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

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0
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1answer
63 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
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1answer
849 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
134 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
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1answer
99 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 ...
2
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1answer
114 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
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3answers
304 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
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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
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0answers
113 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
74 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?
2
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0answers
174 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
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1answer
471 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
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1answer
184 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
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1answer
273 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
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1answer
155 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
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0answers
137 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
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2answers
10k 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, ...
2
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1answer
93 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
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3answers
338 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
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0answers
148 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
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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
540 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) \} ...
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0answers
83 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
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2answers
491 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
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5answers
368 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 ...
1
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1answer
132 views

Ways to project arbitrary Fractals on 2D objects and 3D objects w different dimensions?

I am trying to create a house/texture in 3D and in 2D with fractals, perhaps related. My friend said that fractals can have different dimensions such as 1.74, 1, 4.71111... and pretty much anything. ...
4
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0answers
106 views

Help understanding this 'Fractal' I've just made?

I was messing around in C++, making an image where the pixels change depending on the the rectangle's dimensions and whether or not the space bar is down, and I formed this image: Could anyone ...
0
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0answers
41 views

Addressing/traversing an infinite 2D grid using a Z-line?

I'm looking for a method to map an infinite 2D grid using a line, so that I would have just one integer from which I would compute the X and Y. I know something like that exists, but can't recall the ...
1
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0answers
96 views

doubt in a book proof from 'The Geometry of Fractal Sets'

I am reading the proof of existence of positive finite $H^s$-measure (Theorem 5.4) on page 67-68 of The Geometry of Fractal Sets.I am not quite convinced that $E_k$ are closed set by the construction ...
2
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0answers
52 views

Show that Hausdorff measure is semifinite

I am currently reading a book about fractals and the author states the result that Hausdorff measure is semifinite. Can someone tell me how to prove or provide a hint for me?
2
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0answers
64 views

Do there exist periodic fractals $A_f$ of this type?

Let $z$ be a complex number. Meromorphic here means meromorphic on all of the complex plane $C$. Lets define a fractal $A_f$ on the complex plane as the result of iterating a meromorphic function ...
4
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0answers
73 views

Is the measure induced by the Mandelbrot set computable on rational rectangles?

Is there a computable function that, given a positive rational number $\epsilon$ and a rectangle with rational corners $A$ returns a number $f(A,\epsilon)$ such that $|\mu(A \cap ...
2
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1answer
61 views

Countability of “center” points of line segments in complement of Cantor set

So, start with the set [0,1] of the real line. Remove the middle third, and keep removing the middle thirds of the remaining line segments as usual when making the Cantor set. Each time you remove a ...
6
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1answer
374 views

Hausdorff Dimension of Arbitrary Julia Set

I am looking to find an exact solution to the Hausdorff dimension of a Julia set $J(f)$ for a polynomial $f: z \mapsto z^2 +c$ given an arbitrary $c$. I know this question is known for a number of ...
3
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1answer
266 views

Is this Perlin Noise?

http://freespace.virgin.net/hugo.elias/models/m_perlin.htm This method involves getting a random dataset, sampling it at various resolutions, and adding together the result. I've heard it claimed ...
9
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4answers
522 views

How to draw a fractal from $z \mapsto z^2 + c$ explained for a mere mortal?

I am interested in: 1) Understanding in detail how fractals are draw. 2) Coding a computer program to draw a simple fractal. Can someone with good explaining skills take care of 1) for me? I don't ...
4
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1answer
149 views

fractal structure of the sum of squares function

The sum of squares function came up at a job interview, corrected for signs and symmetry. $d_2(n)=\#\{(x,y): x^2 + y^2 = n\}$ However, want $(x,y)\sim (\pm x, \pm y) \sim (y,x)$. The first ...
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1answer
195 views

Buddhabrot Sewing machine [closed]

The Buddhabrot fractal traces the orbits of the points outside the Mandelbrot set. What design considerations need to be taken into account to create a computerised sewing machine that traces out ...
7
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1answer
743 views

Properties of the Mandelbrot set

Are there any properties of the Mandelbrot set that can be analysed without a knowledge of complicated topology? Considering the fact that the set is based on a quadratic function, are there any ...
8
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2answers
465 views

Why should Gaussian noise have fractal dimension of 1.5?

In a paper I'm trying to understand, the following time series is generated as "simulated data": $$Y(i)=\sum_{j=1}^{1000+i}Z(j) \:\:\: ; \:\:\: (i=1,2,...,N)$$ where $Z(j)$ is a Gaussian noise with ...
0
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1answer
238 views

Unexplainable noise graph function.

I'm sorry for the ambiguity here but I've recently discovered a function which plots, what seems to be either a fractal or simply noise in a selected area. Can anyone explain this function: ...
0
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1answer
130 views

Mandelbrot precision target the center of a pixel?

I read this question and I don't understand the answer: http://stackoverflow.com/questions/8381675/how-to-perform-simple-zoom-into-mandelbrot-set?rq=1. Especially how can I aim for the center of the ...
3
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2answers
419 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 ...
4
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1answer
183 views

Mandelbrot bulb's countable?

Are the Mandelbrot set's bulb's countably infinite? My daughter asked me this question, after I pointed out that some Julia sets are a Cantor dust. For a point not in the Mandelbrot set, the ...
5
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0answers
91 views

How difficult is it to impose a differential structure on a fractal?

Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply ...
13
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1answer
5k views

Odd and even numbers in Pascal's triangle-Sierpinski's triangle

Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed. I recently learned that when the Pascal's triangle is reduced ...
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0answers
177 views

Area fractal pentagrams II

A simple fractal. How to find the area of it? (only the arms of the star) Working with pentagrams is quite complicated, I can not solve this.
4
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3answers
452 views

Area fractal pentagrams I

When I saw this image I was a little curious. How can I find the area of this fractal?
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0answers
124 views

Can happy numbers be made into a fractal image?

I've just learnt of Happy numbers, from Doctor Who and I was wondering if they could be used, if applied to numbers in the complex plane, to make a fractal image, like the Madelbrot set? Or do happy ...
45
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2answers
2k views

A new kind of fractal?

http://www.gibney.de/does_anybody_know_this_fractal Is this some known kind of fractal? Update: This one got a lot of great feedback from around the net. I summarized it here: ...
6
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1answer
693 views

any idea what fractal algorithm might generate this shape?

I Found this image around, and i'm curious what algorithm generates this kind of shape In particular, i'm curious how the flow lines are generated, since usually the Mandelbrot iteration just ...