Questions tagged [fractals]

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

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Geometry of Set of States of Random Process

Consider the following random process in the unit circle. Starting at $(0,0)$, move up, down, left, right a distance $1/2$ with equal probability. At the next step there are three possibilities: stay ...
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43 views

What is the average distance between two randomly chosen points on a Sierpiński triangle?

Since a Sierpiński triangle is made up of three smaller versions of itself, an algorithm made to pick random points on the triangle might go like this: Select one of the three Sierpiński triangles ...
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Difference between small and large inductive dimension

As I understand it, the small and the large inductive dimension differ only in the second property (taken from here). The second property of the small inductive dimension is as follows: Definition: $...
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22 views

Lindenmayer system for Pólya space filling curve

I am considering the special case of an isosceles right triangle. The pattern seems that starting with $+F$ for odd depth of recursion and $F$ for even depth of iteration where + means rotate ...
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33 views

3D spiral that follows a fractal spiral geometry

I want to model a spiral mathematicaly like this: A spiral like that of the 3d spiral of a single planetary motion through space except I want the trajectory of it (the direction of the spiral running ...
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80 views

Can one representatively sample the Julia set

Frankly, I am not sure what it means to representatively sample such a fractal set, that is my question. In order to gain some understanding, I looked at large unstable cycles of the function $f:z\...
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1answer
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Is this a probability measure on the cantor set?

$c$ is the Cantor function, $C$ the Cantor set, and $\rho$ the Lebesgue measure We consider $\mu$ as $\mu(A) = \rho(c(A \cap C))$ for each each element of the tribute Is $\mu$ currently defining ...
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1answer
50 views

What is the difference between the small inductive dimension, the large inductive dimension and the lebesgue covering dimension?

I am studying fractals autodidactically (I have never had any topology discipline) and according to Mandelbrot: a fractal has a Hausdorff dimension that exceeds the topological dimension. I understood ...
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22 views

Borel probability measure that takes value zero on all subsets with lower Hausdorff dimension

Let $X\subseteq \mathbb{R}^n$ be a compact subset, assume $X$ has Hausdorff dimension $0<s<n$. I want to find condition on $X$ in order to guarantee that there exists a Probability Borel measure ...
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1answer
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Do fractals necessarily have to be self-similar? What is the definition of a fractal? Is this figure a fractal? [duplicate]

I am doing a dissertation in Materials Engineering. I obtained the following images regarding crystal growth: In the literature they usually call these structures "fractals" and calculate ...
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How to optimally adjust the probabilities for the random graph-directed IFS algorithm?

In this related question it is shown that the optimal probabilities for an iterated function system of similarities (that satisfy the open set condition) with contraction ratios $r_k$ and fixed point ...
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1answer
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What is the dimension of the “Julia set” generated by inverse iteration and why do I get numbers different from Hausdorff dimension

On the Julia set, the iterating the function $f:z\mapsto z^2+c$ generates a sequence of points on the Julia set. Because roundoff errors would be expected to cause the sequence to fall into an ...
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How does the ABC-conjecture sound?

One version of the abc-conjecture can be stated in terms of the positive definite kernel over the natural numbers: $$K(a,b) = \frac{\gcd(a,b)^3}{ab(a+b)}$$ Based on this kernel, I wrote a small python ...
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(fractals) What is the link between my bifurcation diagram of a population and the mandelbrot set?

This is my first time here as I'm investigating fractals. I started of by writing about the similarity between a bifrucation diagram of a population, using formula $z_{n}=a*z_{n-1}*(1-z_{n-1})$. I ...
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1answer
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Are the invariant sets of all iterated function systems necessarily fractal?

An iterated function system is defined as a finite set of contraction mappings, defined over a complete metric space $X$, and iteration is defined as sequential composition of these contraction ...
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Jarnik's theorem in Falconer's book: what number?

Simple reference question: What is the theorem number/page number of Jarnik's theorem in Kenneth Falconer's Fractal Geometry (third ed.)? Full reference: Falconer, Kenneth, 2014. Fractal geometry: ...
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How does this alternate construction of the Sierpinski triangle differ from the usual one?

