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

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380 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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1answer
107 views

rearrange $z \mapsto z^2 + c$

The Mandelbrot Set: $Z \mapsto Z² + C$ (or more precisely) $Z_{i+1} = Z_i ^2 + C$ Where $Z$ and $C$ are complex numbers. Can this well-established equation be rearranged to determine things that ...
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1answer
23 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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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. ...
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2answers
126 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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51 views
+50

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 ...
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1answer
283 views

Is The *Mona Lisa* in the complement of the Mandelbrot set.

Here is a description of how to color pictures of the Mandelbrot set, more accurately the complement of the Mandelbrot set. Suppose we have a rectangular array of points. Say the array is $m$ by $n$. ...
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2answers
34 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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70 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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1answer
41 views

Intuitive meaning of fractal dimension.

I'm studying M. Barnsley's book 'Fractals Everywhere', but I'm stuck in the chapter 'Fractal Dimension'. Suppose $(X, d)$ is a complete metric space and let $A \in \mathcal{H}(X)$ be a nonempty ...
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2answers
53 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 ...
2
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2answers
99 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
24 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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2answers
62 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 ...
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0answers
21 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 ...
0
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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 $K$. ...
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0answers
37 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 ...
3
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1answer
36 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
39 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
48 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 ...
0
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1answer
122 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 ...
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1answer
43 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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0answers
26 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 ...
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1answer
70 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 ...
2
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1answer
167 views

Mandelbrot set and riemann hypothesis

Has anyone tried to make a connection between the Mandelbrot set and the non-trivial zeros the zeta function? Looking at the Mandelbrot set, it appears that all points are to the left of the line 0.5 ...
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1answer
15 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?
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2answers
49 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 ...
4
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0answers
109 views

Is there a simplification for the coefficients generated with the Mandelbrot iteration rule?

The Mandelbrot Set is obtained using the equation $z_n=z_{n-1}^2+c$ for some constant $c \in \mathbb{C}$ with $z_0=0$. Therefore, $z_1=c$, $z_2=c^2+c$, $z_3=c^4+2c^3+c^2+c$, etc. I have a function ...
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0answers
20 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 ...
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1answer
36 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 ...
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2answers
43 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
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1answer
36 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 ...
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0answers
32 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, ...
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0answers
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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 ...
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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 ...
3
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1answer
50 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 ...
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1answer
40 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 ...
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0answers
50 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 ...
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2answers
28 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 ...
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1answer
35 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 ...
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1answer
51 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
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2answers
86 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 ...
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3answers
89 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 ...
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1answer
48 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 ...
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2answers
95 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. ...
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2answers
66 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) ...
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0answers
13 views

Identify rules for fractal L-system for plant representation using lengths and angles

As some of you already knew, to develop an L-system fractal you need some rules for angles and lengths. L-system also is well known for its application in plants. So i had a plant in the ground and ...
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1answer
24 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 ...
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1answer
33 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 ...
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0answers
52 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 ...