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

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On a curve every point of which is a point of ramification

The title of my post is the same as the title of a known article written by Sierpinski where he introduced its famous triangle. In the book Handbook of the history of general topology by Lowen said ...
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52 views

Proving that we have $A \subset K$ where $K$ is self-similar

Let $f_1, \ldots$, $f_N:X\to X$ be contractions in the complete metric space $X$, and $K$ the self-similar set with respect to the $f_i$. If $A\subset X$ is compact and $$A\subset ...
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Relations between the 2-disc operad and fractals?

As you can see, as of late I opened a thread on n-disc operads: Clarification regarding little n-discs operads The thing is, those drawings there could somehow be construed in the real world as ...
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37 views

Are fractal image generators one-way functions?

Is it hard to calculate the coordinates and zoom factor that was used to generate a fractal image of, say, the Mandelbrot set? If you know the rest of the parameters, like how many iterations where ...
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Finding parameter paths for beautiful fractal animations

So I just got renewed interest in fractals and especially animations with fractals. To make an image or a frame, we usually need to evaluate a fractal for a subset of it's parameters. However for many ...
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Basins of attraction for Newton-Raphson fractal colouring

What's the general strategy/approach for defining the basins of attraction within the Newton-Raphson(NR) function in the complex plane? I would like to understand where these are to colour-in a NR ...
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355 views

A question about a fractal like iteratively defined function

I am trying to figure out what the following function $f:\Bbb{R}^3-\{\mathbf{0}\}\to\Bbb{R}$ defined below (in pseudocode) does: function $f(\mathbf{v}\in \Bbb{R}^3-\{\mathbf{0}\})$ { ...
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Hausdorff metric and $\varepsilon$-thickenings

Let $h(A,B)$ be the Hausdorff metric defined by: $$ h(A,B)=\inf\{\varepsilon >0 \; | \; A \subseteq B_\varepsilon, B \subseteq A_\varepsilon \}, $$ where $A_\varepsilon$ is the ...
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How does one turn an object into fractal? [closed]

I've seen a lot of digital art made using fractals e.g landscapes, flowers, trees and the like, but I was wondering how does one turn an object into a fractal? Like say....if I wanted to make a ...
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155 views

Choosing an appropriate sequence of $\{1,2,3\}$

Let $f_1,f_2,f_3$ be the contracting maps $f_i:x\mapsto \frac{1}{2}(x+p_i)$ from $\mathbb{R^2}$ to itself and $p_i\in \mathbb{R}^2$. Denoted by $S$ the attractor Sierpinki gastek of the iterated ...
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Could this odd insight help explain part of the difficulty in proving the Collatz Conjecture?

Background: Here's a crash course on the Collatz Conjecture. Basically, you take a number and if it is even you divide it by two. If a number is odd, you multiply it by three and then add one. You ...
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77 views

formal definition of “fractal” or standardized categories?

fractals are many decades old and come up in a wide variety of contexts and can be generated in so many different ways. however, a formal definition of fractal seems really slippery/ difficult. are ...
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62 views

The mandelbrot fractal and it's relations to algebras and groups

I have been fooling around with Mandelbrot fractal to and fro for many years. One of the latest years I learned some general algebra with some of the most basic groups, like cyclic groups, dihedral ...
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52 views

Are Square and equilateral triangle the only convex regular ngons that can be composed to smaller versions of themselves?

While I was trying to make an analogous question to Select $n^2 + 1$ points in the unit square. Show that at least two points are no more than a distance $\frac{\sqrt{2}}{n}$ apart, using equilateral ...
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53 views

The unit ball is not auto similar.

I want read a prove that the unit ball $B$ is not auto similar. I mean that there is not similarities $f_1,...,f_n$ with contracting constants <1, such that $$B=\bigcup_{i=1}^n f_i[B] $$ Anyone ...
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1answer
14 views

What can we do on $S$ in order that $H(S)$ be compact?

Let be $S$ a metric space. We define the hyperspace $H(S)$ as the metric spaces consisting of every no empty compact subset of $S$ and the Hausdorff metric. I want that $H(S)$ be compact imposing ...
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What methods are known to visualize patterns in the set of real roots of quadratic equations?

