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

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4
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3answers
376 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
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0answers
113 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
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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
votes
1answer
610 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
180 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
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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
158 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
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0answers
212 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
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1answer
221 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
347 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
298 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
130 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
102 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
175 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
232 views

What's the analogue of Sierpinski triangle to disk?

What's the (closest) analogue of Sierpinski triangle to disk?
7
votes
2answers
662 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
53 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
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
280 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
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0answers
128 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
184 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
352 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
100 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 ...
8
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1answer
263 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
220 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
351 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
308 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 ...
27
votes
5answers
720 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 ...
2
votes
2answers
131 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
145 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, ...
-1
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1answer
185 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 ...
0
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1answer
88 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 ...
3
votes
0answers
82 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
votes
1answer
402 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
128 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 ...
0
votes
1answer
228 views

Finite sets are dense with respect to Hausdorff distance

Let $(X,d)$ be a complete metric space and consider \begin{align*} BC(X)&= \lbrace C\subset X\;|\;C\neq\emptyset\text {, closed and bounded} \rbrace\cr \mathrm{Fin}(X)&= \lbrace ...
0
votes
1answer
2k views

Representing a 3D Hilbert Curve as an L-system

A 2D Hilbert curve can be represented as the following L-system: A → -BF+AFA+FB- B → +AF-BFB-FA+ where F denotes a step ...
3
votes
0answers
195 views

Is the Hausdorff semi-distance Lipschitz?

Let $X$ be Banach (with metric $d$) and let $H(X)$ be the set of closed bounded subsets of $X$. Define for $A,B\in H(X)$ $$\delta(A,B)=\sup_{a\in A}\inf_{b\in B}d(a,b)$$ be the Hausdorff semi-distance ...
0
votes
1answer
217 views

Relation between Hausdorff metric of and Hausdorff measure of non-compactness

Let $(X,d)$ be a metric space and let $$K(X)=\lbrace Y\subset X\colon Y\text{ is non-empty and compact}\rbrace.$$ Endow $K$ with the Hausdorff metric (which is the natural metric on this space, see ) ...
3
votes
1answer
129 views

Snow Flake Problem: Limit of perimeter & area at $\infty$

I am supposed to find the limits as $n\rightarrow\infty$ of the perimeter & area of a snow flake. $$N_n = \text{Number of sides} = 3\cdot 4^n$$ $$L_n = \text{length of side} = \frac{1}{3^n}$$ ...
0
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1answer
45 views

$L$-Systems: Order of Substitution

I am working the a subject guide on involving $L$-Systems and have the alphabet $A = \{a, b, c\}$. The initiator is the string $a$ and the rules of substitution $a \to ba$, $b \to ccb$, $c \to a$. ...
18
votes
2answers
1k views

Do Integrals over Fractals Exist?

Given, for example, a line integral like $$ \int_\gamma f \; ds $$ with $f$ not further defined, yet. What happens, if the contour $\gamma$ happens to be a fractal curve? Since all fractal ...
2
votes
2answers
381 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 ...
0
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1answer
95 views

rearrange $z \mapsto z^2 + c$

Mathematics, some of its magic is that a lot is known about how to rearrange its statements (equations). Given the Mandelbrot Set: $z \mapsto z² + c$ (or more precisely) $z_{i+1} = z_i ^2 + c$ ...
4
votes
1answer
218 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
votes
1answer
83 views

understanding if a point is inside or outside a Koch fractal curve

I continue this post because i have a problem: understand if a point is inside or outside a Koch curve. I can find the third point of a equilateral triangle (and i need it for build a koch curve) but ...
3
votes
1answer
536 views

How should I assign RGB colors to points in the Mandelbrot Set?

I decided to learn about the Canvas object in javascript by implementing a display of the Mandelbrot Set. I am mimicking the Mandelbrot psuedocode found on wikipedia. The thrust of it is that the ...
1
vote
1answer
95 views

Two questions on fractal

Suppose we have two given fractals $K_1$, $K_2$ of dimension $d_1$,$d_2$ respectively. What can we say about the dimension of $K_1 \cap K_2$, and $K_1 \cup K_2$? Is there any technique to describe a ...