Can you plot the Sierpinski triangle on the Cartesian plane and define the "removal of triangles" in terms of the points being removed? Imagine an equilateral triangle with vertices at $(-1, ...
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Equations for Mandelbrot bifurcation diagram

The Mandelbrot set is the set of all complex numbers $c$ that cause the function $z_{n+1} = z_n^2 + c$ to remain bounded within a circle of radius 2 when iterated from $z_0 = 0$. Looking at only the ...
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28 views

Union of deletion Cantor sets

I know that it is possible to have deletion Cantor set which are of non-zero measure (fat Cantor sets) furthermore it must never of measure one since it would result in a contradiction (as Cantor is ...
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Test to determine if a point is inside a cardioid whose cusp is not at the origin

The Mandelbrot set is the set of all complex numbers $c$ that cause the function $z_{n+1} = z_n^2 + c$ to remain bounded within a circle of radius 2 when iterated from $z_0 = 0$. Plotted in the ...
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Notion of fractal dimension which distinguishes point and square from square

I am new to fractal dimensions and am looking for a notion of fractal dimension which would help me distinguish between the following two sets on the cartesian plane: \begin{align}A&=[0,1]\times[0,...
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1answer
36 views

Simplified expression for centers of period-three Mandlebrot bulbs

The Mandelbrot set contains three regions (two bulbs and a cardioid) with periods of three. These regions each contain a fixed point which is a root of the expression $x^3 + 2x^2 + x + 1$. The fixed ...
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Finding attractors / fixed points for the circumference of the main bulb of the Mandelbrot Set

The Mandelbrot set is the set of all complex numbers $c$ that cause the function $z_{n+1} = z_n^2 + c$ to remain bounded when iterated from $z_0 = 0$. Plotted in the complex plane it includes a main ...
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How to estimate distance from root to nearest immediate basin boundary for Newton's method in one complex variable?

Context: I want to check that the atom domain size estimate is smaller than the inradius of the Newton immediate basin, for centers of hyperbolic components in the Mandelbrot set, and thus justify ...
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1answer
52 views

Is there any way to tell if any givin Number will either blow up or tend towards a Value when squared?

Ok so as a Set Up, i want to do the all time classy, the Mandelbrot set. As far as i know, the equation $$Z \rightarrow Z²+C$$ Just asks the question "Ok givin any Complex Number, if you Square ...
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Can different fractals have the same fractal dimensoins?

I first noticed that fractals can be defined by real numbers from this youtube video: https://youtu.be/gB9n2gHsHN4 My question is that: Are these fractal dimensions unique to the fractals, Can two ...
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A generalization of contraction mapping principle

Let $f_1,\dots,f_n:R^d\rightarrow R^d$ be diffeomorphisms. $\exists 0<c<1$ s.t. $\parallel Df_i(x)\parallel\leq c,\forall x\in R^d,\forall i$. Where the norm is the operation norm in $L(R^d)$. ...
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Continuous (Smooth) Fractal Zoom

This is my first ever post, I made an account to ask this question. Could have put it on a code forum but thought this challenge would be better suited to a mathematician with a programming foundation....
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When can the $n$-cycles of $z\mapsto z^2+c$ be distinguished by their sums?

Question: Which pairs $(n,c)\in\mathbb{Z}_{\ge 1}\times\mathbb{C}$ have the following property: Let $\sigma$ be the usual cyclic permutation on $1,2,\ldots,n$, and let $z_1,\ldots,z_n,w_1,\ldots w_n\...
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Map formulas or terms (naturally) into 2D plane

Is there known any (natural) way of mapping terms of any formal language onto (or into) a 2D real plane? Am aware of Gödel numbering or Binary combinatory logic and the question is a bit similar to ...
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Sierpinski Triangle as Finite Union of Dendrites

Can the Sierpinski Triangle be written as a finite union of dendrites? If so, can it also be verified what the minimal number is (assuming you can't do it with just two)? This is a small piece from a ...
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Matching an image with a fractal, advice needed

Introduction I'm trying to find a fractal function that spells out a name on the complex plane. I'm doing this by numerically 'training' a program to match the desired image / name. I don't expect any ...
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54 views

Integration of function using Hausdorff measure of Cantor Set

I am learning geometry on fractal shapes, along with how fractional calculus can relate to said geometry. At the moment I am trying to understand integration over a Hausdorff measure. According to ...
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Where does these nice patterns come from?

