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

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9
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3answers
395 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 ...
5
votes
1answer
395 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 ...
4
votes
1answer
164 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
104 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
111 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
339 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
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1answer
292 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 ...
2
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1answer
187 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 ...
8
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3answers
512 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
263 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
149 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 ...
8
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0answers
307 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
226 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
votes
2answers
98 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 ...
1
vote
1answer
379 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
votes
2answers
120 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
votes
1answer
604 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
votes
1answer
102 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 ...
1
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1answer
94 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 ...
8
votes
1answer
314 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 $n$. ...
6
votes
2answers
284 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
121 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 ...
1
vote
1answer
154 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 ...
6
votes
1answer
193 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 ...
0
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1answer
120 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. ...
6
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1answer
383 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 ...
0
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1answer
64 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 ...
6
votes
1answer
953 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 ...
1
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1answer
144 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 ...
1
vote
1answer
102 views

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 ...
3
votes
1answer
118 views

Does there exist a set in the plane such that topological dimension 2 with empty interior?

I consider as follows, but i could not proceed it. The topological dimension 2 of a set means that there is a base for the open sets of the set consisting of sets U with topological dimension of ...
5
votes
3answers
319 views

Are there mini-mandelbrots inside the julia set?

I've seen a julia set zoom but it is not nearly as interesting as a mandelbrot zoom. I also have not seen corresponding julia sets for zooms in the mandelbrot deeper than the original image. I'm ...
2
votes
0answers
76 views

What is the name of this metric: Why is $(\mathcal{M}, L)$ complete

I am reading section 4 of this article about invariant measures: http://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf Let $(X,d)$ a complete metric space, ...
4
votes
0answers
115 views

The Tribonacci constant and the Dragon

Let $x = \frac{\ln T}{\ln 2} = 0.879146\dots$ where $T$ is the tribonacci constant, then x solves the transcendental equation, $$4^x(2^x-1)=(2^x+1)$$ Let $x = \frac{\ln y}{\ln 2} = 1.523627\dots$ ...
2
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0answers
77 views

what part of a m-set fractal showing spiral behaviour?

What part of a fractal Mandelbrot Set showing spiral behaviour like this one: what is it's direct equation?
3
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0answers
179 views

Fractal Analysis

Is there any way to compare two fractals and analyse the difference between the two. I'm doing a project on fractals and It'll be very easy if there is a module which can be used to analyse and ...
8
votes
1answer
508 views

Regular open set whose boundary has nonzero volume.

I found this question quite interesting, but its answers were disappointingly non-geometric. I'd be interested to know whether there exists a geometric example. To be precise about what I mean by a ...
2
votes
1answer
190 views

Heighway dragon and twindragon relation

The Heighway dragon F is defined as the limit set for the iterated function system $\begin{cases}f_1(z)=\frac{1+i}2 z\\f_2(z)=1-\frac{1-i}2z\end{cases}\quad$ starting from the two points 0 and 1. The ...
4
votes
1answer
289 views

Is the Fractal Dimension of a Space-Filling Curve in a Plane Always 2?

I have been playing around with space-filling curves that completely fill the unit square. All of them that I have seen have a fractal dimensional of 2. Makes sense that it would be 2, but a Google ...
1
vote
1answer
160 views

Reverse Hölder Continuity and Hausdorff dimension

Let $f$ be a function on $[0,1]$. Say that $f$ is reverse Hölder continuous of exponent $\beta > 0$ if there is a $C >0$ such that for any $s<t\in [0,1]$, there exists $s',t'\in [s,t]$ such ...
2
votes
0answers
145 views

L-systems and Sierpinski Triangle

I was just shocked when I saw these consecutive outcomes of an L-system converging to the Sierpinski triangle (shown in this picture). I'm interested to know how can one arrange the rules of an ...
19
votes
2answers
11k 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, ...
3
votes
1answer
106 views

Notation Clarification of Koch Curve

I am having trouble making sense of the notation used to describe the Koch Curve in the book Getting Aquanted with Fractals. The link will take you to a preview of the book which describes the ...
1
vote
3answers
381 views

Cantor Set and Fractals

I have read that the Cantor set is considered a fractal. I am referring to the Cantor set in which the middle third of a real line is removed recursively. I see that this is recursively defined, but ...
3
votes
0answers
155 views

Fractal derivative of complex order and beyond

Is there some precise definition of "complex (fractal) order derivative" for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would ...
0
votes
2answers
68 views

Is $g(z)=\frac{1}{z}+\frac{1}{z^2+1}+\frac{1}{(z^2+1)^2 +1}+…$ analytic for $|z|>2$?

Let $z$ be a complex number. Let |.| denote be the absolute value. Let $n$ be a positive integer. Let $f_1(z)=z^2+1$. Let $f_n(z)=f_1(f_{n-1}(z)).$ Is ...
5
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0answers
576 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) \} ...
1
vote
0answers
93 views

what is the fractal dimension of the henon map?

I have some questions about the Henon map that are not clear for me. I have seen that the correlation dimension of the Henon map is approximately 1,21, is that measure similar to its fractal ...
13
votes
2answers
524 views

H0w have group theory and fractal geometry been combined?

Has there been a significant tie made between group theory and fractal geometry? What are some ways that they have been tied together? I've been inspired to ask this question by this image of a free ...
22
votes
5answers
384 views

Fractals reference

I want to present an elementary lecture about Fractals in the Nature. So, I am searching open or online references with good pictures like the following one: I prepared a good program that makes ...