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Questions tagged [fractals]

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

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Cantor Problem— Finding the sum at $n=20$

This problem is driving me absolutely nuts. You have a line segment equal to $1$. You take out the middle third. Now you have line segments from $0-1/3$ and from $2/3$ to $1$. What is the sum of the ...
0
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1answer
42 views

Normalise exponential numbers between 0 and 1

I'm creating a fractal visualisation. I want the colour per point to be based off the iteration final value, $f(z_{n})$, instead of the traditional: number of iterations before reaching a cut-off (...
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20 views

Methods to draw fractals [closed]

What are the known methods to generate fractals mathematically other than iterated function systems and how to use these methods for programming?
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15 views

Does a curve with a box-counting dimension greater than 1 have to have infinite length?

If I have a curve that occupies a finite space (e.g. the unit square) and it has a box-counting dimension > 1, can it still have a finite length? If not, is there a proof of this?
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41 views

The methods to create fractal objects mathematically [closed]

I have read that fractals does not have a mathematical definition as for now. So how do we understand a given object can be considered as a fractal? I know iterated function systems theory is a method ...
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55 views

How to prove fractal is differentiable at no point?

this is a problem from Elementary Classical Analysis. Author constructs a function which is continuous everywhere but is differentiable at no point, and i don't how to prove the second result. Here is ...
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2answers
67 views

Julia set fractal generator created a Poincaré disk?

This happened to me when I was playing around on this site did I stumble upon a link between the Julia set and the geometry of a Poincaré disk? Does anyone know if there are documented occurrences of ...
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0answers
25 views

Book recommendation: fractals

I’m looking for a book to learn a bit more about fractals. I have to say I am a theoretical physicist, not a mathematician so it’d rather be mathematically simple (I don’t know what kind of maths are ...
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24 views

What is the fractal dimension of the critical scaling factor that just avoids forming a black hole?

In one of Scott Aaronson's lectures he mentions that attempting to scale up any three or even two dimensionally laid out hard drive will eventually result in a black hole. That is, if you've got some ...
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55 views

Twin dragon behaviour under different scaling factors

I observed a weird behaviour plotting the twin dragon for different scaling factors $\alpha$. I used this mathematica code to generate them ...
3
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1answer
40 views

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 $2 \times 3/2 = 3$, ...
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14 views

Evaluating this limit in Fourier analysis

In the study of Fourier transform of the standard Cantor measure, I came across the following problem: For $k\geq 1$, let $$ S_k=\sum_{m=3^{k-1}}^{3^k}\,\,\,\prod_{j=1}^\infty \cos^2\left(\frac {2m}{...
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40 views

What is the name of this fractal?

I can't find the official name of this in my textbook. Can anyone help me out?
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2answers
51 views

What are space filling curves used for in the real world?

I recently watched 3Blue1Brown's video on the Hilbert Curve and Fractals, and I was wondering if the concept behind space-filling curves like the Hilbert Curve and Flow Snake could be applied in real ...
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43 views

proper description for a sequence with a specified fractal property

I have a sequence A with the fractal pattern that the innermost values x, where length(x) = length(A)/2, has sum(x) = sum(A)/2. Also this is a fractal pattern so that for sequence x with the fractal ...
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27 views

Explicit formula for recursive equation

I would like to find the explicit formula for $f(a)$, where the recursive formula is $f(a)=(a^2+r^2)p-q^2-f(r)$, $p=\lfloor 1+\log_2 a \rfloor$, $q = 2^p$, and $r=q-a$. The base case is $f(1) = -1$. ...
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27 views

Python/R library for fractal analysis

There are several methods for fractal/multifractal analysis such as DFA, MFDFA, Wavelet-leaders, power spectrum exponent, etc. I am going to analysis signals using these methods (or at least most ...
4
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1answer
75 views

A quartic whose Julia set includes two quadratic Julia sets

The abstract for "Quartic Julia sets including any two copies of quadratic Julia sets" in Discrete & Continuous Dynamical Systems - A,36,4,2103,2112,2015-9-1, by Koh Katagata, states [...] for ...
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38 views

Why are Hausdorff dimension and Minkowski (box) dimension not equivalent?

