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

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Identify rules for fractal L-system for plant representation using lengths and angles

As some of you already knew, to develop an L-system fractal you need some rules for angles and lengths. L-system also is well known for its application in plants. So i had a plant in the ground and ...
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1answer
10 views

Infinite number of points in the Sierpinski Triangle

I have basic background in mathematics (Linear Algebra, Calculus) and I've been reading up on fractals, because I find them fascinating. I can't understand one thing in basically all of the fractals ...
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37 views

Symmetric Icon Fractals

I have always been fascinated by fractals. But most of all I like the Symmetric Icon fractals. There is a nice book about these fractals, written by Michael Field, called Symmetry in Chaos. I'm ...
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15 views

Is there a name for the relation between Menger Sponge and Vicsek Fractal?

Both the Menger Sponge and the Vicsek Fractal in 3D can be constructed by starting with a cube, dividing it into 27 smaller cubes (3x3x3 grid), removing some of these cubes, and then applying the ...
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1answer
23 views

Finding external angles for Misiurewicz points in the Mandelbrot set

In the Mandelbrot set for the quadratic polynomial $z \to z^2 + c$, rational external angles with even denominator are pre-periodic and have corresponding external rays which land at Misiurewicz ...
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1answer
21 views

A zero-dimension set and self-referencial equation

Let $K$ be a compact set in $\mathbb{R}^2$. Let $f_1,..., f_n$ be contracting similarities of $\mathbb{R}^2$ to itself. Suposse $K$ satisfies the self-referencial equation ...
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1answer
51 views

Bounding dimension of IFS

Given the IFS $\{\frac x {2+x},\frac 2 {2+x}\}$ ($0\le x \le 1$) with attractor K prove that $0.53<\dim_HK<0.8$ I thought using the results from my last question by saying ...
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98 views

Proving ineqalities for the similarity dimension

a. Let $K$ be the attractor of the IFS $\{f_1,\dots f_n\}$ which satisfies SSC (i.e $f_i(K)\cap f_j(K)=\emptyset\forall i\neq j$) where for all $i, c_i$ such that $ 1\le i\le n, \space ...
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3 views

Reference request: Generalized Hurst Exponent

I'm looking for some references on the Generalized Hurst Exponent of a time series, more than what is on wikipedia. The Generalized Hurst Exponent, $\mathbb{H}_q$ is defined by ...
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25 views

Compact metric space implies that the hyperspace is compact

I need a hint to the following problem: If $S$ is a compact metric space then the hyperspace $H(S)$ is compact. I don't know how to begin this problem. Thanks!
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1answer
26 views

Upper Minkowski content and finite Hausdorff measure

Does someone know an example of a set $E$ with positive finite $s$-Hausdorff measure, Minkowski dimension $s$, and infinite $s$-dimensional upper Minkowski content ? The $s$-dimensional upper ...
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1answer
24 views

Minkowski dimension behaviour of sets with positive finite Hausdorff measure.

It is (rather) well known that the set \begin{equation*} E=\{k^{-1},k\in\mathbb{N}^{*}\} \end{equation*} has box-dimension $1/2$ and Hausdorff dimension $0$. However $H^{0}(E)=|E|=+\infty$. Is it ...
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1answer
21 views

Solving inverse problem related to Iterated function systems?

I generated a Barnsley's fern fractal using details in this link with the aid of MATLAB. My doubts are as follows : How do we justify the shape generated from those equations? Is it possible to ...
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2answers
78 views

Examples of bounded continuous functions which are not differentiable

Most often examples given for bounded continuous functions which are not differentiable anywhere are fractals.If we include probabilistic fractals exact self-similarity is not required. Are their ...
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1answer
42 views

Prove carpet has positive Hausdorff measure in its dimension

Given $D\subset\{0,1,2,\dots n-1\}\times\{0,\dots,m-1\}$, let $$K(D)=\{\sum_{k=1}^\infty(a_kn^{-k},b_km^{-k}):(a_k,b_k)\in D\forall k\}.$$ Show that if $D$ has uniform horizontal fibers (i.e. the ...
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1answer
120 views

Finding the location of an image of the Mandelbrot set

I've got an image of a segment of the Mandelbrot set that I generated with an iPhone app a long time ago (I use it as my background image). I now have no idea where in the set the image came from. ...
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18 views

Proving that $\dim_M{K_{\lambda,\gamma}}=\dim_HK_{\lambda,\gamma}$

Define $K_{\lambda,\gamma}$ to be the attractor of the IFS $$\bigg\{\bigg(\matrix {\lambda&0\\0&\gamma}\bigg)\bigg(\array{x\\y}\bigg),\bigg(\matrix ...
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29 views

Structure of Mandelbrot bug antennae

I have analysed the structure of of Mandelbrot set and I have understood something, but I still have some questions, mainly about antennae and little bugs. Mandelbrot set consists of the main ...
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1answer
48 views

Self-similar fractal dimension of unsymmetrial fractal

As far as I know, the following fractal has a self-similar fractal dimension of $D = -\log(3) / \log(1/2) = 1.5850$ But what is the fractal dimension of the following fractal (4 times the fractal ...
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5answers
481 views

If $f(x)=x^2-x-1$ and $f^n(x)=f(f(\cdots f(x)\cdots))$, find all $x$ for which $f^{3n}(x)$ converges.

Let $f:\mathbb{R}\to\mathbb{R}$ be the polynomial defined by $$f(x)=x^2-x-1$$ and let $$g_0(x)=f(x),\quad g_1(x)=f(f(x)),\quad\ldots\quad g_n(x)=f(f(f(\cdots f(x)\cdots)))$$ The positive root of ...
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3answers
370 views

Simplifying $\sqrt {1+\sqrt 5}$

I considered simplifying $\sqrt {1+\sqrt 5}$. So I started $(a+b \sqrt 5)^2 = 1 + \sqrt 5$. This gave me $a\color{blue}{^2} + 5b^2 =1 , 2ab = 1$ so the result was $ \sqrt{1+\sqrt 5} = \sqrt{1/2 - ...
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2answers
45 views

How do I prove the circumference of the Koch snowflake is divergent?

How do I prove that the circumference of the Koch snowflake is divergent? Let's say that the line in the first picture has a lenght of $3cm$. Since the middle part ($1cm$) gets replaced with a ...
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Fractal Generation Project: How to Continue after L-Systems?

Three months ago, we defined a programming project for some gifted middle school students: Writing code to generate simple fractals. Two of them have finished writing Pascal programs already! The ...
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1answer
104 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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55 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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33 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 ...
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1answer
43 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
29 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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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
34 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
46 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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35 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
34 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 ...
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1answer
112 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 ...
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2answers
104 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
74 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 ...
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1answer
46 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 ...
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1answer
94 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 ...
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1answer
39 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: ...
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1answer
48 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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1answer
22 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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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 ...
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2answers
91 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 ...
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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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21 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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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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38 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 ...
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1answer
59 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 ...
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1answer
178 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 ...
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2answers
62 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 ...