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

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Can we prove the Mandelbrot set is a fractal? Which maps/processes produce fractals?

So, as you probably noticed, I have two questions. The second leads on from the first. Can we prove the Mandelbrot set is a fractal? It is very easy to see that something like the Sierpinski triangle ...
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Perturbation of Mandelbrot set fractal

I recently discovered very clever technique how co compute deep zooms of the Mandelbrot set using Perturbation and I understand the idea very well but when I try to do the math by myself I never got ...
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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 $z_0=c$. Therefore, $z_1=c^2+c$, $z_2=c^4+2c^3+c^2+c$, etc. I have a function $f(n,x)$ ...
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Supremum of all y-coordinates of the Mandelbrot set

Let $M\subset \mathbb R^2$ be the Mandelbrot set. What is $\sup\{ y : (x,y) \in M \}$? Is this known? To be more descriptive: What is the supremum of all y-coordinates of all black points in the ...
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What types of fractals have a closed-form interior formula?

I was looking at the Menger Sponge earlier, and I realized it has a neat property: Let x, y, and z be spatial dimensions, each between 0 and 1 (inclusive.) Express them as ternary floating point ...
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2answers
39 views

Mathematical background for one wishing to study Chaos/Complexity Theory

I don't have a very strong mathematics background. In fact I quite abhorred mathematics during my Middle/High School years. I'm currently applying for PhD programs in the field of literature as that ...
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96 views

Name of this fractal

I am writing my final paper in the field ob computer enginering my work are on fractals. Some time ago, I found this fractal. Now I need to refer to it in my work but i have no clue what is it called. ...
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65 views

Why such iteration leads to fractal?

I saw a piece of codes like: ...
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64 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 ...
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86 views

Integral over filled Julia sets

Defining the usual quadratic Julia set iteration $f_c(z)=z^2+c$ for complex $c$, and its $n$th iteration $f^n_c(z)=f_c(f_c(\cdots f_c(z)\cdots))$, you can define a function of 4 variables ...
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Mandelbrot sets and radius of convergence

While watching this Numberphile video on Mandelbrot sets, it's more or less stated that the fractal will "blow up" if it's radius of convergence is greater than 2. What is the mathematical basis for ...
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47 views

Behavior of Pascal's triangle in $n\mod m$ where $m>2$, any fractals?

If Sierpinski Triangles are found in Pascal's Triangle under modulo 2 what happens when we view Pascal's Triangle under modulo $m$ where $m>2$? Do fractals appear and if so for which numbers? ...
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Can Wiener process on a fractal random graph be reduced to a levy flight?

Weiner process on small-world graphs is a Levy flight. But does the condition still hold for a random graph that connects the edges of a fractal?
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62 views

Why must fractals be self-referential?

Having an idea of what a fractal is, by example, etc., then seeking the actual definition is, at first, both obvious and imprecise. You'll see it defined as an object that is self-similar in some ...
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How to calculate line-length for fixed width koch fractal?

I am playing with fractals, and drawing them with Python turtle I am using this rules to create l-string for my koch fractal: begin: f f -> f+f--f+f In here, ...
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114 views

Packing infinitely many ellipses into a circle

Given a circle $C$, and an infinite set $S$ of mutually disjoint ellipses which are inside and tangent to $C$, prove that there must exist a disk $D$ which lies inside $C$ but outside every ellipse. ...
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Henon Map Parameter

In case of Hennon map two parameters $a$ and $b$ to be set.The Hénon map takes a point $(x_n, y_n)$ in the plane and maps it to a new point $x_{n+1} = 1-a x_n^2 + y_n$, $y_{n+1} = b x_n$. The map ...
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Reference - formal characterization and analysis of Koch curve

I am studying the Koch curve but most resources I have seen do not describe the Koch curve formally and are similar to the Wikipedia page on the subject. For example, I have looked at books like ...
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60 views

Creating fractals through computers

What are some beginner softwares for creating fractals on computers?
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77 views

Points in a general Cantor set

We often look at the Cantor set with the construction that keeps removing the middle thirds of the remaining line segments at each iteration. Corresponding to this construction, we can determine ...
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54 views

Drawing a nested epicycloid

I would like to learn how to draw this kind of pictures (possibly with Mathematica, as it is the only language I would be comfortable to code such a thing in): There is something similar on the ...
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2answers
38 views

Does there exist a Lipschitz map from the unit interval onto the unit square?

It is well-known that continuous space-filling curves exist. But can they be Lipschitz? Specifically, is there a Lipschitz map from [0,1] onto [0,1]x[0,1]?
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Generalisation of Vitali's covering lemma

In "The geometry of fractal sets", Falconer gives the following generalisation of the Vitali covering lemma as an exercise: Let $\mu$ be any measure on $\mathbb{R}^{n}$ and $E$ a set with ...
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Is the Mandelbrot set computable?

This is a weakened version of Is the measure induced by the Mandelbrot set computable on rational rectangles? ; Given a (computable, or rational) rectangle in the complex plane, is it computable ...
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1answer
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Can we construct a Koch curve with similarity dimension $s\in[1,2]$?

We can make a Koch curve $K$ with similarity dimension $s\in \mathbb Q \cap [1,2]$ by writing $s=\frac{p}{q}$, and constructing such a generator that by scaling with the factor of $2^q$, we'd find ...
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Minkowski content of a Cantor-like fractal

Let $K_0 = [0,1]$. Split $K_0$ into 4 intervals and remove the middle half. This gives $K_1 = [0,1/4] \cup [3/4, 1]$ and so on and set $K = \cap K_i$. I computed the upper and lower Minkowski content ...
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Is a similarity map necessarily affine linear?

