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

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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
231 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 ...
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1answer
92 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 ...
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132 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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24 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
79 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
145 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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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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435 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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146 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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343 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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37 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 ...
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1answer
69 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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1answer
91 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
155 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 ...
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2answers
197 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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2answers
106 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-- ...
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212 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. $\forall S \subset X$, ...
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129 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
362 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 ...
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1answer
281 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 ...
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1answer
137 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 ...
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95 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} ...
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1answer
102 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.
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232 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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213 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 ...
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128 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 ...
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436 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 ...
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247 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\} $
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2answers
135 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 ...
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243 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 ...
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187 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 ...
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2answers
92 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
287 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 ...
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116 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 ...
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1answer
447 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 ...
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1answer
95 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
89 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 ...
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239 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 $n$. ...
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260 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 ...
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2answers
226 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 ...
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100 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
115 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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1answer
176 views

Can we check whether a Cantor set is self-similar or not?

Given a Cantor set $C$ on the real line, do we have some ways to determine whether it is self-similar or not? In particular, how can we check that $C$ is not self-similar? Edited: Definition: Let ...
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99 views

Mandelbrot set incorrect picture

I'm writing an algorithm to generate the Mandelbrot set in Java. However, the final picture is incorrect. It looks like this I was wondering if the algorithm was incorrect. ...
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267 views

Why is the bailout value of the Mandelbrot set 2?

For the past few days I've been studying the Mandelbrot set, and many say that if the iterations of a point stay within a magnitude of 2, the point converges. A very natural question of "why is the ...
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59 views

Correspondence between fractal sets and trees

In Hillel Furstenberg's series lectures on ergodic theory in fractal geometry, he mentioned his search on finding a one-to-one correspondence between fractal sets and trees, however, I couldn't not ...
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1answer
638 views

How to prove Mandelbrot set is simply connected?

In this lecture note of Harvard, it is proved that Mandelbrot set is connected, a result due to Douady and Hubbard. However, I lack necessary knowledge to comprehend it. Then in the same note it is ...
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111 views

Mandelbrot set approximation

Is there a function $f:\mathbb N\to \mathbb R$ such that $\lim_{n\to\infty} f(n) = 0$ and for every $c\in\mathbb C$: If $z_0=0$, $z_{n+1}=z_n^2+c$ and $|z_k|<2$, then there exists a point $c'$ in ...
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Name of this “cut 'n slide” fractal?

Can you identify this fractal--if in fact is has a name--based either upon its look or on the method of its generation? It's created in this short video. It looks similar to a dragon fractal, but I ...