Questions on fractals, irregular-looking mathematical objects that display the property of self-similarity.

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Why is there periodicity in the output of Richard Voss' fractional Brownian motion?

I am trying to figure out why the output of fractional Brownian motion (fBm) as described by Richard Voss (Random fractal forgeries. In: Fundamental Algorithms for Computer Graphics, R. A. Earnshaw ...
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0answers
19 views

Hausdorff dimension of Sierpinski triangle

https://en.wikipedia.org/wiki/Hausdorff_dimension#Behaviour_under_unions_and_products Wikipedia page says that if $ \underset{i \in I}{\cup} X_i = X$ and $I$ is countable then $dim_{Haus}(X) = ...
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1answer
104 views

Roots of iterations of polynomials

Let $f \in \Bbb Q[X]$ a polynomial, and let denote by $f^n$ the composition $\underbrace{f \circ \cdots \circ f}_{n \text{ times }}$. Let $R(f^n) \subset \Bbb C$ the roots of $f^n$. I'm interested in ...
4
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0answers
129 views

The Hausdorff dimension of the set of solutions of a system of coupled differential equations

I am interested in the relationship between non-linear differential equations and the Hausdorff, or fractal, dimension of the set of solutions. For example, the Lorenz Attractor, which is a standard ...
4
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0answers
77 views

Properties of King's Dream fractal

My question is focused on the King's Dream fractal, which can be defined as follows (nice pictures can be found here) : $$ \Omega = \{f^n(0.1,0.1) \;\vert\; n \in \mathbb N \} \quad ...
3
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1answer
81 views

Can a plane be split into three connected sets so that $\epsilon$-neighbourhood of any point of any one set also contains points of two other sets?

Math SE. This question was a shower thought of mine. I tried to come up with an answer by twisting comb spaces and cantor sets, but to no avail. I was educated as experimental physicist, so I ...
3
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1answer
224 views

How to draw a Mandelbrot Set with the connecting filaments visible?

The M-Set is connected. But the M-Set viewers I’ve found create cool pictures that don’t really show the connecting filaments. This mini-Mandel beetle should be connected to a larger min-Mandel by a ...
7
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2answers
409 views

Simplest way to determine if a number is a member of the Mandelbrot set?

I'm writing JavaScript code to plot the Mandelbrot set on an HTML5 Canvas element. (That's probably not relevant to the answer to this question). A core part of the problem is to write a simple ...
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2answers
403 views

Is The *Mona Lisa* in the complement of the Mandelbrot set.

Here is a description of how to color pictures of the Mandelbrot set, more accurately the complement of the Mandelbrot set. Suppose we have a rectangular array of points. Say the array is $m$ by $n$. ...
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0answers
139 views

How to find Misiurewicz Points without solving huge polynomials? (Mandelbrot Set)

Here is a plot of 17,723 Misiurewicz Points. The code below generates a set of polynomials u[m,n], the roots of which have periodicity (m-n) starting at iteration n. I stopped at 17,723 points ...
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1answer
203 views

Is the Hausdorff dimension less than the box counting dimension?

I have been asked to prove that for a bounded set $F\subset\mathbb{R}^n$, $\dim_H F\le \underline{\dim}_B F \le \overline{\dim}_B F$ where $\dim_H F$ is the Hausdorff dimension, ...
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1answer
171 views

Continuous path inside the Mandelbrot set connecting i to zero?

This relates to another challenge Question about drawing Mandelbrot filaments. It is possible to compute a formula for a continuous path inside the Mandelbrot Set connecting c=i to c=0? Obviously, ...
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0answers
30 views

Show that $E_\mu$ has no periodic points that are not fixed points

Problem statement: Consider $E_\mu(x)=\mu e^x$, where $0<\mu<1/e$. Show that $E_\mu$ has no periodic points that are not fixed points. It is in my understanding that what we need to show is ...
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2answers
15k views

Has anyone found a “pattern” in prime numbers?

Yesterday I was having some fun trying to look for some patterns in primes; and I think I found something interesting (to me at least). I still have not found any lists of patterns already found, ...
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0answers
41 views

Lower bound on dimension for nearest neighbor classifier to fail at k=1 and pass at k=3

What is the minimum dimensionality of a dataset of a finite number of points where 1-NN has an accuracy of 0% but 3-NN has an accuracy of 100%. This is certainly possible in 3 dimensions and my ...
5
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2answers
159 views

Does interval spacing effect Hausdorff dimension of Cantor set?

