Questions on fractals, irregular-looking mathematical objects that display the property of self-similarity.
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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
42 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, ...
3
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0answers
43 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 $x = \frac{\ln y}{\ln 2} = 1.523627\dots$ ...
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0answers
11 views
what part of a m-set fractal showing spiral behaviour?
What part of a fractal M-Set showing spiral behaviour like this one:
M-Set spiral behaviour
what is it's direct equation?
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0answers
27 views
What is M-set of this complex plan? [closed]
What is M-set where complex plane is:
$$\cos\theta+i\sin\theta=e^{i\theta}$$?
this is M-set in general:
$z_{n+1}=z_n^2+c$
1
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0answers
75 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 ...
6
votes
1answer
80 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 ...
1
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1answer
73 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 ...
3
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1answer
30 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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0answers
32 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 ...
11
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2answers
373 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, ...
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1answer
52 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 ...
2
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3answers
75 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 ...
1
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0answers
28 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 ...
4
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0answers
98 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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2answers
53 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 ...
3
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1answer
130 views
coloring the inside point for Julia Fractal
I am trying to continuous coloring the inside point for a fractal image,such as $z \to z^2+C$. For those outside point, we can use the escape iteration to determine the color, just as the description ...
4
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2answers
113 views
Julia Set of polynomials
If $f$ is a polynomial and $z\in\mathbb{C}$, show that either $f^n(z)\rightarrow\infty$ or $\{f^n(z) : n\geq 1\}$ is a bounded set.
Here, $f^2(z)=f(f(z))$ and $f^n(z)=f(f^{n-1}(z))$ for $n\geq 2$
...
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2answers
107 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 ...
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0answers
25 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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3answers
111 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 ...
0
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1answer
43 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. ...
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0answers
60 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 ...
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0answers
17 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 ...
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0answers
9 views
recommendation about multifractal analysis book
I have finished reading the classic Strogatz book about Nonlinear Dynamics and Chaos, but the problem is that the part of multifractal analysis is very short. Only it mentions its definition in half ...
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0answers
67 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 ...
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0answers
36 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?
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0answers
24 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 ...
2
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0answers
39 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 ...
5
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1answer
116 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 ...
5
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1answer
195 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 ...
2
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1answer
35 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 ...
8
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4answers
308 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 ...
3
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1answer
76 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 ...
4
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1answer
73 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 ...
0
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1answer
106 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 ...
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2answers
117 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 ...
4
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2answers
256 views
Is a 3D Mandelbrot-esque fractal analogue possible?
I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties).
Regardless, I'm wondering if there might be a 'trick' to create a 3D ...
2
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1answer
176 views
Is an Inverse Menger Sponge a fractal?
Is the Inverse of a Menger Sponge a fractal? I know a Menger sponge is fractal in nature, and it seems to me that the inverted form of it would be fractal as well, but I don't know.
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1answer
112 views
Integration on fractals [closed]
Who can explain the proof of the formula (2.12) given here:
J. Phys. A: Math. Gen. 20 (1987) 3861-3875. Printed in the UK
...
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1answer
87 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 ...
10
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1answer
2k 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 ...
0
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1answer
74 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:
...
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1answer
44 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 ...
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0answers
21 views
Density of a multifractal distribution
I am trying to grasp the concept of density of a multifractal. So I start from the simple case of a line. Let's assume I have a uniform distribution of points on a line and I center a cubic box in the ...
3
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1answer
151 views
Every basin of attraction contains a critical point?
Years and years ago, back when I first became interested in fractals [but didn't know much about anything], I vaguely remember coming across an interesting theorem. The gist of it was that "every ...
8
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2answers
182 views
Quadratic Julia sets and periodic cycles
Consider the function $f_c(z) = z^2 + c$. Applying this function repeatedly, we get the familiar quadratic Julia sets that fractal enthusiasts burn compute cycles plotting.
Infinity is always one ...
3
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1answer
72 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 ...
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0answers
56 views
fractal string problem
Let the zeta function of a fractal string be $ Z(s)= \sum_{n}g(n)(l_{n})^{s} $
Here $ g(n) $4 is the degeneracy of the strings and $ l_{n} $ are the lengths of the string.
In order to evaluate the ...
2
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2answers
306 views
Undiscovered fractal sets?
I'm interested in the topic of fractals, such as those created by the borders of the Mandelbrot and Julia sets.
My question is if there are other, not yet discovered fractal sets, which one could ...
