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

learn more… | top users | synonyms

46
votes
4answers
5k views

Does this Fractal Have a Name?

I was curious whether this fractal(?) is named/famous, or is it just another fractal? I was playing with the idea of randomness with constraints and the fractal was generated as follows: Draw a ...
14
votes
3answers
234 views

What is the moment of inertia of a Gosper island?

We know that regular hexagons can tile the plane but not in a self-similar fashion. However we can construct a fractal known as a Gosper island, that has the same area as the hexagon but has the ...
0
votes
0answers
24 views

Is there a there a non intersecting mapping to unit square.

Is there a way to go from the fat cantor set to a half unit square in a non intersecting way using Hilberts curve? How would I go about constructing a non intersecting space filling curve of non zero ...
3
votes
1answer
104 views

Is a hypersphere of non-integer dimension a fractal?

Thanks to the gamma function the formula for the surface of a unit http://mathworld.wolfram.com/Hypersphere.html $$ S(n) = \frac{2 \pi^{n/2}}{\Gamma(n/2)} $$ allows to calculate the surface of a ...
0
votes
0answers
15 views

Sierpinski gasket is the closure of a set of its vertices

Why Sierpinski gasket is the closure of a set of its vertices? Let $V_0:=\{0=p_0, p_1, p_2\}$ be vertices of an equilateral triangle, and let $\hat {\mathcal H} := \bigcup_{i=0,1,2}(- + p_i)/2$ ...
1
vote
1answer
30 views

IFS which construct this fractal and have affine transformation only

[Image updated] Is there an IFS which construct this fractal and have affine transformation only? (I think there must be a restriction, which is not an affine transformation. Can it be proved?)
1
vote
0answers
15 views

When the self-similar dimension and the Hausdorff dimension are different?

By the en.wikipedia, for the self-similar sets in a metric space, the self-similar dimension and the Hausdorff dimension are often the same, but not always. Is there a known sufficient-necessary ...
6
votes
1answer
83 views

What is the shortest path to a “little Mandelbrot” from $i$?

As you all already know, the Mandelbrot set has little "copies" of itself strewn throughout the boundary region (some of them distorted somewhat), and these are all connected. The point $i$ (or $x = ...
2
votes
1answer
25 views

Exact value of Hausdorff measure of middle-third Cantor set

Is there any result about the exact value of $\log_3 2$-dimensional Hausdorff measure of the middle-third Cantor set? And is there any fractal (in $\mathbb R^n$) which is not contained in a ...
0
votes
0answers
35 views

What´s the dimension of a Sierpinski fractal?

I know the dimension of a Koch snowflake (log4/log3), but what numbers do I have to put in to obtain the dimension of a Sierpinski fractal?
-1
votes
0answers
14 views

What´s the dimension of a Sierpenski fractal [duplicate]

I know the dimension of the Koch snowflake (log4/log3), but what numbers you have to put in for the Sierpenski fractal?
2
votes
1answer
34 views

What is the condensation set of a fractal?

Is there a definition of the condensation set of a fractal that is both clear and rigorous? I've been searching around to get a sense of what exactly the condensation set of a fractal is - I've ...
6
votes
1answer
66 views

Can the fractal dimension of a surface be less than 2?

I have two surfaces represented as raster images with heights as grayscale values. One is natural landscape elevations; the other is just distance from a line. I have computed Minkowsky D = 2 - H ...
1
vote
0answers
29 views

How was one derivied from the other?

In the geological paper entitled The power–law relationship between landslide occurrence and rainfall level by C. Li et al, a power-law cumulative probability distribution is derived. However, I don't ...
0
votes
1answer
31 views

How to determine constant $C$ in $p(x) = Cx^{-D}$?

Given a distribution obeying the power-law (fractal) relation, such as the cumulative distribution function $L_{cf}(> X) = CR^{-D}$, if $X$ is given, how does one find the constant $C$ from a given ...
1
vote
0answers
53 views

Scaling factor closest to 1 in an infinite sequential rectangle packing

The Ammann Chair can be used in an infinite dissection of a rectangle, where the pieces have a scaling factor of $ k = 1/\sqrt{\phi} = 0.786151...$. The largest piece has area $\sqrt{5}$ and longest ...
0
votes
2answers
51 views

hyperbolic spaces and fractals

Is there a relation between hyperbolic spaces and fractals? In group theory, if we take the Cayley graph of a free group on two generators, we get a fractal quaternary tree, which I'd like to think as ...
4
votes
1answer
72 views

Is this a valid definition of “self-similar fractal”?

