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

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6
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2answers
51 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
votes
1answer
54 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
19 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
vote
1answer
32 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
votes
0answers
24 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
vote
0answers
72 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, ...
0
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0answers
14 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
votes
1answer
37 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 ...
1
vote
1answer
45 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 ...
0
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0answers
36 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
votes
1answer
40 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
votes
2answers
163 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 ...
5
votes
0answers
68 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 ...
1
vote
0answers
29 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 ...
1
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0answers
23 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
votes
2answers
57 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
votes
2answers
127 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 ...
1
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0answers
35 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
vote
1answer
34 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 ...
5
votes
2answers
150 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 ...
1
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2answers
44 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 ...
2
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0answers
73 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
votes
2answers
111 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 ...
1
vote
1answer
28 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 ...
0
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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 ...
1
vote
2answers
67 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 ...
6
votes
0answers
77 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 ...
3
votes
1answer
43 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 ...
3
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0answers
39 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 ...
0
votes
1answer
18 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
41 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 ...
1
vote
1answer
61 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
vote
1answer
44 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 ...
1
vote
1answer
77 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 ...
0
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0answers
27 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 ...
1
vote
2answers
60 views

How does one determine the containing boundary of a fractal?

In the Mandelbrot set, the fractal is said to be contained in the circle of radius 2. $$ z_{n+1} = {z_{n}}^{2} + c $$ I did read about a proof that showed values of 'c' beyond this circle are not ...
0
votes
1answer
129 views

How does one estimate the Hausdorff measure for arbitrary fractals, and does the constant c in $N=c\epsilon^d$ provide a good estimate?

Background: When one finds the fractal dimension of a fractal in real life, they will generally use the relation $N=c\epsilon^d$ to do so. However, the constant c is almost always neglected in ...
1
vote
1answer
17 views

Is there a hilbert curve equivalent for circles?

Is there a space-filling curve that has the same properties of a hilbert curve (two points close in 1D are close in 2D) but grows in a circular shape instead of a rectangular one?
1
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0answers
28 views

multifractal scaling exponent tau(q) - concave up or down?

I have read some conflicting information from two reliable sources regarding the scaling exponent in multifractal systems - tau. On the Yale website devoted to fractals, they say "Tau is a decreasing ...
1
vote
1answer
37 views

What are properties of dynamical systems in non-integer dimension spaces?

A 1D dynamical system (R1) exhibits convergence to a fixed point, or escapes to infinity. A 2D dynamical system (R3) can produce oscillations, spiral-shaped trajectories, etc. A 3D dynamical system ...
3
votes
2answers
49 views

Explicit formula for IFS fractal dimesnion

Is there an explicit formula for finding the box counting dimension of an arbitrary IFS fractal, such as the IFS fern or any other random IFS fractal? If not, is there at least a summation, or ...
2
votes
1answer
45 views

Integral over Fractals with respect to fractal dimension

I understand that there is type of integral with respect to measures that can return values when evaluated over an integral. But is there an Integral that returns d dimensional volume where d is the ...
1
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0answers
33 views

Are the iterates of this function bounded?

I have the function $f(z) = \sqrt z + C.$ For the value of $C = i$ (complex number), would the iterates be bounded or not? Iterating from $z = 0: f(0) = i, f(i) = \sqrt i + i$ and it goes on, ...
0
votes
0answers
15 views

Comparing fractals plant representations

After collecting data , by measuring angles and lengths of branches on some plants, i tried to represent them with L-system fractals. Let's assume that we have two plants (see below) . Those two ...
1
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1answer
30 views

Prove that $\mathscr{H}^0(F) = |F|$.

As stated above, I'd like to prove that the 0-dimensional Hausdorff Measure of a set $F \subset \mathbb{R}^n$ is the cardinality of $F$. In other words, that $\mathscr{H}^0 (F) = |F|$, or the number ...
1
vote
1answer
44 views

Mandelbrot Set - Predict which value of c will give bounded results?

I have been looking into the Mandelbrot set a little bit lately, and I have a question. Given the equation: $$z(n+1) = (zn)^2 + c$$ where $c$ is a complex number of the form $a+bi$ is there an easy ...
2
votes
0answers
54 views

interior distance estimate for Julia sets - getting rid of spots

From wikibooks colouring the Julia set, the distance estimate $\delta(z)$ can be calculated by: $$\begin{aligned} \delta(z) &= \lim_{n \to \infty} \frac{|z_n| \log ...
2
votes
2answers
53 views

test for membership in mandelbrot bulb of period n

Is there a efficient test (formula or inequality) of whether a given point is in a bulb of period n? In other words, something other than running the iteration a lot of times to see if it converges ...
3
votes
1answer
55 views

Numerical computation of unlimited small Julia set details

I've read the claim of a fractal image application to be able to show infinite levels of zoom for Julia sets for the classic iteration formula $z_{i+1}:=z_i^2+c$. The application has a realtime ...
1
vote
1answer
42 views

Generalizing the Apollonian Gasket to other closed curves

An Apollonian Gasket is a fractal set constructed out of tangent circles. The first stage is three mutually tangent circles (which are not all tangent at a single point). At each step, we can take a ...