I am putting the finishing touches on my master's essay for graduation this semester and I want to end my paper with a proof of why Hausdorff dimension and Minkowski (box) dimension are different. ...
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0answers
42 views

The box dimension of the graph of a continuous function is $\ge 1$

In the book Interpolation and Approximation with Splines and Fractals by Prof. Massopoust one studies the following theorem with regard two the box dimension of affine fractal interpolation functions: ...
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23 views

Computing the lipschitz constant of an affine IFS

In Massopoust, Interpolation and Approximation with Splines and Fractals, page 184, the author gives the construction of certain "fractal interpolation functions" as follows: Let $X = [a,b] \times \...
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56 views

How can I generate this pattern?

I've been working on a problem (full problem at the bottom) and as a result I've generated some binary matrices with a peculiar pattern. Here is the pattern for $n=29$ (the largest $n$ before my ...
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1answer
38 views

A pointless characterization of the relation between a fractal and its code space.

In Massopoust's Interpolation and approximation with splines and fractals, one can read this theorem: Given an hyperbolic IFS $(X,\{f_i:i=1,\ldots,N\})$ and denoting its code space by $\Sigma_N$ ...
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1answer
31 views

The relation between a fractal and its code space.

I'm having problems understanding a proof in the following references: Massopoust, Interpolation and Approximation with Splines and Fractals, page 159. Barnsley, Fractals Everywhere, page 124. The ...
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2answers
42 views

What are Fractals - Beginner's version

I know some math. Whenever I come across the word fractals, I see cool colourful pictures. So I googled it and can't understand a word about it in Wikipedia. Can anyone explain me what fractals are in ...
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28 views

Multifractal analysis - q parameter

If I am viewing a certain area of a time series that I have performed a multifractal analysis $(h(q,s))$ of how do I determine which $q$ parameter I should look at to determine this area's Hurst ...
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1answer
30 views

Explicit homeomorphism between product of Cantor sets onto the Cantor set

I want to find an explicit homeomorphism $\varphi: C\times C \longrightarrow C$ where $C$ denotes the Cantor set. The hint is to use the base $3$ expansion of the elements of the Cantor set. My two ...
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30 views

Self-similarity dimension for IFS or attractor of IFS

We can have same attractor from different iterated function systems. So i wonder about self-similarity dimension concept is for IFS or its attractor. We know that when IFS satisfies the open set ...
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7 views

Hölder exponent and its relation to multifractality

Suppose that for a discrete set of x=0,1,2,...,N, I have f(x). They show multi-fractality properties: $P_q = \sum_{x=1}^N f(x)^{2q} \propto N^{-\tau}$ $\alpha = \frac{d\tau}{dq}$ $f(\alpha) = q \...
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1answer
93 views

ODE solutions - how un-differentiable can they be?

Suppose we have a first order Ordinary Differential Equation, $y^\prime(x)=f(x,y)$. On the face of it, it looks as though any solution $y$ should be differentiable throughout its domain, but this may ...
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0answers
14 views

Hölder exponent and its relation to multifractality

Suppose that for a function $f(x)$, I have discrete set of numbers for $x$ with size $N$ (and the corresponding values for $f(x)$). They show multi-fractality properties: $P_q = \sum_{x=1}^N f(x)^{2q}...
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Is the function $\sinh(x)/x$ fractal at small values of $x,y$ or am I seeing rounding errors in computation?

I asked Wolfram Alpha to give me a solution to an integral function https://www.wolframalpha.com/input/?i=(integral+exp(-mx)+dx+between+x-a+and+x%2Ba+)%2F(2+a+exp(-mx)) and it gave me an expression ...
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0answers
18 views

Biased estimates of Hurst exponent in R/S analysis

I've used the standard R/S algorithm for estimating the Hurst exponent in Mathematica*, and tested it on fBm and fGn for $H\in\{0.05,0.1,\ldots,0.95\}$, generating 1000 time series for each $H$. The ...
6
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1answer
107 views

What is the smallest number of $45^\circ$–$60^\circ$–$75^\circ$ triangles in non-trivial substitution tiling?

Let base = $45^\circ$–$60^\circ$–$75^\circ$ triangle. Over at What is the smallest number of bases that a square can be divided into? it was determined that 23 base were needed to make a $45^\circ$–$...
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1answer
54 views

Information on this fractal?