I came across a previous awesome question about the visualization of the distribution of polynomial roots and tried to do a simpler version applied to the set of real roots of quadratic equations ...
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127 views

Can a fractal be a manifold?

Here it is said that it is not possible: Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower? But I am confused about this. What about the invariant ...
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How to create a new formula for a fractal-type image?

(If this is the wrong place to ask, then PLEASE tell me where to take the question instead of chewing me out over this.) I have been learning how to write out SVG by hand, and in the process made a ...
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What is this pattern found in the first occurrence of each $k \in \{0,1,2,3,4,5,6,7,8,9\}$ in the values of $f(n)=\sqrt{n}-\lfloor \sqrt{n} \rfloor$?

Learning how to generate the Mandelbrot set, I came across the definition of the "escape condition" which is the one that decides the color that is applied to each point of the plane where the ...
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Determine coordinates for Mandelbrot set zoom.

I am writing a computer program to produce a zoom on the Mandelbrot set. The issue I am having with this is that I don't know how to tell the computer where to zoom. As of right now I just pick a ...
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Is it possible to construct a smooth curve with fractional Hausdorff dimension?

It is known that fractal curves have fractional Hausdorff dimension. These curves are not smooth and have undefined length. However, is the converse true? If a curve has a fractional Hausdorff ...
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is the Buddhabrot well-defined?

Define the Mandelbrot set $M = \{ c \in \mathbb{C} : P_c^n(0) \not\to \infty \text{ as } n \to \infty \}$ where $P_c(z) = z^2 + c$. Define the complement of the Mandelbrot set $\overline{M} = ...
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39 views

Functions differentiable on “small” sets

I was recently looking again at functions like the Cantor staircase, the modified Dirichlet, etc., and something occurred to me. The modified Dirichlet is interesting because it's continuous almost ...
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63 views

Relation between Power Laws and Fractals

Are all power laws (i.e., of the general form $y=cx^{\alpha}$) fractal (exhibiting some form of self-similarity)? Does the scalability of power laws also mean by definition that they are also ...
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100 views

Area of a fractal?

I wanted to know that how can one find the area of the Mandelbrot set or any fractal for that matter ?
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Is this a valid example of a non-euclidean Sierpinski attractor?

I am learning the basic concepts about the Chaos Game (I did a previous question about the same topic here), the method to create fractals elaborated by professor Michael Barnsley. The basic example ...
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105 views

Box-Counting Dimension with finite resolution

Does the method of determining dimension of a shape via the Box-Counting dimension (Minkowski–Bouligand dimension) have to be performed on fractals (objects that look the same at all scales), or can ...
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1answer
80 views

Construction of Rauzy Fractals with substitutions without a fixed point

The formal definition of a Rauzy fractal can be found at the beginning of this paper Using Sage-math-cloud, I can generate Rauzy fractals of substitutions that I choose. Should I choose the ...
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1answer
45 views

Question from book 'Indra's Pearls' about limit set arising from infinite words (compositions of maps)

The book considers mappings $a, b, A,$ and $B$ where $A = a^{-1}, B = b^{-1}$. It goes on to say that words represented by compositions of these maps (e.g. $abbA$) correspond to points. I ...
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How can I calculate the formula of this fractal-like structure?

I did the following fractal-like structure manually, and I was trying to convert it to a formula (or an algorithm including formulas) to compute some parts of the drawing, but I get lost due to the ...
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Why does the boundary of the Mandelbrot set contain a cardioid?

In a comment to a previous answer it has been mentioned that the boundary of the Mandelbrot set contains the cardioid $$ c = e^{it} \, \frac{2 - e^{it}}{4} $$ but how can we prove this?
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Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower?

So it is weekend! and I am reading a nice book, "The Poincaré conjecture", written by a mathematician (Donal O'Shea, topologist). The book introduces step by step basic concepts of Topology, and talks ...
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132 views

How to make an icosahedron from 20 tetrahedra?

