sorry if this is not the right place. I have picture of this fractal: I have not seen this fractal among other more popular ...
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Intersection of random fractals; i.e. the range of $\alpha$-stable subordinators

I am thinking about the following: https://users.math.yale.edu/public_html/People/frame/Fractals/FracAndDim/DimAlg/intersection.html states that most placements of two fractals whose dimensions add to ...
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$ \tan{\left\lfloor{x}\right\rfloor}$ self-similarity?

As you can see here, there is plenty of what appears to be self-similarity within the graph of $ \tan{\left\lfloor{x}\right\rfloor}$. Every $7\pi$ interval along the x-axis there appears to be a ...
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What's the topological dimension of Sierpiński's Triangle?

I am aware that Sierpiński's Triangle is a fractal, with Hausdorff dimension $1.5850$. Therefore my intuition leads me to believe it's topological dimension is 1 (as the topological dimension must be ...
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Find extrema of $y=?(x)-x$ with the Minkowski Question Mark function

The Goal: is to figure out the global extrema of the Minkowski Question Mark function $?(x)$. Here is the graph of: $$?(x)-x:$$ The $y$ value of the global maximum was found by systematically ...
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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 ...
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Relationship between two objects with the same fractal dimension.

I've been doing some research into fractal dimension (namely Minkwoski-Bouligand dimension) for my university dissertation and have found two seemingly unrelated objects that have the same Minkowski-...
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Irreducible factors of Mandelbrot polynomials

Let $p_n$ be the $n$th Mandelbrot polynomial, so $p_1:=c$ and $p_{n+1}:=p_n^2+c$. Then each $p_n$ is divisible by all $p_k$ for which $k$ is a proper divisor of $n$; to get rid of those factors, we ...
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Understanding Falconer's Example 4.3 (Hausdorff dimension of Cantor $\times [0,1]$)

In the above calculation, I think I understood most of it, except the last bit. Why is $\mathcal{H}^s(F_1) > \dfrac{1}{2}$ instead of $\mathcal{H}^s(F_1) \ge \dfrac{\mu(F_1)}{2}$ as Mass ...
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$\frac{\max(1,xy)}{y} = x$: Did I just create a fractal? [closed]

https://www.desmos.com/calculator/6hfyqc6ks9 Did I just create a fractal? Again, the function is $\frac{\max(1,xy)}{y} = x$. Even uncannier version, from the alternative form with "sgn" ...
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Are exponential functions fractals?

I have been trying to find fractals in some data I am working with and (using box-counting), I think I have found one. However, when I visualize the data, it looks like an exponential, which got me ...
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2 votes
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Does the Mandelbrot set contain all possible images?

Since the Mandelbrot fractal has infinite complexity and we can zoom inside of it infinitely long one might ask if any combination of pixels eventually emerge and thus It contains all possible images (...
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Similarity Between Basilica Julia Set Structure And Phi-4 Feynman Diagram

I have found the same vertex-edge graph in two seemingly disparate places and am wondering about any possible connection between the two (beyond just pure coincidence, of course). The relevant Feynman ...
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Motivation for the concept of dimension

I am currently learning about Hausdorff measure and dimension. In several of the resources I have encountered, the author provides several different definitions of dimension, and they are often ...
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Questions regarding the open set condition of a self similar set

A collection of similarities $\{\varphi_n\}$, $n\in\{1,...,N\}$ satisfies the Open Set Condition if there is a nonempty open set $U$ such that $\varphi_n(U)\subseteq U$ $\forall n$ and $\varphi_n(U)\...
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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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Induction over Rham piecewise linear function

The following exercise is inspired in Daubechies - Lagarias: Two-Scale Difference Equations, Existence and Global Regularity of Solutions. Define the following functions: $$f_0(x)= \left\{ \begin{...
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What's the name of the property that leads to Newton's Fractal? For rational maps, regions contain either 1 limiting behavior or all

3Blue1Brown has 2 great videos on Newton's fractal. In the second video he describe a property that makes it inevitable. Does this property/theorem have a name? For any rational map, if you were to ...
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1 vote
1 answer
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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 ...
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Are there known points on the boundary of the Mandelbrot set which iterate forever?

Where $f(z)=z^2+c$ is the Mandelbrot iteration function, are there any known complex numbers $z$ such that iterating $z\to f(z)$ to infinity retains $z$ on the boundary (i.e. it does not explode to ...
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Are there "half step", or other subdivisions, of the Mandelbrot iteration function?

The Mandelbrot Set is generated by iterating $f(z)=z^2+c$. Is there a "half-step" function $g(z)$ such that $g(g(z))=f(z)$? What is the method for finding such half-iteration functions? ...
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Julia Set Fixed Points And The Golden Ratio

When calculating fixed points for the Basilica Julia set ($z_{n+1}=z^2+c$, $c=-1$), the fixed points are given by $$ z^*_0(-1)=\frac{1 \mp \sqrt{1-4(-1)}}{2}=\frac{1 \mp \sqrt{5}}{2} $$ This is ...
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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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n-dimensional ln and exp functions

I have some quaternion functions, like ln and exp. If I want to go to 5-component numbers, I can easily migrate the ln and exp functions by adding in an extra imaginary component. Since 5-component ...
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Is speaking about a fraction of the boundary of the mandelbrot set meaningful?

Sorry if my question is vague, as I have very little background with fractals and measure theory. My question is inspired by a tweet, where a light shone onto the mandelbrot set, and certain rays were ...
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