# Questions tagged [fractals]

For questions on fractals, which are irregular, rough, or "fractured" sets that often possess self-similar structure.

915 questions
Filter by
Sorted by
Tagged with
10 views

### What is the theorem for construction from projections attributed to Falconer?

I heard in a talk that there is a theorem that makes it possible to construct objects from their projections. The theorem seems to be attributed to an individual by the name of Falconer. Does anyone ...
33 views

### Fractal construction and IFS (iterated function system): defining a specific construction with IFS

The setting: We consider the plane $\mathbb R^2$ with its canonical euclidean structure. The canonic base is written $(e_1,e_2) = ((1,0)^\top,(0,1)^\top)$. Let's consider $I = [a,b]e_1$ (with $a<b$...
19 views

### What is the box dimension of the image of a continuously differentiable function on $[0,1]$?

Im going through some notes for fractal geometry and the following exercise is stated: "Let $f:[0,1]\rightarrow{}\mathbb{R}$ be a continuously differentiable function with $f(0)\neq{}f(1)$. Show that ...
32 views

### Set with Hausdorff dimension $s = \log 2 / \log 3$ but $H^s = \infty$?

I am trying to solve the following problem: Find a set $X \subset \Bbb R$ s t.$\dim_H (X)= s$ where $s = \frac{\log 2}{\log 3}$, but $H^s (X) = \infty$. Here I am using the notations from ...
26 views

### Space filling curve which is a closed map

Does there exist a continuous surjective closed map $f : [0, 1] \to [0, 1]^d$? That is, does there exist a space-filling curve which is a closed map?
51 views

### is it possible to embed a fractal inside another fractal

I have been looking at one-dimensional (1D) cellular automata (CAs) which generate two-dimensional (2D) fractal patterns. Among the 256 1D elementary CAs, I tried to list down the fractal generating ...
27 views

### Tetration fractal algorithm

In the code example at http://code.activestate.com/recipes/577917-tetration-fractal/ the author seems to implement the operation of a complex number, $z=x+iy$, raised to the power of itself, $z^z$, as ...
25 views

### Find a metric on a compact space so that a transformation becomes a contraction mapping

I am struggling with exercise 6.12 of Chapter III from Barnsley's Fractals Everywhere, 2nd edition. The exercise is as follows: Let $(X, d)$ be a compact metric space. Let $f : X \to X$ have the ...
25 views

### Why is the conjugate alteration of the Mandelbrot set (the tricorn) also known as the “Mandelbar” set?

I am currently working with a shader rendering that deals with higher powers of the Mandelbrot set, . I understand that the conjugate of the first parameter of this set,, also known as the tricorn, is ...
36 views

### Finding the dimension of the sphere cube

If you take an $2r\times 2r\times 2r$ cube, and divide it to 27 equal cubes, and then remove all the "axis" cubes (all the cubes which are straight left, straight right, straight up, etc. from the ...
34 views

### Prove that non-attracting periodic orbits of $x \mapsto x^2 + c$ are in the Julia set [closed]

Let $c$ be complex number and let \begin{align} f_c : \mathbb{C} &\to \mathbb{C} \\ x &\mapsto x^2 + c \end{align} be the quadratic map. It is to be shown that if $O$ is a periodic orbit that ...
31 views

### Check that this transformation is a contraction

Let $T_0, T_1:\mathbb{C}\to\mathbb{C}$ be given by $T_0(z) = rz\exp{\frac{i\pi}{4}}+i$ and $T_1(z) = rz\exp{\frac{-i\pi}{4}}+i$ respectively, for $0<r<1$. I need to show that these are ...
42 views

### Are there Lie groups in fractional dimension?

Consider the Sierpiński triangle. It has dimension $\log_23$. So does it have any rotational group associated with it? e.g. a Lie group $SO(\log_2 3)$ ? Or are there any such things as Lie groups in ...
44 views

### Can we tell if a given rational point is a point on the Sierpiński triangle?

Stated precisely, is the indicator function for the Sierpiński triangle restricted to rational points in the plane a computable function? My intuition is telling me no, but maybe the fractal folks ...
21 views

### How do I calculate the Hausdorff dimension of a self-affine fractal (like the Barnsley Fern)?

The fractal I am concerned with has an infinite number of self-affine copies of itself, and all scaled to different dimensions. And all but one of them are rotated too. I know this may sound way more '...
5k views

### Why can't Antoine's necklace fall apart?

Antoine's necklace is an embedding of the Cantor set in $\mathbb{R}^3$ constructed by taking a torus, replacing it with a necklace of smaller interlinked tori lying inside it, replacing each smaller ...
69 views