I have always been fascinated by self-similarity, particularly in fractals. I was always wanted to find a simple definition of a self-similar fractal. Of course, saying "is self-similar, and is a ...
2
votes
1answer
38 views

Why c>1/4 is not in Mandelbrot set

As title: $f_c(x)=x^2+c$ I got to the step: $f_c(x)>x$ (for all x) But what's next? How to show that after k iterations, $f^k_c \to \infty$ as $k \to \infty$ Thanks,
2
votes
1answer
55 views

Fractal fundamentals

I am a programmer by trade, and am very interested in fractals. To be very basic about the concept, one might say a 'circle of circles' is a fractal. Where each circle is made up of circles, and ...
2
votes
1answer
79 views

Is every basin of attraction completely invariant?

I can't seem to find a definitive answer in the literature. I believe the answer is yes, but my focus has been on the rational maps on the Riemann sphere. At the very least I'm confident that if the ...
0
votes
1answer
59 views

Is there a general metod to construct a fractal?

I would like to construct a fractal (traditional, self-affine, and fat fractal) with a given embedding and fractal dimension, but I don't know how to do it programmatically. The shape of the fractal ...
1
vote
0answers
36 views

How to generate/validate unique fractal?

There are many known fractals that exist such as Mandelbrot, Cantor set, or the Koch curve, Sierpinski Triangle. What I am curious about, is how one could go about creating their own, unique fractal ...
5
votes
1answer
79 views

How is this fractal produced?

It is stated here: Iterating the above optimized map $$f(z)=\frac{1}{4}(1 + 4z - (1 + 2z)\cos(\pi z))$$in the complex plane produces the Collatz fractal. The point of view of iteration on ...
2
votes
0answers
32 views

Is the generalized mandelbrot set a fractal in the $d$ dimension?

The $d$-mandelbrot set is defined as the set of $c$ such that the iterations of $$z \mapsto z^d + c$$ starting with $z=0$ is bounded in absolute value. Here is a picture of the mandelbrot sets from ...
0
votes
0answers
31 views

On a formulation in Hilberts original paper about the space-filling Hilbert curve

I have a question on the famous paper Über die stetige Abbildung einer Linie auf ein Flächenstück (which translates roughly as On the continuous mapping of a line onto a square) by D. Hilbert. Let the ...
2
votes
1answer
52 views

Is there a plane filling function calculator online?

I recently read about the "Hilbert Curve" and found it very interesting. Does anyone know of a place online where I could extrapolate different shapes and explore this field of mathematics?
1
vote
0answers
43 views

Deriving convergence region of iterative formula

A year ago I asked this question about fractal icons, however I didn't get any wiser yet. Now I am trying to understand the convergence of a simplified version of the fractal, to learn more about the ...
17
votes
0answers
153 views

Has this chaotic map been studied?

I have recently been playing around with the discrete map $$z_{n+1} = z_n - \frac{1}{z_n}$$ That is, repeatedly mapping each number to the difference between itself and its reciprocal. It shows some ...
3
votes
2answers
89 views

Different Coloring of Julia Sets

I have known about Julia Sets for a while now, and today I had an idea about the coloring of Julia and Mandelbrot Sets. What if someone were to color them not only by how quickly z diverges, but also ...
13
votes
1answer
166 views

Projection of Antoine's necklace

Antoine's necklace is a pathological embedding of the Cantor set into $\Bbb R^3$. The second iteration looks like this: Interestingly, the complement $\Bbb R^3\setminus\rm A$ is not simply ...
0
votes
2answers
65 views

Maps on the hyperspace of compact sets

In the theory of fractals via iterated function systems, it is well-known that an IFS $\{f_i\}_{i=1}^n$ (being a finite collection of contractions defined on a metric space $X$) induces a single map ...
0
votes
1answer
49 views

Notation of set on $R^2$ (attractor of Cantor dust)

I am studying fractal geometry, and pretty much confused by the notation/meaning below: Quote: The Cantor dust is easily seen to be the attractor of the four similarities on $R^2$ which give the ...
0
votes
0answers
14 views

Maximum principle in harmonic function over finite sets.

I want to understand a proof about laplacian operator on finite sets. This is an analogous result of the well known maximum principle for harmonic functions. Let $V$ be a finite set and let $H$ be ...
1
vote
0answers
13 views

Why is there periodicity in the output of Richard Voss' fractional Brownian motion?