My text on fractal geometry introduces the following definition: A map $S: \mathbb R^n \to \mathbb R^n$ is called a similarity map if $$\exists c>0 \ \forall x,y \in \mathbb R^n: ...
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3answers
151 views

What is the topological dimension of the Peano curve?

The Hausdorff dimension of the Peano curve is know to be two. And I assume it to be a fractal since it's on the List of fractals by Hausdorff dimension. Moreover: According to Falconer, one of the ...
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3answers
127 views

How do we solve $c_1^d+\ldots+c_n^d=1$ for $d$?

The question is motivated by the definition of self-similarity dimension for self-similar sets: Let $M \subset \mathbb R^d$ be self-similar. That is, there are $T_1, \ldots, T_m \subsetneqq M$ and ...
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3answers
132 views

Why is the Koch curve homeomorphic to $[0,1]$?

Henning Makholm has provided a nice proof that the limiting curve is a continuous function from $[0,1]$ to the plane. I was curios if the function is homeomorphism. A quick search gave me many sources ...
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Fractals - when the number of seed shapes that can fit into the scaled-up copy is non-integer.

I've heard people say (for eg. here) that the dimension of fractal patterns (particularly, in this question, Lindenmayer fractals) can be formulated as follows: $$D=\frac{\ln N}{\ln S}$$ Where $N$ ...
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Why do we require a finite number of subsets for self-similarity?

Here is how my text defines self-similarity: We call $M \subset \mathbb R^d$ self-similar if there are $T_1, \ldots, T_m \subsetneqq M$ and similarity maps $\alpha_1, \ldots, \alpha_m$ such that ...
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1answer
420 views

Is the two-dimensional Koch curve space-filling?

Say, we'd like to make a Koch curve with self-similarity dimension of two. A Koch curve with the following generator seems to be two-dimensional, since if we double its size by scaling we'll find ...
6
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271 views

Can monsters of real analysis be tamed in this way?

Consider the Weierstrass Function (somewhat generalized for arbitrary wavelengths $\,\lambda > 0$ ): $$ W(x) = \sum_{n=1}^\infty \frac{\sin\left(n^2\,2\pi/\lambda\,x\right)}{n^2} $$ $W(x)$ is an ...
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2answers
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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 ...
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1answer
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Generlized Entropy compared to Generalized Dimension

I am currently reading the following paper by F.Takens: Multifractal analysis of dimensions and entropies. This paper discusses two different measures. One is generalized entropies and the other is ...
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72 views

Reference - Fractal Geometry

I am looking for textbooks or lecture notes about Fractal Geometry that reach an advance level on the topic and aren't just introductory.
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How do you detect when an iterated function converges / diverges and calculate limit accurate-enough?

See this post for a background on what I'm doing. So how many iterations $N$ of $P_c(z)$ does it take so that if $f(c) = P_c^N(z)$, and $g(c) = $ infinite iterations of $P_c(z)$, then $h \circ ...
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1answer
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How do you graph a fractal on a line, as a function of time and position?

I'm writing ANSI C code for work and just got working my LED Light Show Library. The first project it's going on is a sound bar with 10 volume indicator (1-color (white)) leds that are lined-up. ...
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Why does the Mandelbrot shape show up in other fractals?

In the pictures below, the Collatz map fractal includes parts resembling the Mandelbrot set. Why? Do other fractals do so? The Mandelbrot set From Wikimedia Commons Part of the Collatz map fractal ...
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Singular distributions: Applications and Instances

This is the duplication of the question I asked here. I repeat it here with hope of getting new answers. Singular distributions are special mathematical objects. They have an interesting property ...
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1answer
66 views

How do I create a Hilbert curve that is bounded by a polygon?

All images of the Hilbert curve that I've seen show the Hilbert curve as bounded by the unit square: However, if I have a list of vertices that define a closed polygon, how can I create a Hilbert ...
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1answer
105 views

Demonstrating that the Mandelbrot Set is connected

I know that demonstrating the Mandelbrot Set is connected requires a non-trivial proof, and that Mandelbrot himself was fooled at first. But can it be demonstrated visually that the set is connected? ...
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Does the Mandelbrot fractal contain countably or uncountably many copies of itself?

I've been working on a program that draws fractal images, and I was struck by a question that came to mind. It is clear that the Mandelbrot fractal contains infinitely many copies of itself, but I've ...
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2answers
103 views

Are mini-Mandelbrots known to be found in any fractals other than the Mandelbrot set itself?

This is a generalization of the question Are there mini-mandelbrots inside the julia set? @Hagen raises an issue I was afraid of, which is that even the mini-Mandelbrots in the Mandelbrot set are not ...
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312 views

Relationship between the Weierstrass function and other fractals

Consider the Weierstrass function: $$\sum_{n=0}^{\infty}a^n\cos{b^n\pi x}$$ It is well-known as an example of a function that is everywhere continuous and nowhere differentiable. When reading about ...
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Is this plot of Ford circles actually a fractal?

Is this plot of Ford circles actually a fractal?
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algorithm for traversing a fractal in a “maximally ordered” way

consider a multidimensional fractal that can be "traversed" in an arbitrary order. is there an algorithm for traversing a fractal in a "maximally ordered" way? in other words the algorithm has ...
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Fourier decomposition of the Mandelbrot set

It is not clear that the boundary of the Mandelbrot set is an analytic curve, even though it is connected. Nevertheless, we can approximate the boundary with a curve by iterating a finite number of ...
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Intersections of fractal sets with connected sets

Let $\beta \geq \alpha > 0$. Let $A\subset\mathbb R^n$ be a measurable set with Hausdorff dimension $\alpha$. Must there exist a closed connected set $B$ with Hausdorff dimension $\leq \beta$ ...