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

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7
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1answer
78 views

Convex hull of the Mandelbrot set

What is the convex hull of the Mandelbrot set? I know that the leftmost point is $c=-2$ and I thought the extreme vertical points were $c=\pm i$. Sheldon's answers says they're not. I think that the ...
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0answers
53 views

Every projection of the square of the middle thirds Cantor set contains an interval

Let $C_\lambda$ the cantor set which is defined by the IFS $\{\lambda x,\lambda x+(1-\lambda)\}$ and also let $E=C_\lambda\times C_\lambda$.Suppose $\lambda =\frac 1 3$, we get the standard ...
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0answers
32 views

Proving that plane - cantor - set contains an interval

Let $C_\lambda$ the cantor set which is defined by the IFS $\{\lambda x,\lambda x+(1-\lambda)\}$ and also let $E=C_\lambda\times C_\lambda$. Denote the orthogonal projection of the set from the ...
2
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1answer
37 views

Slightly Chunky Cantor Sets

I'm familiar with the construction of so-called "fat" Cantor sets (e.g. the Volterra construction), where a Cantor-type construction is used to construct a nowhere-dense set of positive Lebesgue ...
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1answer
25 views

Fractal Dimension of $C_{\frac{1}{3}}\times[0,1]$

I wonder what is the dimension of the fractal set given by the product of the unit interval $[0,1]$ by the thirds-cantor-set ($C_\frac{1}{3}=\bigcap_n C_n$ where $C_0=[0,1],C_1=[0,\frac 1 3]\cup[\frac ...
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0answers
21 views

Intuitive meaning of fractal dimension.

I'm studying M. Barnsley's book 'Fractals Everywhere', but I'm stuck in the chapter 'Fractal Dimension'. Suppose $(X, d)$ is a complete metric space and let $A \in \mathcal{H}(X)$ be a nonempty ...
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1answer
29 views

Iterative function with $z_{n+2}$

I'm currently playing arround with my custom fractal renderer and on Math SE in this answer Américo suggested the following function: $z_{n+2}=z_{n+1}^{3}+c^{z_{n}}$ But to get the first value I'd ...
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1answer
43 views

Is this c the same as that c?

Are the highlighted $c$'s the same or should it be $c_1$ and $c_2$.
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31 views

Hausdorff dimension of a ball

Let $\{f_1,\dots,f_m\}$ be an IFs and $E_n$ be the associated self similar set. It's known that $E_n$ is a union of disjoint balls $B(x_i,R\cdot r^n)$ (balls with same radius but not the same ...
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1answer
29 views

Proving that the Bernoulli self similar measure is doubling

Let $\mu_p$ a measure which is the push forward of the bernouli product measure $(p,1-p)^\mathbb N$. Let S=$\{f_1,\dots f_m\}$ an IFS, a system of functions with attractor $K$, means ...
2
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1answer
99 views

Relating convergence theorem for Newton-Raphson method to Newton fractal

I have created a Newton fractal (below) using the Newton-Raphson method to find the five solutions of f = (z^5-1) The convergence theorem of Newtons method say ...
4
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2answers
92 views

Is it possible to prove that some point belongs to Mandelbrot set? Is this an example of Gödel's theorem?

Everybody knows about Mandelbrot set drawing computer programs. Program takes some point, builds sequence from it, and if found that sequence goes out of circle with 2 radius, then knows that this ...
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2answers
72 views

Upper step function of the Cantor set.

Let C be the Cantor set and let $f:[0,1]\rightarrow\mathbb R$ be determined by $$f(x) = \begin{cases} 1, \quad \text {if}\ x\in C\\0, \quad \text{if}\ \ x\notin C\end{cases}$$ Find an upper ...
2
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0answers
29 views

What are the most recent devopments with applying fractals to economics?

I was researching fractals for my senior mathematics presentation and discovered that one of the most recent pioneers in that section of the field was able to apply fractal mathematics to the field of ...
2
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1answer
87 views

The measure generated by the Cantor staircase and the intersection of the Cantor set with its translate

Suppose that $T$ is the shift $\bmod 1$ of the Cantor set by an irrational number $\alpha\in (0,1)$. Consider the measure $\mu$ on the interval $[0,1]$ generated by the Cantor staircase. I'd like to ...
2
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1answer
35 views

What does “The closure of the shift-orbit of the Fibonacci word” mean?