I've played around some with complex numbers in python and created a program that creates beautiful patterns using $E_n = e^{i(2\pi)/(R_n+1)} = A+Bi$ where R is a set of values between $0$ to $2\pi$ ...
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1answer
28 views

Is there a Weierstrass function with asymptotic line? [closed]

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 ...
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1answer
61 views

Looking for an example of a function with a Julia set of positive Lebesgue measure.

I am looking for an example of a function with a Julia set of positive Lebesgue measure. I am certain these exist but cannot find a single example in the papers I have looked at. The function does not ...
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55 views

Why are the formal and informal definitions of fractals equivalent?

A fractal is usually defined to be a self-similar shape (this is the informal definition). But, the formal definition is: A fractal is a set for which the Hausdorff dimension strictly exceeds the ...
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Koch-like fractal of the dodecahedron: Does this object already exist?

Platonic solids are made of squares, triangles and one of them is made of pentagons. If the faces of the cube are taken while preserving its vertices, each face can be divided into smaller squares ...
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1answer
60 views

Proving that the limit of the Mandelbrot polynomial's co-efficients are the Catalan numbers.

According to Donald D. Cross, the characteristic polynomial of the $n^{\mathrm{th}}$ iteration of the Mandelbrot function, $z_{n+1} = z_n^2 + c$ where $z_0 = 0$, gives the Catalan numbers as the ...
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Has anyone studied the set $\Big\{\sum_{k<t}e^{ik^2}\ |\ \forall t\in\mathbb{N}\Big\}$

As a curiosity and a test of the pseudo-random nature of trigonometric functions evalutated at integer-squares, I inquired to see what it would look like to graph the subset of $\mathbb{C}$ descibed ...
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Does any cross section of the Sierpinski carpet yield the Cantor set?

My gut tells me that if you took a vertical cross section of the Sierpinski carpet at an $x$-coordinate in the Cantor set, the cross section would again be the Cantor set. However, I'm at a loss for ...
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Combinatorics / fractal geometry.

I have a question on combinatorial fractal geometry. I drew the dragon curve on a piece of graph paper. For this particular figure the total distance of the sides increases by a factor of squareroot(2)...
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1answer
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Fractal Surface Area of Lung

I'm preparing for an upcoming exam and one of the questions is the following; "A typical human lung has a volume of approximately 5 litres, and a fractal dimension of approximately 2.97. One way ...
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The Coastline Paradox and The Mandelbrot Set

I've been doing some research on the Mandelbrot Set and have discovered that it's boundary is a fractal. I was wondering if the Coastline Paradox can be applied to the boundary of the Mandelbrot Set? ...
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45 views

Parametrizing fractals and space filling curves

I am a 2nd year undergraduate doing a BS-MS in an Indian Institute and am taking a course in Elementary Differential Geometry (studied till the basics of surfaces). Our instructor has assigned a term-...
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1answer
39 views

Is there any way to figure out if a system of two recurrence relations will converge?

Context: I'm trying to write code that generates 2D Iterated Function System (IFS) fractals based on some affine transformations. I want to generate the fractal till convergence when possible. The ...
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1answer
20 views

How to calculate Assouad Dimension of sets

I am trying to work with sets $F$ as subsets of $\mathbb{R}^n$ and their Assouad dimension. I am a bit stuck on how to apply the definition as it asks that We find the infimum of possible $\alpha$ We ...
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Showing that the Sierpinski's Carpet is path-connected.

I'm trying to show the path-connectedness of the Sierpinski carpet and so far all I got is that at any iteration the sides of the carpet are always gonna be there so if there is a path from a point to ...
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Can the fractal dimension of a coastline be less than 1?

I am currently investigating the fractal dimension of the coastline of the island the Palm Jumeirah, including the crescent. Using the Hausdorff method I have reached an answer of 0.879. This is less ...
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Given an analytic function with $f(0) = 0$, $\Vert f'(0)\Vert= 1$, when I can I find an open invariant set?

If $f:\Omega\rightarrow\mathbb{C}$ is analytic, where $0\in\Omega$, and $f(0) = 0$ and $|f'(0)| = 1$, what other conditions are necessary for finding an invariant open subset of $\Omega$? In ...

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