To make an icosahedron out of Sierpinsky tetrahedrons is difficult because regular tetrahedra can't tile in space. The dihedral angle of a tetrahedron is ~70.53. So the first step would be to make ...
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Mandelbrot Set area

If there are an infinite amount of details that can be found in a Mandelbrot set, shouldn't the Mandelbrot Set have an infinite area? Supposedly the area of a Mandelbrot set is 1.5065918849 ± ...
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Example of a function that has fractional derivatives of order less than 1 but not 1

I have recently learned that some fractals can have fractional derivatives of order less than 1, say of 1/2 even if they are not differentiable (have no derivative of order 1). I wonder if there is a ...
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Any relationship between Hausdorff measures

Let $ S_1= ( [0,1], d_1 ) $ and $ S_2 = ( [0,1], d_2 ) $ be two metric spaces, where $ d_1 = |x - y|$ and $d_2 = (1/2^i) $ where binary expansion of x and y matches upto $ i^{th} $ coordinate. Let $ ...
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70 views

Lower Bound of Hausdorff Dimension of Cantor Set

Consider a Cantor set $E$ where the intervals at every level of the construction maintain a minimum spacing and have a finite number of intervals on each level. I have two questions regarding finding ...
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116 views

How many vertices are in the Koch Snowflake?

EDIT: The question was put on hold because I didn't specify what I meant by vertex. In a comment below by Mark McClure, by "vertex" I mean one of the vertices of the standard, polygonal approximations ...
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25 views

Nonincreasing and nondecreasing sequences in Hausdorff metric

For every metric space $(X,d)$ we have the Hausdorff metric space $(\mathcal{H}(X),H)$ that assosiates with it, where $\mathcal{H}(X)$ is the space of nonempty compact subsets of $X$ and $H$ is the ...
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What does “points spanned by powers” mean in the Goffinet dragon definition?

The definition of the Goffinet dragon fractal given by Wolfram Mathworld refers to plotting all points spanned by powers of the complex number p=0.65-0.3i What does it mean for points to be ...
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Categories of fractals

I have a question about classifying a few fractals I've been programming. I understand that there are types of fractals like L-systems (Barnsley's Fern, Fractal plant, ...), IFS systems (Sierpinski's ...
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1answer
51 views

Subsets of set satisfying open set condition

Suppose an iterated function system of similarity transformations $S_1, S_2, \dotsc, S_k:\mathbb{R}^n\to\mathbb{R}^n$ (with unique invariant set $F$) satisfies the open set condition for some ...
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Is the boundary of the Mandelbrot set jagged or smooth?

As the title states, I am wondering if the boundary of the Mandelbrot set is jagged or smooth. If it is jagged, is there some algorithm to find the vertices of any one of them? Are there an infinite ...
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53 views

Showing the attractor of an IFS is either connected or totally disconnected

I came across this execise in a problem set about Iterated Function System (IFS) and fractals: "Show that the attractor of an IFS of the form $\{\mathbb{R};~ax+b, cx+d\}$ where $a,b,c,d \in ...
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Calculate moment of inertia of Koch snowflake

That's just a fun question. Please, be creative. Suppose having a Koch snowflake. The area inside this curve is having the total mass $M$ and the length of the first iteration is $L$ (a simple ...
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203 views

Mandelbrot set of $c \cdot \cos(z)$

I'm given a task to write a program, that determines if a given point $c \in \mathbb{C}$ is in the Mandelbrot set of the function $$f_c(z) = c \cdot \cos (z)$$ That is if the set $\{z_n = f_c^n (0) : ...
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1answer
32 views

Sufficient condition for integer Hausdorff dimension.

It is pretty much in the title: is there a non-trivial sufficient condition on geometrical shapes that forces the Hausdorff dimension to be an integer ? Most fractals look "complicated" in some way, ...
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95 views

Proving basic properties of Hausdorff dimension and measure

I have two questions on basic properties of the Hausdorff measure and dimension which I've taken for granted for a while (I'm revisiting Falconer after about a year), but that I've never actually seen ...
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How can I represent a fractal fraction in a way that can control precision?

I'm looking for shorter ways to represent a fractal fraction where the value can be found at a level of precision ($p_n$), similar to the following example, but without expanding the whole fraction: ...