Im trying to translate an article about rauzy fractal. But since my English is not good enough I cant understand this paragraph: ...
0
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1answer
44 views

Hausdorff Dimension for Brownian motion over [0,1]

I am trying to calculate Hausdorff dimension for the trajectory of Brownian motion over $[0,1]$. I read the book of Morters and Peres and know that the dimension will be $\frac{3}{2}$. I tried to use ...
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13 views

Understanding the expression of fractal dimension in plants

I just finished a small, demo exercise on fractal dimension of a plant by using MATLAB and box-count method. There were two different treatments. A plant treated with a specific hormone and a plant ...
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0answers
26 views

Finding countable compact set s.t $\underline{\dim}_M(K)\lneq\overline{\dim}_M(K)$

Im trying to find a countable compact set such that $$\underline{\dim}_M(K)\lneq\overline{\dim}_M(K)$$ I tried thinking about Koch curve, sierpinskii gasket and carpet, Bedford-McMullen carpet and ...
6
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2answers
87 views

Count with only certain digits allowed - And feel a fractal

I have a friend ~200 years old mathematician who has forgotten some digits and now he counts things in very strange manner: when he is going to count the $n$-th thing and $n$ contains a digit he ...
0
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0answers
42 views

What are the most advanced mathematics that apply to architecture?

I am doing an investigation about the mathematics in architecture, yet I still haven't quite discovered a higher-level mathematical concept that is actually used in the design and construction of ...
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0answers
18 views

mid-point free sets of maximal dimension

Does there exist a subset $S$ of the unit interval of Hausdorff dimension greater than $\frac{\log2}{\log3}$ (i.e. 'larger than' the Cantor set) such there do not exist three points in S one of which ...
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20 views

fractal dimension estimation for an image

Apparently it is possible to estimate the fractal dimension of an image by using the following method: from: Fractal-based texture analysis, K L Chan, 16-20 Nov 1992, pp. 102 - 106 vol.1, in ...
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33 views

Do fractals really happen in nature?

We live in a 3 dimensional world. So, line and plane as 1 and 2 dimensional objects do not exist in reality although using these concepts are useful for modeling some problems such as motion in one ...
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0answers
33 views

Defining strict self-similarity

I have been reading through John Hutchinson's paper "Fractals and Self-Similarity" and some other things, and I haven't really found a definition of strict self-similarity to work with that makes much ...
7
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1answer
50 views

Is every point on a Menger Sponge visible from the outside?

Choose an arbitrary point on the surface of a Menger Sponge. Can you find a straight line starting at that point and extending beyond the sponge that doesn't intersect the sponge anywhere else? That ...
13
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1answer
153 views

Koch snowflake versus $\pi=4$

The only proof I could find of the Koch snowflake having infinite perimeter was by calculating the perimeter $P_n$ after the $n$th iteration $$P_n = 3s\left(\frac{4}{3}\right)^n,$$ where $s$ is the ...
2
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2answers
49 views

How does mathematics fit into fractal generation for computer graphics?

I have to do a research paper on any mathematical concept. The mathematical concept must be complex, so I thought fractals would be a good choice (I was told it was a complex idea). I have been ...
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0answers
28 views

Is there a natural interpretation of automorphic forms in terms of fractal geometry?

Disclaimer : This question is rather vague and thus might not be suitable for MathOverflow, so I prefer to ask it here. According to Wikipedia, an automorphic form is, roughly speaking, a ...
1
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1answer
57 views

Comparing fractals

Is there a way to compare if two fractals are "isomorphic"? I'll give an example of what I mean. Consider the following two fractals. First we have the Sierpinski triangle, and next we have the ...
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1answer
39 views

Hausdorff dimension of a Modified Cantor like set

Suppose you have the unit interval $[0,1]$. For the first iteration you remove the segment $(1/5,3/5)$. So you are left with two intervals of lengths $1/5$ and $2/5$. You now repeat the process on the ...
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0answers
37 views

Show the Hausdorff dimension of a set of numbers with digit 5

I can show heuristically that the answer is $log(9)/log(10)$ but I am struggling to prove this rigorously. This is using the construction that after the first iteration we have $9$ intervals, length ...
1
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1answer
33 views

Relationship between the Hausdorff dimension and the Box-counting dimension

In Fractal Geometry by Falconer the author writes: If $1<\mathcal H^s(F)=\lim_{\delta\to0}\mathcal H_\delta^s(F)$ then $\log N_\delta(F)+s\log\delta>0$ if $\delta$ is sufficiently small. ...
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23 views

