Questions on fractals, irregular-looking mathematical objects that display the property of self-similarity.
2
votes
0answers
44 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
votes
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 $x = \frac{\ln y}{\ln 2} = 1.523627\dots$ ...
1
vote
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?
-1
votes
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
vote
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
81 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
vote
1answer
74 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
votes
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
vote
1answer
54 views
+50
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 ...
1
vote
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
votes
2answers
375 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, ...
0
votes
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
votes
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
vote
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 ...
0
votes
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 ...
4
votes
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) \}
...
1
vote
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 ...
11
votes
2answers
108 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 ...
13
votes
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
votes
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. ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
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?
1
vote
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
votes
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 ...
2
votes
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 ...
5
votes
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 ...
3
votes
1answer
77 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 ...
8
votes
4answers
309 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
votes
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
votes
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 ...
5
votes
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 ...
6
votes
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
votes
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
...
0
votes
1answer
75 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
votes
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 ...
0
votes
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 ...
1
vote
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 ...
3
votes
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 ...
4
votes
0answers
53 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 ...
10
votes
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
votes
0answers
57 views
Area fractal pentagrams III
how can I find the area of these two fractal?
I've been try to solve some geometry exercises here but this and some other are so much difficult.
0
votes
0answers
66 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
votes
3answers
216 views
Area fractal pentagrams I
When I saw this image I was a little curious.
How can I find the area of this fractal?
0
votes
0answers
76 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 ...
39
votes
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: ...
4
votes
1answer
200 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 ...
1
vote
0answers
24 views
Determining the roughness of a multidimensional optimization surface
Is there a way to determine the roughness of an n-dimensional optimization surface (n > 3)?
Preferably a method that uses measures from fractal geometry/chaos theory...
