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

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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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7 views

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
33 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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3answers
106 views

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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1answer
232 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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Basic concept books [closed]

I would like to refresh my basics & I would like to know the best books for the following topics [basic/beginner level with practical real life applications] Curves, Splines, Surfaces & ...
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2answers
72 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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2answers
258 views

Hausdorff's distance of some sets

We define $H^{n}$ for the set of all compact subsets of $\mathbb{R}^n$. Define the metric $\Delta$ in $H^{n}$ as following.Let $A,B \in H^{n}$ then define $d(x,B):= \min \lbrace d(x,y): y \in B ...
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0answers
43 views

Is this plot of Ford circles actually a fractal?

Is this plot of Ford circles actually a fractal?
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16 views

How to… quotient set on a fractal continuous curve.

I'm really not good at math so I can't really formulate my problem in a closed form :) There is a curve $C$ in $R^2$. There are some rulers of length ${L1,L2,L3,L4,L5,....}$ I need to find a way to ...
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14 views

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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1answer
38 views

Reference for a Cantor set in the plane formed from series of roots of unity

This is a long shot, but I'm looking for a particular article that I once read, and I'm trying to find it again. It deals with a certain Cantor set in the plane. The set could be written as something ...
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1answer
29 views

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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0answers
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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 $\leq \beta$ ...
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1answer
43 views

Cantor sets and drawing figure-8s in the plane

I know that the Cantor Set is uncountable (this is a well-known result), so I know that there must be something wrong with the following method for counting its elements, but I'm not sure where the ...
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24 views

Hausdorff dimension is less than box counting dimension?

I have been asked to prove that for a bounded set $F\subset\mathbb{R}^n$, $dim_H F\le \underline{dim}_B F \le \overline{dim}_B F$ where $dim_H F$ is the Hausdorff dimension, $\underline{dim}_B ...
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3answers
298 views

Quadratic Julia sets and periodic cycles

Consider the function $f_c(z) = z^2 + c$. Applying this function repeatedly, we get the familiar quadratic Julia sets that fractal enthusiasts burn compute cycles plotting. Infinity is always one ...
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5answers
2k 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 ...
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2answers
50 views

How to interpret the OEIS function for the “even fractal sequence” A103391 (1, 2, 2, 3, 2, 4, 3, 5, …)

I'm interested in a particular integer sequence that is the same as itself when you remove all of the even-indexed members of the sequence. It begins (1, 2, 2, 3, 2, 4, 3, 5, ...). I looked it up in ...
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2answers
468 views

What's wrong with this 'open cover' of the Koch Snowflake?

This question is to help me find peace. First, the question of the Snowflake's compactness has been tackled here on this site: Is the Koch Snowflake a Compact Space? Is Koch snowflake a continuous ...
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1answer
48 views

Box counting fractal dimension

S: 1,0.3,0.22,0.12,0.06 N: 7,201,478,2595,17950 (no idea how to put this in a tally) I've got a question here where S is not shrunk by the same fraction throughout, I know how to work out the ...
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1answer
70 views

From 'The Joy of x' book: John Hubbard and problems with multiple roots

My math skills are super rusty. In an effort to get some vigor back I started some reading and picked up The Joy of x based on its rave reviews.. I just couldn't make any sense out of the following ...
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2answers
4k views

Has anyone found a “pattern” in prime numbers?

Yesterday I was having some fun trying to look for some patterns in primes; and I think I found something interesting (to me at least). I still have not found any lists of patterns already found, ...
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1answer
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Self similar set which does not fulfill the open set condition

Informally, a set is considered self similar if it consists of smaller copies of itself. If this set fulfills the so called open set condition, one can easily calculate the Hausdorff Dimension (see ...
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Why are fractal curves nowhere differentiable?

I am a highschool student who stumbled upon fractals when doing a math project. In my research about fractals, I have found that they are nowhere differentiable. Can someone explain this in simple ...
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1answer
259 views

Every basin of attraction contains a critical point?

Years and years ago, back when I first became interested in fractals [but didn't know much about anything], I vaguely remember coming across an interesting theorem. The gist of it was that "every ...
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1answer
88 views

integral apollonian sphere packing

Can a sequence of cotangent spheres be packed inside a sphere so that the reciprocals of all of the radii are integers, like the integral apollonian circle packings on ...
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1answer
70 views

Area of filled Julia set

This is a vague question, and I know nothing about this area. We fix some $c\in\mathbb C$ and iterate the map $z\mapsto z^2+c$. This gives some filled Julia set, i.e. the set of points $z\in\mathbb ...
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1answer
196 views

Mandelbrot set and prime numbers

I have written a simple program in C to generate Mandelbrot set. Wherever I zoom in, it seems to me that I see prime numbers, most often 11, 17, 19. For example the object on the attached image has 11 ...
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1answer
73 views

How to generalize a Moore Curve to 3 dimensions?

