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

learn more… | top users | synonyms

0
votes
0answers
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 ...
3
votes
1answer
34 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 ...
1
vote
1answer
54 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? ...
6
votes
3answers
107 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 ...
2
votes
2answers
73 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 ...
7
votes
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 ...
2
votes
0answers
43 views

Is this plot of Ford circles actually a fractal?

Is this plot of Ford circles actually a fractal?
0
votes
0answers
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 ...
1
vote
1answer
30 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 ...
1
vote
0answers
16 views

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 $\leq \beta$ ...
5
votes
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 ...
3
votes
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 ...
0
votes
0answers
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 ...
27
votes
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 ...
0
votes
0answers
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 ...
0
votes
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 ...
6
votes
2answers
469 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 ...
0
votes
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 ...
1
vote
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 ...
8
votes
2answers
115 views

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 ...
1
vote
1answer
31 views

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 ...
3
votes
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 ...
0
votes
0answers
19 views

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 ...
2
votes
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 ...
6
votes
1answer
136 views

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( ...
0
votes
0answers
45 views

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 ...
1
vote
0answers
32 views

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 ...
18
votes
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 ...
1
vote
0answers
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. ...
0
votes
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.
3
votes
1answer
32 views

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 ...
2
votes
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 ...
2
votes
1answer
53 views

Behavior of Hausdorff dimension under homeomorphisms

Let $X$ and $Y$ be metric spaces, $f : X\rightarrow Y$ a homeomorphism. Denote by $\dim_{\mathcal H}$ the Hausdorff dimension. I know that it is possible that $\dim_{\mathcal H} Y < \dim_{\mathcal ...
1
vote
1answer
93 views

What is the area of the apollonian gaskets?

I searched for the internet, but found nothing relavant to the area. The areas in each intermediate step form a bounded increasing sequence, so there is a limit. But wil it eventually fill in almost ...
5
votes
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 ...
3
votes
2answers
78 views

Length of a Coastline

When B. Mandelbrot's typical example of measuring the length of a coastline is referenced, they mention how at every scale the length increases. In pure mathematics, I can imagine this quite well-- ...
0
votes
1answer
107 views

Hausdorff Measure and Hausdorff Dimension

Could someone explain the intuition behund the Hausdorff Measure and Hausdorff Dimension? The Hausdorff Measure is defined as the following: Let $(X,d)$ be a metric space. $\forall S \subset X$, ...
5
votes
0answers
95 views

Fractal dimension of Gaussian white noise is infinite?

I read in this paper that the fractal dimension of Gaussian white noise is infinite. The paper does not prove it nor give a reference to support it. I failed to find a reference from online searching. ...
9
votes
3answers
322 views

Geometrical objects whose volumes are fractional powers of their sizes

While studying properties of foams (imagine bubbly soap or microscopical grids/networks), I started wondering on the relationship between the volume occupied by the matter $V_s$ itself and the overall ...
4
votes
1answer
201 views

We know the dimension of the Koch snowflake's perimeter, but does it have a measure?

I start with an equilateral triangle with side three meters. I can define a Koch snowflake by the following sequence of figures. Starting with that triangle, produce the next figure by replacing the ...
3
votes
1answer
92 views

Fractal dimension of the boundary of a fractal

Sorry if this is a stupid question, but I'm a physicist, not a mathematician, and fractals are pretty new to me. Is there a simple relationship between the fractal dimension of a set and the fractal ...
0
votes
1answer
84 views

Can someone help me find the sum of the following series?

I am working on one of the fractals and finding its convergent area. $$\begin{align} S & = 1+3\left(\frac{1}{9}+4(\frac{1}{9^2})+4^2(\frac{1}{9^3})+...\right)\\ & = 1+3*\sum_{i=0}^{\infty} ...
2
votes
1answer
83 views

Help with fractals

Let $f(z)=z^2+4z+1$. Is the filled Julia set (denoted $F_f$) connected? I'm not sure to show how its connected. The only thing I know how to do is verify whether a given point is in the set.
0
votes
0answers
19 views

Pascal's triangle and Sierpinski triangle [duplicate]

Is there a formal proof that Pascale's triangle, coloring only the odd numbers, converges to a Sierpinski triangle? In other words: Is there any formal proof that the graph of Rule 90 Wolfram ...
2
votes
1answer
119 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 ...
1
vote
1answer
103 views

Calculate points of Koch Curve

I'm having struggles with what I believe to be simple trig equations to find the points of the little triangle on the base segment of a koch curve. If we divide the line segment up into 5 points ...
0
votes
0answers
61 views

Fractal geometry of literature: First attempt to Shakespeare's works

I found this article on arxiv.org. It has been written by some unknown guy named Ali Eftekhari. Apparently, he is a chemist with original works and publications in chemistry and nano-technology. ...
1
vote
0answers
85 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 ...
7
votes
3answers
350 views

Mandelbrot boundary

Is there a sequence of parameterized expressions for the border of all the major bulbs of the mandelbrot set? By major meaning all bulbs with diameter greater than 0.01 for example. I am interested ...
4
votes
3answers
243 views

Discuss the convergence of $ \left \{ a_n \right\} $ where $ a_{n+1}=\frac{a_0}{2}+\frac{a_n^2}{2},n\geq 1 $

Let $$ a_{n+1} = \dfrac{a_0}{2} + \dfrac{a_n^2}{2} $$ where $ a_1 = \dfrac{a_0}{2} $ and $ n\geq 1 $ Discuss the convergence of $ \left\{a_n\right\} $