Let $C=\bigcap_{j=0}^{2^n}C_j$, $C_0=[0,1]$, and the intervals in the construction of each stage of $C_j$ consists of removing the center 1/3 from the $j-1$ stage intervals. In other words, the ...
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0answers
11 views

What is a hurst exponent in simple terms and what is the relation of it to fractals.

So I read this blog/paper recently and it is talking about the hurst exponent and it mentions that it can be used as an indicator of the fact that the time series can be predicted or has some sort of ...
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0answers
38 views

Are the vertices of a Voronoi diagram obtained from a Sierpinski attractor also a kind of attractor?

Trying to understand how the Voronoi Diagrams work I did a test generating the Voronoi diagram of the points obtained from The Chaos Game algorithm when it is applied to a $3$-gon. The result is a set ...
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0answers
49 views

converting to math from economics major

Recently, i'm majoring in honour track of economics taking econometrics statistics courses and minoring in mathematics taking advanced calculus, real analysis ,linear algebra courses. Upon research on ...
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1answer
36 views

Proving that a sequence is unbounded without knowing the sequence explicitly

Given that $f(x)=x^2+\frac{1}{4}$, there exists the iterated sequence ${f^{\circ n}(x)}_{n=1}^\infty$ (where $f^{\circ n}(x)$ is defined as $\underbrace{f(f(f...(x)...))}_{n\text{ times}}$), which is ...
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2answers
33 views

Proving that a sequence is bounded without knowing the sequence explicitly

Given that $f(x)=x^2+\frac{1}{4}$, there exists the iterated sequence ${f^{\circ n}(x)}_{n=1}^\infty$ (where $f^{\circ n}(x)$ is defined as $\underbrace{f(f(f...(x)...))}_{n\text{ times}}$), which is ...
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0answers
20 views

Random process theory: probability distribution of height vs summits

Imagine I have a matrix of height values ($z$), e.g. a surface height topography. This surface is a random process: randomly rough isotropic surface with Gaussian distribution. What is the difference ...
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32 views

Calculating Hausdorff dimension from the definition

I am currently approaching the study of fractals and I have a bit of a problem understanding Hausdorff dimensions. In detail, I don't understand how I do use the definition to calculate the dimension ...
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1answer
19 views

Hausdorff Dimension of $f(F)$

I am currently working through several problems regarding the following fact: For $F \subset \mathbb{R}$, $f: F \rightarrow \mathbb{R}$, we have that $dim_H(f(F)) \leq dim_H(F)$ I am fine with ...
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0answers
26 views

Simulation of brownian motion and fractional brownian motion

It's easy to simulate a path of a brownian motion with the method explained in Wiener process as a limit of random walk: ...
1
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1answer
27 views

Generalizing the Cantor Set to the $n$-dimensional plane

I am interested in how to describe an $n$-dimensional cantor set. I think that it may be a good idea to develop the Cantor Set on the two-dimensional plane at first, but I am having issues figuring ...
2
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1answer
35 views

Discrete systems with complicated basin boundaries?

I am trying to come up with the strategy to write my Master's thesis in mathematics. At the moment it is as follows: Finding a (preferably) discrete dynamical system that possesses at least 3 ...
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0answers
49 views

Area of 2D fractal?

Some fractals have a whole fractal dimension, can their measure be calculated? For example if you start with a tetrahedron of a given size and recursively remove the central octahedron leaving 4 ...
3
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0answers
108 views

What is asymptotics of this oscillatory double sum? (Fractal Dimension problem)

The term Gibbs Phenomenon refers to the peculiar way Fourier Series behave at sharp changes in a function's value. However, this problem becomes particularly annoying to deal with when trying to ...
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1answer
13 views

Finding the similarity dimension of a variation of the Cantor Set.

If we take the Cantor set and instead of removing the interval $[1/3, 2/3]$, we remove the open interval $[x,1-x]$, with $0<x<1/2$, will the similarity dimension change? What I think is that we ...
3
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1answer
37 views

Centroid of Mandelbrot Set

How to find the geometric centroid of Mandelbrot Set?
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1answer
45 views

Asymptotic rendering time for koch snowflakes

I posted a similar question on stack-overflow, but this may be a more proper forum since it is more math-related than programming related: I'm currently working through the online course material for ...
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0answers
46 views

Relations between the 2-disc operad and fractals?

