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

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2
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72 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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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 ...
5
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1answer
166 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
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 ...
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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
48 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
19 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 ...
3
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1answer
221 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 ...
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0answers
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 ...
1
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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 ...
1
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0answers
37 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
24 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 ...
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 ...
1
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1answer
26 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
57 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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0answers
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 ...
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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 ...
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 ...
0
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1answer
39 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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0answers
70 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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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 ...
0
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2answers
39 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 ...
0
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1answer
20 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!
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 ...
2
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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 ...
0
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1answer
16 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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2answers
28 views

There exist fractal with similarity dimension between 0 an 1?

How to prove that there exist a fractal with similarity dimension D = x, where x is between 0 and 1?
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1answer
12 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 ...
0
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1answer
75 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 ...
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2answers
17 views

Set characterization

I need to understand what are the elements of the set $\left[0,1\right]$ whose non-terminating decimal expansions contain only the digits 3, 5 and 7. I suppose it is an auto-similar set, up to an ...
1
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1answer
24 views

Interpreting a nonstandard definition of a tree

Definition: A tree is a triple $(T,\sigma,\pi)$ where $T$ is a set and $\sigma$ is a so-called successor function from $T$ to the set $T^*$ of all nonempty subsets of $T$, together with a surjective ...
2
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0answers
33 views

Showing a point is a limit point of a set

For sake a clarity, when I say a "limit point", the definition I am using is: $x$ is a limit point of a set $A$ if there exists a sequence $\{x_k\}$ such that $x_k \neq x$ and $x_k \rightarrow x$ ...
3
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2answers
65 views

Fractals with Moduli in Pascal's Triangle

I'm working through a problem for my graduate math class and am hitting a wall. Here's the problem: For the first 10 lines of Pascal's Triangle, replace the odd numbers by black squares and the even ...
1
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1answer
59 views

Box dimension of $\{\frac{ 1}{5^n} : n \in \mathbb{N}\}$

I am working through a first course in Fractal Geometry, and have encountered a problem that has asked me to calculate the box-counting dimension of $F=\{ \frac{1}{5^n} : n \in \mathbb{N}\}$ However, ...
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3answers
96 views

Union of two Cantor sets is also a Cantor set

To prove that a set is Cantor, we have to prove that it's closed and bounded (compact), contains no intervals of positive length, and is perfect. The union of two Cantor sets would also be compact ...
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1answer
21 views

Cantor set and ternary expansions

I'm trying to show that if $x \in [0,1]$ has a ternary expansion consisting only of $0$'s and $2$'s, then $x$ is in the Cantor ternary set. The proofs I've seen typically rely on induction. Is it ...
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2answers
38 views

Are there methods to recursively calculate the decimal expansion of real numbers?

Using the concept of self-similarity, it's possible to encode the decimal expansion of a number as a sort of 'fractal' object. For instance, consider the sequence, $$(1) \quad C_0=0.1, \ C_1=0.101, \ ...
2
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1answer
42 views

Definition of Minkowski dimension

I'm trying to understand the definition of Minkowski dimension given by Wikipedia here. In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting ...
3
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1answer
66 views

Is there a rigorous mathematical definition of the Koch curve?

Is there a rigorous mathematical definition of the Koch curve? Wikipedia says that mathematics is not given a rigorous formal definition of a fractal in general. And also I have not found a strict ...
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0answers
38 views

Non-empty interior of a Mandelbox

I have a question which has been interesting me for some time, namely when a Mandelbox has a non-empty interior. A definition of the Mandelbox may be found here. Essentially, it is defined by a ...
8
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2answers
188 views

How to avoid overlap in circle fractals?

I had asked this on reddit and someone suggested that I try here: Assuming that the pattern in the image below continues infinitely, how much would each generation of circles have to decrease to ...
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0answers
23 views

On a curve every point of which is a point of ramification

The title of my post is the same as the title of a known article written by Sierpinski where he introduced its famous triangle. In the book Handbook of the history of general topology by Lowen said ...
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1answer
52 views

Proving that we have $A \subset K$ where $K$ is self-similar

Let $f_1, \ldots$, $f_N:X\to X$ be contractions in the complete metric space $X$, and $K$ the self-similar set with respect to the $f_i$. If $A\subset X$ is compact and $$A\subset ...