# Questions tagged [fractals]

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

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### Hutchinson's Notation in Fractals and Self-Similarity

I have been examining https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf for my thesis but couldn't find an explanation for two notations in the paper. ...
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### Hypersphere of Fractal Dimensions

Is the concept of a fractal dimension when applied to questions like a fractal dimension hypersphere volume nonsense? For example: Can a 4.5 dimensional hypersphere exist and have a 4.5D volume. And ...
1 vote
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### Reflecting a fractal

Assume you have a certain IFS (iterated function system - a finite set of contractions) given by affine transformations. As reflections are affine transformations, any reflection of an IFS fractal ...
• 139
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### Hölder regularity and Hölder condition

Is there any textbooks or papers introducing Hölder regularity and Hölder condition? When I read the paper(https://www.ams.org/journals/tran/2021-374-11/S0002-9947-2021-08489-5/home.html), I found ...
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### Cover the set $\{0,1,1/2,1/3,1/4,\dots\}$ with the least amount of closed intervals of length $\delta>0$.

Let $\delta>0$ and define $N(\delta)$ to be the least number of closed intervals of length $\delta$ necessary to cover the set $S:=\{0,1,1/2,1/3,\dots\}$. For example, it is clear that $N(1/2)=2$. ...
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### Contraction mapping theorem for statistical self-similarity

The notion of self-similarity that I was given is the following: Let $(X, d)$ be a metric space, and for each $1 \leq i \leq N$, let $f_i: X \rightarrow X$ be a contraction. Then there exists a unique ...
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### Do all >1D box fractals really have lines in them?

Consider an $n \times n$ square split up into $n^2$ cells the natural way. Now suppose we fill in $1 \le k \le n^2$ of these cells s.t. $\log_n(k) > 1$. Is it always the case that there will be a ...
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1 vote
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### Iterated function system with a fractional number of contractions

Iterated function systems can be used to generate fractals. One starts off with a simple geometric figure and applies the IFS infinitely many times to obtain a fractal. For example, in the case of the ...
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### Need help naming a fractal/image generated by exponential

I'm taking an introductory comp-sci course this semester, and my professor pulled up this graphic while explaining mergesort (the graphic shows blocks of size $2^{-n}$). For some reason, the vertical ...
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### Correctness implementation perturbation theory on Mandelbrotset

I'm trying to wrap my head around the perturbation theory paper by K.I. Martin. I've made a toy Python script to try it out using float64 as 'high precision' for the reference and calculating the rest ...
1 vote
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### Easy to understand proof of Self-similar IFS fractal dimension

I'm preparing for a seminar talk on fractals, the topic is Self-similar IFS fractal dimension, proving the main theorem used: Given $IFS=\left\{ \mathbb{R}^{n};w_{1},...,w_{N}\right\}$ with ...
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### 3d fractal helix modeling

I'm trying to build a 3d visual to illustrate a concept. Imagine a circular helix. We could define a cylinder that contains that helix. But now imagine this cylinder takes the helicoïd shape too ! We ...
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### Is the Mandelbrot set sufficiently self-similar to non-rigorously visually recreate procedurally from warped, nested structures?

While not strictly self similar, the Mandelbrot set shows significant similarity in its pixelated visual samplings. Are the overall shapes approximating each region well understood enough, not in a ...
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### List of open conjectures on fractals [closed]

I want a list of open conjectures on fractals. Preferably easy to understand and low dimensions. Before you say Eremenko's conjecture ( https://en.wikipedia.org/wiki/Escaping_set ): https://arxiv.org/...
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### Best way to fit a huge balanced binary tree into a rectangle

Does anyone know if there is a known algorithm/fractal to draw a very large balanced binary tree nicely, with minimal empty space when you put in on a piece of paper? Obviously, simplicity is also ...
• 210
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### deriving/explaining the Distance Estimation Method (by gradient) for rendering Julia & Mandelbrot sets

This question is about the derivation / explanation of the Distance Estimation Method (DEM) for rendering Julia and Mandelbrot fractals. I have not succeeded in finding an explanation online so I ...
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### Why do fractal-like patterns appear in this sequence?

I came across this sequence called Digital River, where the next number in the sequence is defined as the sum of the digits of the previous number, plus, the previous number itself. It caught my ...
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### Are co-ordinate systems within a fractal inhertly compressible due to them having non integer dimensions?

This is a thought that I came up with while I was building a giant fractal in minecraft. If you have a space of some sort you can also create a coordinate system for it. At first I defined that as ...
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### Is there a Julia fractal that contains uncountable many copies of itself?

We know that the Mandelbrot fractal contains a countable number of copies of itself. See : Does the Mandelbrot fractal contain countably or uncountably many copies of itself? Where that is explained. ...
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1 vote