I understand the concept of a Moore curve in 2D: However I find it a bit tough to conceptualize and generalize it to 3D or higher dimensions. Can someone kindly help me out by providing some ...
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6answers
70 views

conjecture: a supremum property of the cosine fixed point?

in a previous question a composition of circular functions was defined for each binary string of finite length. this question will use the same terminology. if the existence of a fixed point is ...
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1answer
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Definitions of Sierpinski Carpet and Higher Dimensional Analogues

We define the Cantor Set as: $Let \mathscr{J} := \{ 0, 2, \ldots , 3^{m-1} -1 \}$ for $m \in \mathbb{N}$, then $$C = [0,1] \setminus \bigcup_{m \in \mathbb{N}} \bigcup_{k \in \mathscr{J}} \Big( ...
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203 views

Zoom out fractals? (A question about selfsimilarity)

It is well known that if we zoom in on the Mandelbrot set we get selfsimilarity. So I wonder if $g$ is a fractal (in the complex plane) generated by a nonperiodic nonpolynomial entire function $f$ ...
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information dimension and correlation dimension, what do they really mean?

If I have measure the information dimension and correlation dimension of a couple of fractals, I would like to know what these measures really stands for. For example, lets suppose: fractal 1, inf ...
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1answer
333 views

Is a 3D Mandelbrot-esque fractal analogue possible?

I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties). Regardless, I'm wondering if there might be a 'trick' to create a 3D ...
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Cantor sets, Sierpinski carpets, and Menger sponge

How does one distiguish between iterations near infinity? Naturally there is an empty feeling about saying that the iterations $\forall k \in \mathbb{N} < \infty$, $\infty + k$ and $\infty - k$ are ...
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On the Legitimacy of Grossone [duplicate]

A paper describing grossone used to measure such things as the sierpinski carpet here:http://arxiv.org/abs/1203.3150 I'd like to discuss the legitimacy of grossone. What is the general consensus ...
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192 views

coloring the inside point for Julia Fractal

I am trying to continuous coloring the inside point for a fractal image,such as $z \to z^2+C$. For those outside point, we can use the escape iteration to determine the color, just as the description ...
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2answers
345 views

Mini Mandelbrots, are they exact copies?

(This one was found by magnifying 280,000,000 times.) In popular "zoom movies" of the Mandelbrot set the last image is often what appears to be an exact copy of the original set. This is always ...
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4answers
603 views

Why does the Hilbert curve fill the whole square?

I have never seen a formal definition of the Hilbert curve, much less a careful analysis of why it fills the whole square. The Wikipedia and Mathworld articles are typically handwavy. I suppose the ...
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104 views

Sequential Algorithm to generate Fractal (Koch's snowflake)

As part of an assignment I had developed a sequential algorithm to generate a Koch's snowflake. Algorithm I have encountered so far have been recursive and iterations generate closer approximations. ...
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1answer
104 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
118 views

Some questions about Fractals and software

Ever since I read this article on math.SE I have been amazed by the wonder of fractals. I have been trying to learn what are fractals and how to write an equation for one, and I am truly confused, I ...
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4answers
2k views

Why does the Mandelbrot set contain (slightly deformed) copies of itself?

The Mandelbrot set is the set of points of the complex plane whos orbits do not diverge. An point $c$'s orbit is defined as the sequence $z_0 = c$, $z_{n+1} = z_n^2 + c$. The shape of this set is ...
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2answers
334 views

Classification of points in the Mandelbrot set

I am trying to understand the classification of points in the Mandelbrot set. There are an infinite number of baby Mandelbrots, each associated with a defined set of landing rays. There are the pre ...
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935 views

Do Integrals over Fractals Exist?

Given, for example, a line integral like $$ \int_\gamma f \; ds $$ with $f$ not further defined, yet. What happens, if the contour $\gamma$ happens to be a fractal curve? Since all fractal ...
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Let $A$ and $B$ be fractals with box dimension of $x$ and $y$ respectively. Then prove:

Let $A$ and $B$ be fractals with box dimension of $x$ and $y$ respectively. Then prove that the Cartesian product $A \times B$ has box dimension $x+y$. Any hints to start out? (note that box ...
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1answer
54 views

Prove that the box dimension of $\{0,1,\frac{1}{2},\frac{1}{3},…\} $is$ \frac{1}{2}$

I'm supposed to consider the difference $\frac{1}{n+1}-\frac{1}{n}$ and let it equal to $\epsilon$. Hence $\epsilon=\frac{1}{n(n+1)}$. But how do I show that the number of boxes of size $\epsilon$ to ...
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2answers
151 views

Simplest way to determine if a number is a member of the Mandelbrot set?

I'm writing JavaScript code to plot the Mandelbrot set on an HTML5 Canvas element. (That's probably not relevant to the answer to this question). A core part of the problem is to write a simple ...
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292 views

The Cantor Space as $\{0,1\}^{\mathbb{N}}$ and as $[0,1]$.

The Cantor-Space is defined as the space of all infinite binary sequences, i.e. the space $\{0,1\}^{\mathbb{N}}$. It has a natural metric, $$ d(x,y) = \inf\{ 2^{-|w|} : w \in pref(x) \cap pref(y) \} ...