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

learn more… | top users | synonyms

1
vote
1answer
38 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 ...
3
votes
1answer
52 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 ...
1
vote
1answer
40 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 ...
1
vote
1answer
52 views

Is this c the same as that c?

Are the highlighted $c$'s the same or should it be $c_1$ and $c_2$.
1
vote
0answers
52 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 ...
1
vote
1answer
35 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
votes
1answer
170 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
votes
2answers
133 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 ...
1
vote
2answers
78 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
votes
1answer
54 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
votes
1answer
101 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
votes
1answer
46 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
votes
1answer
72 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 ...
0
votes
1answer
35 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 ...
1
vote
0answers
29 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
votes
2answers
111 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
votes
0answers
22 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 ...
1
vote
0answers
46 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 ...
1
vote
0answers
108 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 ...
8
votes
1answer
67 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 ...
14
votes
1answer
225 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
votes
2answers
85 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 ...
1
vote
1answer
113 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 ...
1
vote
1answer
69 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 ...
1
vote
1answer
65 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. ...
0
votes
1answer
54 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 $2^j \leq \sum_i 2^j3^s|U_i|^s$
0
votes
1answer
21 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 ...
3
votes
1answer
83 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 ...
1
vote
1answer
106 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
votes
1answer
93 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 ...
1
vote
0answers
38 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 ...
1
vote
1answer
58 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 ...
1
vote
1answer
57 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?
3
votes
1answer
133 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 ...
1
vote
1answer
212 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 ...
4
votes
0answers
111 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 $z_0=0$. Therefore, $z_1=c$, $z_2=c^2+c$, $z_3=c^4+2c^3+c^2+c$, etc. I have a function ...
21
votes
3answers
626 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
vote
1answer
52 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 ...
1
vote
2answers
122 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 ...
5
votes
2answers
193 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. ...
3
votes
0answers
108 views

Why such iteration leads to fractal?

I saw a piece of codes like: ...
3
votes
1answer
75 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 ...
14
votes
1answer
173 views

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 ...
0
votes
1answer
238 views

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 ...
2
votes
1answer
67 views

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? ...
1
vote
0answers
31 views

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?
1
vote
1answer
88 views

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 ...
0
votes
1answer
53 views

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, ...
10
votes
1answer
169 views

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. ...
0
votes
1answer
93 views

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 ...