As you can see, as of late I opened a thread on n-disc operads: Clarification regarding little n-discs operads The thing is, those drawings there could somehow be construed in the real world as ...
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1answer
21 views

Hausdorff dimension of a countable set

I don't understand why the Hausdorff dimension of a countable set in $\mathbb{R}^n$ is $0$. Can someone please give me a hint? Thank you!
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1answer
27 views

Hausdorff dimension calculation related to Jarnik's theorem

Let $$F=\{x \in R:||qx||\le2q^{1-\alpha}\log q \text{ for infinitely many } q \in \mathbb{R}\}$$ Show for $\alpha>2$, $\dim_H F\le 2/\alpha$. Jarnik's theorem (By Falconer) says: Suppose ...
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1answer
58 views

fractal curve and fractal set

Would it be correct to say that all fractal curves are fractal sets, but not all fractal sets are fractal curves? If that is correct, what would be an example of a fractal set that is not a fractal ...
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1answer
40 views

Does this simple pattern show that all integers are part of a fractal set?

The simple pattern Count up through nine, then on ten you wrap to the next line and keep going until you've counted nine more integers - then wrap to the next line. Repeat ad infinitum. ...
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44 views

Hausdorff dimension calculation of union of sets

$F$ is a Cantor set in $(0,1)$, $\dim_HF=1/5$. What's the $\dim_HE$ where $E=(F×R)\cup(R×F)$? By the product properties, I know that and $\dim_H(F×[0,1])=6/5=1+1/5$, which is the sum of hausdorff ...
2
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1answer
47 views

Fractal signal analysis

What kind of results can be proven about continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which present some sort of fractal behaviour / self-similarity? Do you have some textbook ...
1
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2answers
81 views

Revolution of fractal

How to find the volume and surface area of a shape which made from revolution of Koch Snowflake? (I think the surface area will be an infinity, because length of the Koch snowflake is infinity.) And ...
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2answers
40 views

Adding Examples to Math Paper

I'm writing a paper on the Mandelbrot set and want to add some examples of iteration to it to show values that are members of the set and to show values that are not members of the set. What's the ...
3
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0answers
71 views

Is it known whether the boundary of the Mandelbrot set is not continuous?

I might be missing something obvious here, but my understanding is that nobody currently knows whether the boundary of the Mandelbrot set is a Jordan curve because otherwise we would know that the ...
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2answers
1k views

Properties of the Mandelbrot set, accessible without knowledge of topology?

Are there any properties of the Mandelbrot set that can be analysed without a knowledge of complicated topology? Considering the fact that the set is based on a quadratic function, are there any ...
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3answers
67 views

Looking for fractals which are computationally demanding and preferrably parallelizable.

Oh hello guys. I am in the middle of challenging myself to putting my computer and math skills together, trying to build a small hobby computational cluster. Being interested in fractals for a long ...
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1answer
586 views

How to convert a right angled triangle into a equilateral triangle?

I want to use the Apophysis program to make a right angled sierpinski triangle into an equilateral triangle. But how can i do so? i have tried the second picture one but that is not correct.
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4answers
97 views

Creating fractals through computers

What are some beginner softwares for creating fractals on computers?
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2answers
333 views

How to compute a negative “Multibrot” set?

The Mandelbrot set is defined as follows: given the function f(z, c) = z2 + c, a number z in the complex plane is in the Mandelbrot set if and only if the sequence ...
2
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0answers
21 views

Showing that points are in the Mandelbrot set

I am given ( a simplistic definition I think ) of Mandelbrot set: M- set of complex numbers $c \in \mathbb{C}$ s.t. the sequence $(z_n)$ is bounded where $z_0=0 , z_{n+1}=z_n^2+c $ Need to show ...
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1answer
18 views

Cantor set and triadic expansions

I'm trying to prove that the Cantor set is equal to a certain set of 'escape points' for a mathematical feedback system. In this proof I'm going to need the fact that every element of the Cantor set ...
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1answer
77 views

Why is the Mandelbrot Set to the power of 2?

I've been looking at the Mandelbrot set and playing around with the powers on it. In the sense that I know the equation is $z^2 +c$, and I've been changing up the $2$. As it happens gradual increases ...