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

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3
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0answers
57 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 ...
13
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3answers
550 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

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
57 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
48 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
40 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
23 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
30 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
67 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
21 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
32 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 ...
2
votes
1answer
35 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
37 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 ...
0
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1answer
39 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 ...
10
votes
2answers
90 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 ...
7
votes
2answers
157 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) : ...
3
votes
1answer
20 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, ...
1
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1answer
41 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 ...
0
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0answers
26 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: ...
1
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0answers
81 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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0answers
24 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: ...
2
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1answer
48 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
48 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
42 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 ...
0
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1answer
42 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 $d(A,B)$? ...
4
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2answers
197 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 ...
4
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0answers
78 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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0answers
31 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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0answers
25 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 ...
4
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2answers
68 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 ...
6
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2answers
139 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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0answers
36 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. ...
1
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1answer
45 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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votes
2answers
164 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 ...
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2answers
57 views

Examples of Fractals From Simple Algorithms

You all know the Barnsley Fern and The Sierpinski Triangle. I tried to find something similar (to the Sierpinski Triangle) in the disk but all I got was this ring: What would be some other ...
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0answers
76 views

We all know about compositions of functions, but what about decomposition. Is there a way with math, not just heuristics?

The composition operator is a well know and quite often used method in integration and differentiation, think u-substitution. However, given a composition like $$f(f(f(...f(x)...)))$$ Where there are ...
2
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2answers
118 views

Dimension of a Two-Scale Cantor Set

I have a Cantor Set where I begin with a unit interval $[0,1]$. I will remove a middle piece, and the remaining pieces are scaled by $r_1 = \frac{1}{9}, r_2 = \frac{3}{9} $ I am trying to determine ...
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1answer
29 views

Differences in defining the packing (outer) measure

The definition of a packing measure in Falconer's Fractal geometry is given by I am assuming that $\mathcal{P}^s(F)$ as defined in 3.24 is an outer measure (this is not stated in the book). Now ...
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0answers
23 views

Peano curve and Peano's original paper

I have a question regarding the original paper of Peano (here is a link), where he defined his curve in terms of ternary expansions and a mirroring operator. In short, he describes there a continuous ...
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2answers
72 views

Sierpinski triangle game for 3 players

The players are red, green and blue. The game is played on a n-deep Sierpinski triangle. Each player colors a (black) triangle, starting at one of the main vertices. They then take turns to color an ...
5
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0answers
82 views

Distance and Coordinates in fractional dimensions and the creation of functions with non-integral numbers of paramters.

Background: The Euclidean distance between two points in $n$ dimensions, where $n$ is a positive integer, and position can be described by a vector is given by... $$D_E=\left(\sum_{k=1}^n ...
2
votes
1answer
46 views

Proof of fractal dimension of Thomae's function

Thomae's function is defined to be $0$ if x is irrational. Its defined to be $1 \over q$ where $x={p \over q}$ in lowest terms and $q \gt 0$. Its measure is $0$ since the set of rational numbers is ...
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0answers
43 views

the 2D fractional Gaussian noise as derived from the 2D fractional Brownian motion

Let $X_n$ be a 1D discrete fBm. Then, its 1st order difference, $W_n=X_n-X_{n-1}$ is fractional Gaussian noise (fGn). This case is simple. But what happens in 2D? Let $Y(m,n)$ be a 2D fBm, then we ...
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1answer
19 views

Smallest integer $N(\epsilon)$ such that $K\subset \bigcup_{n=1}^{N(\epsilon)}B(x_i,\epsilon)$

In a metric space, a set $K$ is said to be totally bounded if for each $\epsilon>0$ there exist a finite number of balls $B_1,B_2\dots B_{N(\epsilon)}$ with radius $\epsilon$ which covers $K$. ...
0
votes
0answers
44 views

Greek cross fractal

I need some code to generate a Greek cross fractal. Example: http://commons.wikimedia.org/wiki/File:Greek_cross_3D_1_through_4.png It must be made of increasingly smaller panels, but the panels may ...
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1answer
70 views

Taylor series of mandelbrot bulb boundaries

What I am looking for is a way to find an approximation to the boundaries of hyperbolic components of the Mandelbrot set. I would like to be able to write a program to find the equations which ...
1
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1answer
46 views

Bisecting a fractal area

Simple case It is well-known that if we have a regular hexagon on a plane, then every line that passes through the centre of the circumscribed circle bisects the area of the hexagon. Extension ...
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1answer
87 views

Can you help me find a fractal drawing program?

In a previous course on chaos, the professor had us experiment with a program. The program allowed you to draw a base image (with microsoft paint like tools), then it would iterate that image under ...
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0answers
28 views

Can you help find me a particular fractal drawer? [duplicate]

In a previous course on fractals and chaos, the professor had us experiment with a program. The program allowed you to draw a base image (with microsoft paint like tools), then iterate that image ...