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

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0answers
73 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
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5answers
3k 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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0answers
17 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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2answers
60 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
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2answers
488 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
74 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
90 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
184 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 ...
2
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1answer
73 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
103 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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0answers
21 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 ...
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6answers
74 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
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1answer
140 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( ...
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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
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1answer
314 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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0answers
121 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
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1answer
158 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
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1answer
35 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
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1answer
61 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
84 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 ...
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1answer
139 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
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2answers
177 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 ...
4
votes
2answers
98 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
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1answer
153 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
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0answers
114 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. ...
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3answers
351 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
241 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
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1answer
118 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
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1answer
87 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
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1answer
96 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.
2
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1answer
188 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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1answer
177 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
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0answers
74 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. ...
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0answers
108 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
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3answers
392 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
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3answers
245 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\} $
2
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2answers
127 views

M-set interior point probability on the real axis

For the real axis, the Mandelbrot set consists of points from $[-2,0.25]$. Some of these points are in the interior of the m-set, and some are on the boundary. Those points in the interior are ...
7
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0answers
199 views

About devil's staircases

We say that a function $f:\left[a,b\right] \to \mathbb{R}$ is a singular function or a devil's staircase if $f$ satisfies the following properties: $f$ is continuous; $f(a) < f(b)$; $f$ is ...
5
votes
2answers
177 views

Are there an infinite number of minibrots on the real line?

This is at 25 zooms using Fractal Extreme. The red circle indicates that there are more minibrots inbetween the small one and the large one. The pattern of super big (bottom right), medium-sized ...
3
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2answers
82 views

Filling the plane with a sequence

I am not sure if this is the right stack to ask this question, but since there is a definite fractal dimension to it, I thought I'd give it a go. The problem I am facing is one of calculating an ...
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1answer
257 views

Hausdorff dimension of the set of rational numbers within a certain interval?

Intro: The Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative real number associated with any metric space. In general the Hausdorff dimension ...
4
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2answers
115 views

A function that is not contractive with respect to any metric

I am struggling with this homework question with is related to iterated function system and fixed point theory. The question is: Let $\Delta \in R^2$ be a filled non-degenerate triangle with ...
7
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1answer
398 views

Self Study of Fractals

I am looking for a book to self-study fractals with a certain criteria. I have checked out Getting Aquainted with Fractals. Note that Getting Aquainted with Fractals does not include ...
2
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1answer
79 views

Is it possible to generate an $M$-order Hilbert Curve without consuming $O(M^2)$ memory?

This question is admittedly very programming related, but I felt that it is better suited to the Mathematics crowd than Stack Overflow. I would like to generate Hlibert walks through the pixels in ...
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1answer
82 views

Describe attractors of a finite family of contraction mappings

The question is to describe the attractor of iterated function system $\mathcal{F}=\{R^2,f_1,f_2\},$ where $f_1,f_2$ are the two affine transformations$\begin{bmatrix} 0 & 0.8\\ -0.5&0 ...
7
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0answers
223 views

Is The *Mona Lisa* in the complement of the Mandelbrot set.

Here is a description of how to color pictures of the Mandelbrot set, more accurately the complement of the Mandelbrot set. Suppose we have a rectangular array of points. Say the array is $m$ by $n$. ...
8
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0answers
241 views

Kakeya Needle problem video

I'm intruiged by the Kakeya Needle problem, described here on Wikipedia. Wikipedia has a nice animation of a needle turning through a hypo-cycloid: What I'm searching for is a visualisation of the ...
6
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2answers
208 views

How to figure out the starting point for this Mandelbrot?

My answer for another math stackexchange question, asked by Gottfried, involved observing Mandelbrot bifurcation for the iterated function in question, $f(z)\mapsto z-\log_b(z)$. In particular, for ...
4
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2answers
96 views

Good sources on studying fractals (the mathematical, and not just the pretty pictures version)?

Particularly, I'm interested in learning about the dimensions (whether it's always possible to find them, and if so, a concrete way of calculating them) of different types of fractals (given by the ...
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1answer
93 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 ...