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

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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
71 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
545 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
299 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
642 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
238 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
228 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 ...
10
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2answers
1k views

Properties of the Mandelbrot set, accessible without knowledge of topology?

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 ...
9
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2answers
672 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": $$Y(i)=\sum_{j=1}^{1000+i}Z(j) \:\:\: ; \:\:\: (i=1,2,\ldots,N)$$ where $Z(j)$ is a Gaussian noise ...
0
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1answer
267 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
134 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
506 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
227 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 ...
6
votes
0answers
107 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
7k 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
254 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
600 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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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
843 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 ...
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0answers
40 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...
5
votes
1answer
330 views

How is the 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
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6answers
4k 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
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0answers
237 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 ...
3
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1answer
285 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 $f$ ...
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1answer
598 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 ...
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3answers
4k 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 ...
3
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4answers
514 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
194 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 ...
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0answers
115 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 ...
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4answers
229 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
401 views

What's the analogue of Sierpinski triangle to disk?

What's the (closest) analogue of Sierpinski triangle to disk?
7
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2answers
1k 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
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0answers
62 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
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1answer
103 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
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2answers
334 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 ...
3
votes
1answer
270 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?
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1answer
524 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
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1answer
189 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
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1answer
325 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
321 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
563 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 ...
5
votes
1answer
508 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 ...
31
votes
5answers
2k views

Why does the Hilbert curve fill the whole square?

I have never seen a formal definition of the Hilbert curve, much less a careful analysis of why it fills the whole square. The Wikipedia and Mathworld articles are typically handwavy. I suppose the ...
3
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3answers
273 views

Critical points of a function

The literature on Mandelbrot and Julia sets mentions the phase "critical point" quite a lot, but usually doesn't bother to define what it means. As best as I can tell, a critical point is just any ...
3
votes
2answers
195 views

General Mandelbrot iteration formulas

Everybody loves the good old quadratic Mandelbrot set. As you probably know, both it and the corresponding quadratic Julia sets are defined by the iteration $f(z) = z^2 + c$. You might expect, ...
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1answer
212 views

How to find the area. Linked with another question. [duplicate]

Possible Duplicate: Is value of $\pi = 4$? In this question we discussed why the fake proof is wrong. But, what about the area? The process converges to the same area of the circle ...
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1answer
90 views

Fractal walking: well defined case or not?

Please consider the following recursive diagram: diagram Each triangle is connected at the midpoint of a side to the corner of an inner triangle which is 1/4 times the size. The total line length of ...
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0answers
95 views

Lipschitz continuity for an iterated function system

Let $(M,d_M)$ and $(N,d_N)$ be metric and $$ CB(M)=\{\mbox{all closed bounded subsets of }M\}. $$ Let $f: M\to N$ be a Lipschitz map with Lipschitz constant $L$. Define a map $$ F:(CB(M),\rho)\to ...
9
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1answer
584 views

Number of limit points of a continued exponential

Inspired by the work of C. Bender, I recently played with continued exponentials (like continued fractions but with exponential functions ;) ). Given all prefactors are equal to 1, the continued ...
3
votes
1answer
137 views

Is the ball measure of non-compactness a Lipschitz map?

Let $(M,d)$ be a metric space and let $H(M)$ denote the set of closed and bounded subset in $M$. Then $(H(M),d_H)$ is a metric space where $d_H$ denotes the Hausdorff distance. Let $\chi$ be the ...