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

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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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0answers
69 views

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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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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1answer
422 views

We know the dimension of the Koch snowflake's perimeter, but does it have a measure?

I start with an equilateral triangle with side perimeter three meters. I can define a Koch snowflake by the following sequence of figures. Starting with that triangle, produce the next figure by ...
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2answers
244 views

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 ...
4
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1answer
140 views

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 ...
3
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0answers
35 views

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 ...
3
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1answer
88 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 ...
54
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2answers
5k views

Fractal behavior along the boundary of convergence?

The complex power series $$\sum_{n=1}^{\infty}\frac{z^{n^2}}{n^2}$$ has radius $1$ (Ratio Test) and is absolutely convergent along $|z|=1$. Recalling something that my calculus professor (Ray Mayer, ...
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1answer
169 views

How to correctly calculate the fractal dimension of a finite set of points?

The box-counting dimension is defined by: $\lim\limits_{\epsilon \to 0} \dfrac{N(\epsilon)}{1/ \epsilon}$ What works well if you are solving algebraically or if you can recursively generate more ...
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1answer
349 views

About devil's staircases

We say that a function $f:\left[a,b\right] \to \mathbb{R}$ is a singular function or a devil's staircase if $f$ satisfies the following properties: $f$ is continuous; $f(a) < f(b)$; $f$ is ...
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1answer
81 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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2answers
37 views

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 ...
5
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2answers
50 views

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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1answer
36 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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69 views

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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1answer
50 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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389 views

Visualizing the Partition numbers (suggestions for visualization techniques)

So Ken Ono says that the partition numbers behave like fractals, in which case I'd like to try to find an appropriately illuminating way of visualizing them. But I'm sort of stuck at the moment, so ...
0
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1answer
433 views

How to convert a right angled triangle into a equilateral triangle?

I want to use the Apophysis program to make a right angled sierpinski triangle into an equilateral triangle. But how can i do so? i have tried the second picture one but that is not correct.
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0answers
65 views

Greek cross fractal

I need some code to generate a Greek cross fractal. Example: It must be made of increasingly smaller panels, but the panels may not overlap with previous panels. Does anyone know where I might ...
7
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1answer
463 views

Is a 3D Mandelbrot-esque fractal analogue possible?

I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties). Regardless, I'm wondering if there might be a 'trick' to create a 3D ...
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1answer
53 views

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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2answers
65 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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0answers
51 views

Chaos theory in stock market

I am doing an IB Extended Essay on chaos theory and fractals in the consumer stock market. It is a high school level essay (4000 words) and should be understandable for a calculus student. I'm having ...
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1answer
67 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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0answers
155 views

Is the Hausdorff dimension less than the box counting dimension?

I have been asked to prove that for a bounded set $F\subset\mathbb{R}^n$, $\dim_H F\le \underline{\dim}_B F \le \overline{\dim}_B F$ where $\dim_H F$ is the Hausdorff dimension, ...
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1answer
298 views

How is the study of fractals related to Fourier/spectral/harmonic analysis?

In chap. 3 of "Fractal Geometry of Nature" Mandelbrot mentions that "part of the study of fractals is the geometric face of harmonic analysis" (spectral or Fourier, he specifies), but to my dismay, ...
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1answer
32 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 ...
5
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1answer
119 views

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 ...
0
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1answer
94 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 ...
3
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3answers
176 views

Critical points of a function

The literature on Mandelbrot and Julia sets mentions the phase "critical point" quite a lot, but usually doesn't bother to define what it means. As best as I can tell, a critical point is just any ...
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2answers
633 views

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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1answer
76 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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2answers
62 views

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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1answer
48 views

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 ...
2
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1answer
33 views

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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1answer
40 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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1answer
78 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 ...
0
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1answer
20 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 ...
2
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1answer
23 views

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 ...
3
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1answer
38 views

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
38 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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2answers
168 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) : ...
32
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5answers
4k views

What exactly are fractals

I have always been amazed by things like the Mandelbrot set. I share the view of most that it and the Koch snowflake are absolutely beautiful. I decided to get a deeper more mathematical knowledge of ...
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1answer
42 views

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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1answer
44 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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2answers
104 views

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 ...
3
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0answers
101 views

Is this plot of Ford circles actually a fractal?

Is this plot of Ford circles actually a fractal?
3
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1answer
22 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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1answer
52 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 ...