Bounded/Unbounded sets. [Mandelbrot set]

This is the last question from my assignment. For Part a I have: $z_{n+1}=z_n^2+c$ $\Rightarrow c =z_{n+1}-z_n^2$ $\Rightarrow |c|=|z_{n+1}-z_n^2|=|z_{n+1}-z_n^2||-1|=|z_n^2-z_{n+1}|$ ...
0
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1answer
34 views

Problem in the proof of the dimension of the Cantor set

From the proof of the Hausdorff dimension of the middle third Cantor set. I cannot understand the last sentence in this proof. I cannot see how we have counted $2^j \leq \sum_i 2^j3^s|U_i|^s$
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1answer
17 views

Regularity of the surface of a crystal

If I want to model the surface of any random crystal, is it safe to assume that it is the graph of a Lipschitz function. Is there a precise result from physicists? How wrong would it be if I assume ...
1
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1answer
42 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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1answer
96 views

Fractal geometry on the circle, where area exponent and cross section exponent differ by less than 1

I'm looking for a particular class of connected, fractal sets $S_{\epsilon}$, with $0 < \epsilon < 1$ inside the unit disk. The sets are defined such that the circle always belongs to the set ...
0
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1answer
46 views

Hausdorff measure of the middle third Cantor set and Compactness

In the proof of the Hausdorff dimension of the middle third cantor set I cannot understand why we need the following underlined statement. I cannot understand why we need only consider closed ...
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0answers
28 views

Estimating the distance to the Julia set of a rational map

Suppose that $f \colon \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ is a rational map of degree $d \ge 2$. Let $z_0$ be a point in the Fatou set $F(f)$. I'm interested in finding an estimate for the ...
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1answer
52 views

software for studying Newton iterates of complex map $z \mapsto -a + 1/z + b/(1+z)$

I am looking for flexible software for studying complex dynamics (Julia sets, Newton iterations) with user-specified rational functions. Specifically, I wish to study the Newton iterates of complex ...
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0answers
27 views

How to compute the Lyapunov dimension of a time series

I have a time series, for example, $x=\{0.2,0.1,1.2,1.7,0.7,\ldots\}$. I would like to compute the Lyapunov dimension of the sequence of values. Works such as the below: ...
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1answer
41 views

Do 3 Dimensional Fractals exist?

I understand that certain mathematical sets produce fractals. Are there fractals defined by sets with more than 2 variables? Is that possible?
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1answer
53 views

Can we prove the Mandelbrot set is a fractal? Which maps/processes produce fractals?

So, as you probably noticed, I have two questions. The second leads on from the first. Can we prove the Mandelbrot set is a fractal? It is very easy to see that something like the Sierpinski triangle ...
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1answer
63 views

Perturbation of Mandelbrot set fractal

I recently discovered very clever technique how co compute deep zooms of the Mandelbrot set using Perturbation and I understand the idea very well but when I try to do the math by myself I never got ...
2
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0answers
33 views

Is there a simplification for the coefficients generated with the Mandelbrot iteration rule?

The Mandelbrot Set is obtained using the equation $z_n=z_{n-1}^2+c$ for some constant $c \in \mathbb{C}$ with $z_0=c$. Therefore, $z_1=c^2+c$, $z_2=c^4+2c^3+c^2+c$, etc. I have a function $f(n,x)$ ...
20
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3answers
468 views

Supremum of all y-coordinates of the Mandelbrot set

Let $M\subset \mathbb R^2$ be the Mandelbrot set. What is $\sup\{ y : (x,y) \in M \}$? Is this known? To be more descriptive: What is the supremum of all y-coordinates of all black points in the ...
1
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1answer
33 views

What types of fractals have a closed-form interior formula?

I was looking at the Menger Sponge earlier, and I realized it has a neat property: Let x, y, and z be spatial dimensions, each between 0 and 1 (inclusive.) Express them as ternary floating point ...
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2answers
63 views

Mathematical background for one wishing to study Chaos/Complexity Theory

I don't have a very strong mathematics background. In fact I quite abhorred mathematics during my Middle/High School years. I'm currently applying for PhD programs in the field of literature as that ...
4
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2answers
110 views

Name of this fractal

I am writing my final paper in the field ob computer enginering my work are on fractals. Some time ago, I found this fractal. Now I need to refer to it in my work but i have no clue what is it called. ...