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

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7
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2answers
176 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
vote
0answers
45 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. ...
2
votes
1answer
148 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
365 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
vote
2answers
281 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
votes
0answers
88 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 ...
3
votes
2answers
206 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
37 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 ...
1
vote
2answers
88 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
votes
0answers
143 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
76 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
votes
0answers
68 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
24 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 $K$. ...
1
vote
0answers
99 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 ...
1
vote
1answer
103 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
69 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
121 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 ...
1
vote
3answers
82 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 ...
-1
votes
1answer
209 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
44 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
vote
0answers
57 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
49 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 ...
2
votes
2answers
95 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 ...
1
vote
1answer
80 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
vote
0answers
38 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, ...
1
vote
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 ...
2
votes
1answer
83 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 ...
4
votes
0answers
77 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
99 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
139 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
99 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 ...
2
votes
1answer
112 views

Negative Fractal dimension values in plants images

After calculating lengths and angles from a plant i represented it with the help of L-system fractals (see image below). I made that process for many plants and then i went to matlab to calculate ...
2
votes
2answers
230 views

Collage theorem to generate a spiral

I need to answer a question on fractals from the book Fractals Everywhere by M. Barsley and I have been struggling with it for a while: Use collage theorem to help you find an IFS consisting of two ...
5
votes
3answers
209 views

Is there a koch circle?

Is there some fractal like the koch snowflake, but only with many circles around a bigger initial circle, each of them surrounded by smaller circles and so on (but all of them kissing one bigger ...
1
vote
2answers
129 views

Proof of x-intersection of the Mandelbrot Set?

I'm trying to prove that the Mandelbrot set intersects the X-axis on the interval [-2,.25]. I understand and have proven that the Mandelbrot set lies in a radius of 2. Mostly, I'm wondering how to ...
4
votes
2answers
143 views

What is the algorithm hiding beneath the complexity in this paper?

So, I am a computer scientist (at least, I'm working to become one..) and I asked a question on here concerning some mathematics behind the Mandelbrot set. A reply I recieved pointed me to this paper. ...
0
votes
2answers
71 views

Bounded bessel functions in an s-set projection proof

The following is an extract from Falconer's Geometry of Fractal Sets about the proof of: "...Using the definition of a Bessel function $J_0=\frac{1}{2\pi}\int^{2\pi}_0 \cos(u \cos \theta) ...
0
votes
1answer
62 views

Infinite number of points in the Sierpinski Triangle

I have basic background in mathematics (Linear Algebra, Calculus) and I've been reading up on fractals, because I find them fascinating. I can't understand one thing in basically all of the fractals ...
5
votes
0answers
90 views

Symmetric Icon Fractals

I have always been fascinated by fractals. But most of all I like the Symmetric Icon fractals. There is a nice book about these fractals, written by Michael Field, called Symmetry in Chaos. I'm ...
3
votes
0answers
50 views

Is there a name for the relation between Menger Sponge and Vicsek Fractal?

Both the Menger Sponge and the Vicsek Fractal in 3D can be constructed by starting with a cube, dividing it into 27 smaller cubes (3x3x3 grid), removing some of these cubes, and then applying the ...
3
votes
1answer
183 views

Finding external angles for Misiurewicz points in the Mandelbrot set

In the Mandelbrot set for the quadratic polynomial $z \to z^2 + c$, rational external angles with even denominator are pre-periodic and have corresponding external rays which land at Misiurewicz ...
1
vote
2answers
34 views

A zero-dimension set and self-referencial equation

Let $K$ be a compact set in $\mathbb{R}^2$. Let $f_1,..., f_n$ be contracting similarities of $\mathbb{R}^2$ to itself. Suposse $K$ satisfies the self-referencial equation ...
1
vote
1answer
67 views

Bounding dimension of IFS

Given the IFS $\{\frac x {2+x},\frac 2 {2+x}\}$ ($0\le x \le 1$) with attractor K prove that $0.53<\dim_HK<0.8$ I thought using the results from my last question by saying ...
0
votes
1answer
120 views

Proving ineqalities for the similarity dimension

a. Let $K$ be the attractor of the IFS $\{f_1,\dots f_n\}$ which satisfies SSC (i.e $f_i(K)\cap f_j(K)=\emptyset\forall i\neq j$) where for all $i, c_i$ such that $ 1\le i\le n, \space ...
0
votes
1answer
95 views

Upper Minkowski content and finite Hausdorff measure

Does someone know an example of a set $E$ with positive finite $s$-Hausdorff measure, Minkowski dimension $s$, and infinite $s$-dimensional upper Minkowski content ? The $s$-dimensional upper ...
1
vote
1answer
62 views

Minkowski dimension behaviour of sets with positive finite Hausdorff measure.

It is (rather) well known that the set \begin{equation*} E=\{k^{-1},k\in\mathbb{N}^{*}\} \end{equation*} has box-dimension $1/2$ and Hausdorff dimension $0$. However $H^{0}(E)=|E|=+\infty$. Is it ...
1
vote
1answer
69 views

Solving inverse problem related to Iterated function systems?

I generated a Barnsley's fern fractal using details in this link with the aid of MATLAB. My doubts are as follows : How do we justify the shape generated from those equations? Is it possible to ...
3
votes
2answers
175 views

Examples of bounded continuous functions which are not differentiable

Most often examples given for bounded continuous functions which are not differentiable anywhere are fractals.If we include probabilistic fractals exact self-similarity is not required. Are their ...
1
vote
1answer
51 views

Prove carpet has positive Hausdorff measure in its dimension

Given $D\subset\{0,1,2,\dots n-1\}\times\{0,\dots,m-1\}$, let $$K(D)=\{\sum_{k=1}^\infty(a_kn^{-k},b_km^{-k}):(a_k,b_k)\in D\forall k\}.$$ Show that if $D$ has uniform horizontal fibers (i.e. the ...
5
votes
1answer
223 views

Finding the location of an image of the Mandelbrot set

I've got an image of a segment of the Mandelbrot set that I generated with an iPhone app a long time ago (I use it as my background image). I now have no idea where in the set the image came from. ...