I was bored and started thinking of fractals and decided I scribbled what I thought could be one. $$a_{i+1} = (a_i - b_i) / c_i $$ $$b_{i+1} = (b_i - c_i) / a_i $$ $$c_{i+1} = (c_i - a_i) / b_i $$ I ...
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1answer
96 views

Mathematically-Oriented Treatment of Fractals (Text Book)

Can someone suggest me a mathematically-oriented introduction to Fractals? What I am looking for should introduce Fractals as well-defined mathematical objects (classic maths text book style) with a ...
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1answer
44 views

Packing Dimension as a Countable Union of Minkowski Dimension Sets

Is it true that if $X$ has packing dimension $\alpha$, then we can write $X$ as the countable union of sets $X_i$, where $X_i$ has Minkowski dimension $\alpha$. If not, which notion of dimension is ...
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150 views

Solution of advanced functional differential equation

Statement Consider an advanced functional differential equation $$ Lf(x) = f(2x+\pi)+f(2x-\pi),\quad L\equiv\frac{d^2}{dx^2}+1. \tag{1} $$ Let's construct a solution of Eq. $(1)$ with finite ...
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1answer
88 views

Mandelbrot set perturbation theory: When do I use it?

I have read the post on Perturbation of Mandelbrot set fractal. I will also be referring to the PDF by K.I. Martin on this topic. My question is to do with the precision laid out in the Martin paper. ...
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1answer
35 views

Isn't everything in real life fractal? [closed]

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 ...
6
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1answer
91 views

Four Dragon Curves are Edge-covering/Plane-tiling

Four Dragon curves generating outwards from the same vertex will traverse every edge of a grid exactly once (and as a consequence will be plane-tiling as well). I am captivated by this fact, and ...
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0answers
100 views

Partial differential equation with a nowhere differentiable boundary

Consider the Dirichlet boundary value problem of the 2-dimensional Laplace's equation. When the boundary is piecewise smooth, it can be solved by the Green's function for the double layer potential. ...
3
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2answers
69 views

Is there a technique to exactly calculate the Hausdorff dimension of the border of this fractal?

I came up with a self-similar fractal that fits into itself like a jigsaw puzzle. While the surface area is clearly 2-dimensional, it is unclear to me how to compute the Hausdorff dimension of the ...
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1answer
22 views

Definition of Area of a Two Dimensional Line

I learned recently that lines such as the boundary of some Julia sets and the Hilbert curve have area. I was wondering what the strict definition of such an area would be, and I intuitively came up ...
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0answers
25 views

Limit behaviour of the box-counting dimension.

I read this in one of Falconer's books on Fractal Geometry: The definition of box-counting dimension is essentially saying that $N_{\delta}(F) \delta^s \to_{\delta \to 0^{+}} \infty$ if $s < ...
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0answers
38 views

Mandelbrot Boundary Area

I might be entirely off an my assumptions, but the following has led me to a question. The Mandelbrot set is contained by an border of infinite length. Said border is 2-dimensional. The Hilbert space-...
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2answers
411 views

Realistic 3D fractal Christmas tree

I would be interested to see a realistic 3D fractal-generated Christmas tree. The best I could find is the Adobe Stock image below.                   &...
5
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1answer
88 views

Exact value of Hausdorff measure of two dimensional Cantor set

Let $\mathcal{C}$ denote the classical Cantor set, then it is well-known that $\mathcal{C}$ has Hausdorff dimension $\alpha = \ln 2 /\ln 3$, and its $\alpha$-dimensional Hausdorff measure is $\mathcal{...
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67 views

If $s_i < 1$ and OSC holds then $\sum_{i = 1}^N s_i^n < 1$.

Let $X \subseteq \mathbb{R}^n$ is closed and $S_N = \{f_i\}$ is a set of $N > 1$ similitudes with $s_i = Lip(S_i)$. $S_N$ satisfies the open set condition iff there exists a nonempty bounded ...
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24 views

expressivity of graph directed IFS and L-systems

In my answer to Does there exist a L-system for this Pierced Diamond Fractal? I asserted that graph directed iterated function systems of similarities have equivalent expressive power to L-systems. ...