I am trying to figure out why the output of fractional Brownian motion (fBm) as described by Richard Voss (Random fractal forgeries. In: Fundamental Algorithms for Computer Graphics, R. A. Earnshaw ...
1
vote
0answers
35 views

Hausdorff dimension of Sierpinski triangle

https://en.wikipedia.org/wiki/Hausdorff_dimension#Behaviour_under_unions_and_products Wikipedia page says that if $ \underset{i \in I}{\cup} X_i = X$ and $I$ is countable then $dim_{Haus}(X) = ...
8
votes
1answer
131 views

Roots of iterations of polynomials

Let $f \in \Bbb Q[X]$ a polynomial, and let denote by $f^n$ the composition $\underbrace{f \circ \cdots \circ f}_{n \text{ times }}$. Let $R(f^n) \subset \Bbb C$ the roots of $f^n$. I'm interested in ...
2
votes
0answers
145 views

How to find Misiurewicz Points without solving huge polynomials? (Mandelbrot Set)

Here is a plot of 17,723 Misiurewicz Points. The code below generates a set of polynomials u[m,n], the roots of which have periodicity (m-n) starting at iteration n. I stopped at 17,723 points ...
1
vote
0answers
32 views

Show that $E_\mu$ has no periodic points that are not fixed points

Problem statement: Consider $E_\mu(x)=\mu e^x$, where $0<\mu<1/e$. Show that $E_\mu$ has no periodic points that are not fixed points. It is in my understanding that what we need to show is ...
7
votes
1answer
188 views

Continuous path inside the Mandelbrot set connecting i to zero?

This relates to another challenge Question about drawing Mandelbrot filaments. It is possible to compute a formula for a continuous path inside the Mandelbrot Set connecting $c=i$ to $c=0$? ...
1
vote
0answers
45 views

Lower bound on dimension for nearest neighbor classifier to fail at k=1 and pass at k=3

What is the minimum dimensionality of a dataset of a finite number of points where 1-NN has an accuracy of 0% but 3-NN has an accuracy of 100%. This is certainly possible in 3 dimensions and my ...
1
vote
0answers
55 views

converting to math from economics major

Recently, i'm majoring in honour track of economics taking econometrics statistics courses and minoring in mathematics taking advanced calculus, real analysis ,linear algebra courses. Upon research on ...
1
vote
1answer
44 views

Proving that a sequence is unbounded without knowing the sequence explicitly

Given that $f(x)=x^2+\frac{1}{4}$, there exists the iterated sequence ${f^{\circ n}(x)}_{n=1}^\infty$ (where $f^{\circ n}(x)$ is defined as $\underbrace{f(f(f...(x)...))}_{n\text{ times}}$), which is ...
0
votes
2answers
41 views

Proving that a sequence is bounded without knowing the sequence explicitly

Given that $f(x)=x^2+\frac{1}{4}$, there exists the iterated sequence ${f^{\circ n}(x)}_{n=1}^\infty$ (where $f^{\circ n}(x)$ is defined as $\underbrace{f(f(f...(x)...))}_{n\text{ times}}$), which is ...
0
votes
0answers
25 views

Random process theory: probability distribution of height vs summits

Imagine I have a matrix of height values ($z$), e.g. a surface height topography. This surface is a random process: randomly rough isotropic surface with Gaussian distribution. What is the difference ...
4
votes
1answer
243 views

How to draw a Mandelbrot Set with the connecting filaments visible?

The M-Set is connected. But the M-Set viewers I’ve found create cool pictures that don’t really show the connecting filaments. This mini-Mandel beetle should be connected to a larger min-Mandel by a ...
1
vote
1answer
22 views

Hausdorff Dimension of $f(F)$

I am currently working through several problems regarding the following fact: For $F \subset \mathbb{R}$, $f: F \rightarrow \mathbb{R}$, we have that $dim_H(f(F)) \leq dim_H(F)$ I am fine with ...
2
votes
0answers
50 views

Are the vertices of a Voronoi diagram obtained from a Sierpinski attractor also a kind of attractor?

Trying to understand how the Voronoi Diagrams work I did a test generating the Voronoi diagram of the points obtained from The Chaos Game algorithm when it is applied to a $3$-gon. The result is a set ...
1
vote
0answers
49 views

Simulation of brownian motion and fractional brownian motion

It's easy to simulate a path of a brownian motion with the method explained in Wiener process as a limit of random walk: ...
1
vote
1answer
31 views

Generalizing the Cantor Set to the $n$-dimensional plane

I am interested in how to describe an $n$-dimensional cantor set. I think that it may be a good idea to develop the Cantor Set on the two-dimensional plane at first, but I am having issues figuring ...