# Tagged Questions

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 $z_0=0$. Therefore, $z_1=c$, $z_2=c^2+c$, $z_3=c^4+2c^3+c^2+c$, etc. I have a function ...
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### 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 ...
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### 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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### 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 ...
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### 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. ...
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### Why such iteration leads to fractal?

I saw a piece of codes like: ...
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### 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 ...
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### Integral over filled Julia sets

Defining the usual quadratic Julia set iteration $f_c(z)=z^2+c$ for complex $c$, and its $n$th iteration $f^n_c(z)=f_c(f_c(\cdots f_c(z)\cdots))$, you can define a function of 4 variables ...
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### Mandelbrot sets and radius of convergence

While watching this Numberphile video on Mandelbrot sets, it's more or less stated that the fractal will "blow up" if it's radius of convergence is greater than 2. What is the mathematical basis for ...
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### Behavior of Pascal's triangle in $n\mod m$ where $m>2$, any fractals?

If Sierpinski Triangles are found in Pascal's Triangle under modulo 2 what happens when we view Pascal's Triangle under modulo $m$ where $m>2$? Do fractals appear and if so for which numbers? ...
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### Can Wiener process on a fractal random graph be reduced to a levy flight?

Weiner process on small-world graphs is a Levy flight. But does the condition still hold for a random graph that connects the edges of a fractal?
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### Why must fractals be self-referential?

Having an idea of what a fractal is, by example, etc., then seeking the actual definition is, at first, both obvious and imprecise. You'll see it defined as an object that is self-similar in some ...
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### How to calculate line-length for fixed width koch fractal?

I am playing with fractals, and drawing them with Python turtle I am using this rules to create l-string for my koch fractal: begin: f f -> f+f--f+f In here, ...
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### Packing infinitely many ellipses into a circle

Given a circle $C$, and an infinite set $S$ of mutually disjoint ellipses which are inside and tangent to $C$, prove that there must exist a disk $D$ which lies inside $C$ but outside every ellipse. ...
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### Henon Map Parameter

In case of Hennon map two parameters $a$ and $b$ to be set.The Hénon map takes a point $(x_n, y_n)$ in the plane and maps it to a new point $x_{n+1} = 1-a x_n^2 + y_n$, $y_{n+1} = b x_n$. The map ...
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### Reference - formal characterization and analysis of Koch curve

I am studying the Koch curve but most resources I have seen do not describe the Koch curve formally and are similar to the Wikipedia page on the subject. For example, I have looked at books like ...
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### Creating fractals through computers

What are some beginner softwares for creating fractals on computers?
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### Points in a general Cantor set

We often look at the Cantor set with the construction that keeps removing the middle thirds of the remaining line segments at each iteration. Corresponding to this construction, we can determine ...
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### Drawing a nested epicycloid

I would like to learn how to draw this kind of pictures (possibly with Mathematica, as it is the only language I would be comfortable to code such a thing in): There is something similar on the ...
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### Does there exist a Lipschitz map from the unit interval onto the unit square?

It is well-known that continuous space-filling curves exist. But can they be Lipschitz? Specifically, is there a Lipschitz map from [0,1] onto [0,1]x[0,1]?
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### Generalisation of Vitali's covering lemma

In "The geometry of fractal sets", Falconer gives the following generalisation of the Vitali covering lemma as an exercise: Let $\mu$ be any measure on $\mathbb{R}^{n}$ and $E$ a set with ...
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### Is the Mandelbrot set computable?

This is a weakened version of Is the measure induced by the Mandelbrot set computable on rational rectangles? ; Given a (computable, or rational) rectangle in the complex plane, is it computable ...
### Can we construct a Koch curve with similarity dimension $s\in[1,2]$?
We can make a Koch curve $K$ with similarity dimension $s\in \mathbb Q \cap [1,2]$ by writing $s=\frac{p}{q}$, and constructing such a generator that by scaling with the factor of $2^q$, we'd find ...
Let $K_0 = [0,1]$. Split $K_0$ into 4 intervals and remove the middle half. This gives $K_1 = [0,1/4] \cup [3/4, 1]$ and so on and set $K = \cap K_i$. I computed the upper and lower Minkowski content ...