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

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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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1answer
86 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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34 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 ...
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2answers
239 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 ...
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2answers
36 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 ...
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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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68 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
35 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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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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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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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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50 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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2answers
63 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
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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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 ...
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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 ...
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2answers
631 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
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 ...
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1answer
74 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 ...
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1answer
32 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
39 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 ...
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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 ...
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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 ...
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1answer
37 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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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 ...
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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) : ...
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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 ...
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30 views

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: ...
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0answers
88 views

Is this Fractal New?

I developed some equations relating to symmetry. When used recursively, they produce what I believe is a fractal of symmetries. The fractal is procedurally generated like a snowflake or a gasket, ...
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34 views

how to interpret the singularity spectrum?

I have been reading some articles about multifractal analysis and I still do not have the idea pretty clear; for example in the following example found in: ...
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1answer
62 views

What Method is used for Projecting the Rauzy Fractal?

I am trying to construct the Rauzy Fractal (http://en.wikipedia.org/wiki/Rauzy_fractal), I have a Tribonacci word generator and have the stairs constructed but I can't seem to get the projection onto ...
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1answer
51 views

Fractal dimension of a dense subset

Let $M$ be a metric space and $S\subset M$ a dense subset. For vague reasons (below), it seems to me that the upper box-counting dimension of $S$ should be equal to that of $M$, but I don't quite see ...
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0answers
55 views

Proof of an Analogue to Vitali Covering Lemma

Let $\mathscr{C}$ be a family of balls contained in some bounded region of $\mathbb{R}^n$. Then for any $\alpha > 3$, there is a finite or countable disjoint subcollection $\{ B_i \}$ such that ...
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1answer
46 views

Hausdorff distance

Let $A=\{(x,y)∈R^2: x^2+y^2\le4\}$ and $B=\{(0,y)∈R^2:|y|\le3\}$. Determine Hausdorff distance between $A$ and $B$. I wrote $d(2)(B,A)=((0,3),A)=((0,3)(0,2))=1$. What about $d(A,B)$? ...
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2answers
221 views

How to compute a negative “Multibrot” set?

The Mandelbrot set is defined as follows: given the function f(z, c) = z2 + c, a number z in the complex plane is in the Mandelbrot set if and only if the sequence ...
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0answers
90 views

Integral of a function over the Koch Curve. Is it rigourous enough?

(I want to investigate the validity of this approach, as I already know this is the correct result) I present a proof that $$\int_{K} (x+y) \ \mu(x,y)={{9+\sqrt 3} \over 18}$$ Where the region of ...
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32 views

how do i prove that a collection of contractions does not satisfy the open set condition?

I am studying a fractal that is defined by 4 similarities, similar to the Von Koch curve, and I am trying to verify that it does not satisfy the open set condition. The fractal is heavily ...
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27 views

Hausdorf dimension of fractal iterates

For fractals defined iteratively (via subdivision) like the Koch curve or Sierpinsky triangle, what is the Hausdorf dimension of the intermediate iterates? Specifically, for a fractal S defined as ...
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2answers
81 views

Chaos theory and fractal geometry: Constructing from data

I understand that fractal geometry represents behaviour of 'chaotic' system, if I am not wrong. And also, fractals are generated by a recursive function. But, lets say I have random data lying with ...
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2answers
153 views

This one wierd trick integrates fractals. But does it deliver the correct results?

It occurs to me that people most likely already know how to explicitly integrate over fractals, but my method (edit: seems to have been highlighted out in a paper, see comments) seems to vastly ...
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41 views

Intuition for Entropy over Fractals

Is there intuition for "mathematical" entropy. I know that physical entropy tracks the order in a dynamical system, for thermodynamics. As entropy goes up, general randomness and disorder goes up. ...
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1answer
59 views

an example of when Hausdorff and box-counting dimensions are equal?

I am new to fractals and dimension theory, so please excuse any errors in my understanding. For a set $F$, let $dim_b (F)$ be the box counting dimension of $F$, and $dim_H (F)$ be the Hausdorff ...
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2answers
199 views

Sierpinski (Triangle) for Other Polygons

The Sierpinski triangle can be "generated" by the algorihm where you start in the triangle, pick a vertex at random, then move half the distant towards it, draw a dot and then repeat this. I wasn't ...