### Partitioning a Cartesian Product Yields Sierpiński Triangle… ish?

I wanted to find an efficient method to partition a Cartesian product of $n$ sets $S_i$ of varying sizes into maximum size subsets that are defined by all tuples in the partition differing in at ...
38 views

1k views

### Fractal Pattern from Queen's Move Construction

This question relates to the OEIS sequence A279212. Fill an array by antidiagonals upwards; in the top left cell enter $a(0)=1$; thereafter, in the $n$-th cell, enter the sum of the entries of ...
26 views

### Is there such a thing as a reverse Menger Sponge / Carpet?

What I mean is a set similar to the menger sponge/carpet, but expanding, instead of intricate. Iteration $0$, $\ddot m_0$ is the unit cube/square from $0$ to $1$, and the following iterations are ...
60 views

### What are some good resources to start learning about fractals?

I am an undergraduate mathematics major looking for online resources to learn more about fractals and fractal geometry. I have only a basic knowledge of fractals and their properties, so I am only ...
29 views

### Solve $\sum \phi(q) q^{- 2 s} = 1$ for $s$ with rigourous error bounds

I'm trying to solve for $s$ the summation: $$\sum_{q=2}^\infty \phi(q) \left(\frac{1}{q^2}\right)^s = 1$$ where $\phi(q)$ is Euler's totient function. The context is trying to calculate the fractal ...
21 views

### Parametric equation of Mandelbrot curve on k-th iteration?

Is there a way to write the parametric equation of the Mandelbrot-set's boundary curve at every $k^{th}$ iteration?
46 views

### Do you know what fractal this is? [closed]

I want to write a program and draw this fractal but I don't know the recursion step. Does anyone know any information about this fractal?
100 views

16 views

### Hausdorff dimension of graph of composition of functions

Given two functions $f,g$, is there a reasonable bound of the Hausdorff dimension of the graph of $f\circ g$ given the Hausdorff dimensions of the graphs of $f$ and $g$? For example, does it hold that ...
26 views

### Two questions on the Blancmange function.

I have two questions about the so-called Blancmange function (which I'll restrict to having domain $[0,1]$). That is, define: $$B:[0,1]\to [0,2],\quad B(x):= \sum_{k = 0} 2^{-k}s\big(2^kx\big)$$ ...
91 views

### Explain why Mandelbrot set escape radius is 2 to a dummy

I'm curious, in the Mandelbrot set, why is the escape radius $2$? I've seen few proofs of that on the internet, but i can't understand them enough. Why is the bailout value of the Mandelbrot set 2? ...
35 views

### How to find the box counting dimension of line segment [0,1]?

I uploaded this question and no one has answered because there were some flaws in the question but I have the text now. Please can some one explain how we arrive on example of line segment to have a ...
19 views

### Box counting dimension of line segment [0,1]?

I am not understanding this example from Kenneth Falconers book on Fractal geometry. can someone please explain how the inequalities follows from the definitions in the example given?
103 views

### Geometry question and Diameter

Let $F$ be a bounded subset of $\mathbb{R}^2$. Impose a $\delta-grid$ on $F$. (Squares of length and height $\delta$). Prove that any subset $U$ of diameter at most $\delta$ is contained in $3^2$ ...
84 views

### Creating Julia sets using Python

I want to create my own images of the Julia set of the complex function $e^z-2$, similar to the one below: The simple (Python) program: ...
103 views

### Why the cardioid shows up in the Mandelbrot set?

How does the main cardioid appear in the Mandelbrot set? I also wonder why something "weird" happens at a point with coordinate 2 on the actual coordinate line, I mean why is the point a sort of ...
40 views

### logistic map averaged values graphed

For the logistic map: https://en.wikipedia.org/wiki/Logistic_map For a given x, taking the average of n values on the logistic map, gives a converging value. ie for the x domain 3.56 <= x < 4, ...
58 views

### Existence of box counting dimension.

Let $F\subseteq \mathbb{R}^n$ be non-empty and bounded. We say that a countable family of set $\{$ $U_i$ $\}$ is a $\delta-cover$ for $F$ if $F\subseteq \bigcup_{I\in I}U_i$. Let $N_{\delta}(F)$ $=$ ...
43 views

### Help understanding “Peano curve” picture

I am trying to understand a particular version of the "Peano curve." Although definitions will vary from person to person, for the purposes of this question I am talking about the one obtained from ...
619 views

### There's something strange about $\sum \frac 1 {\sin(n)}$.

Clearly, $$\sum_{n=1}^\infty \frac 1{\sin(n)}$$ Does not converge (rational approximations for $\pi$ and whatnot.) For fun, I plotted $$P(x)=\sum_{n=1}^x \frac 1{\sin(n)}$$ For $x$ on various ...