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

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45
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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 ...
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 ...
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 ...
16
votes
3answers
223 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 ...
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 = ...
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 ...
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 ...
1
vote
2answers
58 views

$L$-Systems: Order of Substitution

I am working the a subject guide on involving $L$-Systems and have the alphabet $A = \{a, b, c\}$. The initiator is the string $a$ and the rules of substitution $a \to ba$, $b \to ccb$, $c \to a$. ...
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
2answers
127 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 ...
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
33 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 ...
1
vote
2answers
238 views

How to correctly calculate the fractal dimension of a finite set of points?

The box-counting dimension is defined by: $\lim\limits_{\epsilon \to 0} \dfrac{N(\epsilon)}{1/ \epsilon}$ What works well if you are solving algebraically or if you can recursively generate more ...
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 ...
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 ...
0
votes
2answers
50 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 ...
37
votes
5answers
4k views

What exactly are fractals

I have always been amazed by things like the Mandelbrot set. I share the view of most that it and the Koch snowflake are absolutely beautiful. I decided to get a deeper more mathematical knowledge of ...
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 ...
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 ...
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 ...
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 ...
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?
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 ...
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 ...
4
votes
2answers
427 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 ...
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 ...
3
votes
1answer
53 views

Centroid of Mandelbrot Set

How to find the geometric centroid of Mandelbrot Set?
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 ...
4
votes
0answers
134 views

The Hausdorff dimension of the set of solutions of a system of coupled differential equations

I am interested in the relationship between non-linear differential equations and the Hausdorff, or fractal, dimension of the set of solutions. For example, the Lorenz Attractor, which is a standard ...
4
votes
0answers
117 views

Properties of King's Dream fractal

My question is focused on the King's Dream fractal, which can be defined as follows (nice pictures can be found here) : $$ \Omega = \{f^n(0.1,0.1) \;\vert\; n \in \mathbb N \} \quad ...
3
votes
1answer
81 views

Can a plane be split into three connected sets so that $\epsilon$-neighbourhood of any point of any one set also contains points of two other sets?

Math SE. This question was a shower thought of mine. I tried to come up with an answer by twisting comb spaces and cantor sets, but to no avail. I was educated as experimental physicist, so I ...
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 ...