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

learn more… | top users | synonyms

0
votes
0answers
35 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
92 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
48 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
votes
0answers
56 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
votes
0answers
69 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
55 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
308 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
234 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
votes
4answers
462 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
130 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 ...
-1
votes
1answer
174 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
541 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
votes
2answers
367 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
votes
1answer
212 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
124 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
votes
2answers
368 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
votes
1answer
150 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
votes
0answers
81 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 ...
14
votes
1answer
4k 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
77 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
134 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
382 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
114 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 ...
42
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: ...
6
votes
1answer
626 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
33 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...
3
votes
0answers
187 views

How is study of fractals related to fourier/spectral/harmonic analysis?

In chap. 3 of "Fractal Geometry of Nature" Mandelbrot mentions that "part of the study of fractals is the geometric face of harmonic analysis" (spectral or Fourier, he specifies). But to my dismay ...
12
votes
6answers
2k views

What sorts of problems can fractals solve?

After doing a bit of research on fractals, I was wondering what sort of real-life applications do fractal have and in what way would they be used to help solve a problem. I already know people use ...
3
votes
0answers
160 views

Mandelbrot set's border in parametric form

I've post this question just because I'm curious, Mandelbrot set is defined as: $ z_{n+1} = z^2_n + c $, if $n \rightarrow \infty $ and it doesn't diverge we get the border. This border is unlimited ...
0
votes
0answers
218 views

Simple examples of entire functions that have fractal properties.

Im looking for simple examples of entire functions that have "fractal properties". With "fractal properties" I mean that $|f(z)| < 1$ has a "fractal structure" in the complex plane. With "fractal ...
3
votes
1answer
224 views

Zoom out fractals? (A question about selfsimilarity)

It is well known that if we zoom in on the Mandelbrot set we get selfsimilarity. So I wonder if $g$ is a fractal (in the complex plane) generated by a nonperiodic nonpolynomial entire function $f$ ...
1
vote
1answer
357 views

Complex Numbers in Fractal Algorithms

I am a high school freshman who is undertaking a small development project on fractals. I do not want to get too in depth, but I would love to blow my math teacher's socks off. Having looked through ...
7
votes
3answers
1k views

Is Fractal perimeter always infinite?

Looking for information on fractals through google I have read several time that one characteristic of fractals is : finite area infinite perimeter Although I can feel the area is finite (at ...
2
votes
4answers
308 views

In Need of Ideas for a Small Fractal Program

I am a freshman in high school who needs a math related project, so I decided on the topic of fractals. Being an avid developer, I thought it would be awesome to write a Ruby program that can ...
4
votes
2answers
136 views

Quick Julia/Mandelbrot Testing

I have successfully implemented a realtime Julia/Mandelbrot set generator on the GPU. Primarily out of curiosity, what I'm looking for now is a faster test algorithm. Ideally, I want a boolean ...
1
vote
0answers
104 views

Is there a fractal origami shape that trades volume for area to always keep a flat surface when expanded?

I'm thinking of something like a 2.5D sierpienski type shape. The idea is to enable an lcd type screen that could unfold to "any" size by unpacking space filling elements packed in 3d to a 2d ...
0
votes
4answers
177 views

what is $c$ in Mandelbrot set?

The Mandelbrot Set is an extremly complex object that shows new structure at all magnifications. It is the set of complex numbers $c$ for which the iteration indicated nearby remains bounded. ...
3
votes
3answers
241 views

What's the analogue of Sierpinski triangle to disk?

What's the (closest) analogue of Sierpinski triangle to disk?
7
votes
2answers
707 views

What real numbers are in the Mandelbrot set?

The Mandelbrot set is defined over the complex numbers and is quite complicated. It's defined by the complex numbers $c$ that remain bounded under the recursion: $$ z_{n+1} = z_n^2 + c,$$ where $z_1 = ...
3
votes
0answers
54 views

Fractals vs. “neatness” / order

I've seen a lot of high level videos on fractals, etc, and how they might apply to the real world. So a tree is branches with branches with branches, and our blood vessels branch and then branch ...
2
votes
1answer
77 views

About fractal structures

I read somewhere that we can not measure the length of the Adriatic Coast because it has fractal structure. I want some concrete explanation for the fractal structure
8
votes
2answers
283 views

How to prove a property regarding periodicities of points in the Mandelbrot set?

While studying a visual representation the Mandelbrot set, I have come across a very interesting property: For any point inside the same primary bulb (a circular-like 'decoration' attached to the ...
0
votes
0answers
134 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 ...
3
votes
1answer
192 views

Defining distance in fractal dimensions.

Is it possible define a distance measure in fractal dimensions? namely, what the generalization of $$ D(x,y)=\left(\sum_i(x_i-y_i)^2\right)^{\frac{1}{2}} $$ in fractal dimensions?
5
votes
1answer
354 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 ...
3
votes
1answer
103 views

integral apollonian sphere packing

Can a sequence of cotangent spheres be packed inside a sphere so that the reciprocals of all of the radii are integers, like the integral apollonian circle packings on ...
9
votes
1answer
267 views

The Mandelbrot Set Membership

To define the Mandelbrot Set we consider a sequence of complex numbers $z_0$, $z_1$, $z_2$, $z_3$, with the following conditions: $$ \begin{cases} z_{n+1} &= &z_n^2 + c &\text{ for }n\geq ...
5
votes
2answers
225 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 $n\geq 2$ ...
9
votes
3answers
359 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 ...
4
votes
1answer
321 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 ...