Questions tagged [fractals]

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

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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 ...
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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 $a<b$...
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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 ...
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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 ...
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2answers
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 '...
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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 ...
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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 ...
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Dimension capacity of a set

The question is very direct. How can I calculate the dimension of capacity of the set: $\mathbb{Q}\cap[0,1]$? I know that the dimension of capacity of a set $A\subseteq\mathbb R^n$ ( but I' m not $...
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Is this relationship between Mandelbrot bulbs a coincidence?

I was messing around with the Mandelbrot boundaries on Desmos and came across something interesting, and I don't have enough experience with the math behind this to conclude whether it's a coincidence ...
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Colouring the Mandelbrot Set (Textbook question - Homework)

I don't really understand the solution provided by the textbook for this following question Solution provided by the textbook I used Geogebra to create the sequence for each value of c provided. ...
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Bounds for the box dimension of the Koch curve with different definitions

I am working on Kenneth Falconer's book on fractal geometry. The book gives us 5 different equivalent definitions for the box dimension and these are the two needed for this question: (i) Let $N_{\...
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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 ...
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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 ...
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1answer
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 ...
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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 ...
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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?
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1answer
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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?
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1answer
100 views

Find an explicit formula for $I_n$ in terms of $a$ and $b$.

Let $b > 0$, $0 < a < 1$, and set $f(x) = ax + b$. Moreover, put $I_0 = [0, b]$, $I_n = f^{◦n}(I_0)$, $n = 1, 2, \dots$ and $$I = \bigcup_{n=0}^\infty I_n.$$ Find an explicit formula for $...
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1answer
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Hausdorff dimension for bounded set

Let $F$ be the set of of numbers $x\in [0,1]$ with base 3 expansions $0.a_1a_2...$ for which there exists an integer k such that $a_i\neq 1$ for all $i\geq k$. Find the Hausdorff dimension of $F$. ...
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Confusing pictures about tetration !? [closed]

On the webpage http://tetration.org/Tetration/index.html, We are supposed to get an explanation of tetration, whatever that means exactly. In particular I feel the pictures are not well explained. ...
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Analytically, why does Sierpinski's triangle appear in Pascal's triangle?

It is well-known that if you isolate the even from the odd terms in Pascal's triangle, you obtain an 'approximation' of Sierpinski’s triangle. Better stated, from user Glorfindel in an answer to this ...
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Name of these Sierpinski-like fractals?

A Sierpinski triangle can be created by Starting with a row consisting of a single 1 Each row below is horizontally offset by a half-cell Each cell is the sum-mod-2 of the two cells above it Now ...
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Calculate Minkowski dimension for a square

How can i calculate the Minkowski dimension for a ordinary square? I saw the few examples where dimension is calculated with help fractal dimension - fractal dimension formula In this case D = 2, ...
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Is there a way in finding the area of an island with its coastline's Hausdorff Dimension?

I have applied the Hausdorff method to find the fractal dimension of the coastline of one of my country's islands. Is there any way for me to find the area of the islands with the fractal dimension I ...
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How do you calculate Hausdorff-dimension of real world objects when Hausdorff-dimension needn't to be equal to other dimensions?

In general Hausdorff-dimension is never larger as Minkowski / Box-counting dimension. For some sets like self-similar sets both dimension coincidence. In literature you can read that the Hausdorff-...
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Covering subintervals

Consider subintervals of $[0,1]$ such that $E_k$ which has $9^k$ intervals, each of length $10^{-k}$, with total gaps in between the intervals being greater than or equal to $10^{-k}$. Note that $E_{...
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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 ...
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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: $$ B:[0,1]\to [0,2],\quad B(x):= \sum_{k = 0} 2^{-k}s\big(2^kx\big)$$ ...
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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? ...
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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 ...
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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?
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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 $3^2$ ...
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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: ...
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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 ...
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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, ...
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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 $N_{\delta}(F)$ $=$ ...
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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 ...
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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 ...
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Does not the existence of fractals with fractional Hausdorff dimension prove that there are cardinalities in between countable and continuum?

Is not it the case that the cardinality of fractals of dimension 0.00001 should be greater than countable but less than that of continuum?
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Are convex fractals possible?

Is it possible to have a fractal shape (and by that I mean a shape with non-integer Minkowski dimension) that is convex? I'm pretty sure it isn't, as every example of a fractal I can think of is ...

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