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

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2
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2answers
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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
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2answers
135 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
176 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
85 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
81 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
351 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
125 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
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1answer
208 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 ...
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1answer
1k 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
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0answers
164 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
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1answer
210 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
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1answer
126 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
43 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 $a \to ba$, $b \to ccb$, $c \to a$. ...
17
votes
2answers
931 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
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2answers
364 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
92 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
192 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 ...
3
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1answer
79 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
466 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
88 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 ...
2
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2answers
258 views

Hausdorff's distance of some sets

We define $H^{n}$ for the set of all compact subsets of $\mathbb{R}^n$. Define the metric $\Delta$ in $H^{n}$ as following.Let $A,B \in H^{n}$ then define $d(x,B):= \min \lbrace d(x,y): y \in B ...
42
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1answer
835 views

Fractal behavior along the boundary of convergence?

The complex power series $$\sum_{n=1}^{\infty}\frac{z^{n^2}}{n^2}$$ has radius $1$ (Ratio Test) and is absolutely convergent along $|z|=1$. Recalling something that my calculus professor (Ray Mayer, ...
2
votes
1answer
163 views

Why does my carpet have so many holes?

I always liked the Sierpinski triangle, and happened upon the related article about the Sierpinski carpet. The article is pretty sparse, and states the area of the carpet is zero (in standard ...
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0answers
67 views

question about multifractal analysis

I have a general question about multifractal analysis: Suppose that I have two figures, that are multifractals. The question is, how I can compare how similar they are to each other? Can I do it by ...
2
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1answer
364 views

Fractal dimension of the Cantor Set

How can we get or prove that the 'fractal dimension' of the Cantor set is $\log_{3} (2)$? I know how to prove by evaluating the poles of $f(s)= \sum \limits_{n=1}^{\infty} 2^{n-1} 3^{-sn}$, and then ...
2
votes
1answer
228 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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0answers
138 views

Is the maximal temperature of the curlicue fractal acheived by $e\times\gamma$?

The Curlicue Fractal is defined as follows: Choose an irrational number $s$ and a horizontal unit segment with angle $\phi_0 = 0$. Define $\theta_{n+1} = \theta_{n} + 2 \pi s \pmod{2 \pi}$, with ...
6
votes
2answers
516 views

Interesting but elementary properties of the Mandelbrot Set

I suppose everyone is familiar with the Mandelbrot set. I'm teaching a course right now in which I am trying to convey the beauty of some mathematical ideas to first year students. They basically know ...
3
votes
1answer
206 views

$\dim_\text{topology}(\text{Cantor Sets}) \leq \dim_\text{hausdorff}(\text{Cantor Sets})$?

Please, explain: the Cantor set (a zero-dimensional topological space) is a union of two copies of itself, each copy shrunk by a factor 1/3; this fact can be used to prove that its Hausdorff ...
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0answers
199 views

Julia Sets in Mathematica

stackexchange geniuses! I'm a high school student doing engineering research and am in need of some technical assistance. I'm working on a paper on using fractals in civil engineering and need to ...
0
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1answer
51 views

Why this Mandelbrot program add current point in the Mandelbrot

I read some posts about the Mandelbrot. I read that the Mandelbrot should be defined by $f(z)=z^2+C$. In my understanding, I think, the $C$ should be a constant, like $0.27$ or $2.1+4.5i$. However, in ...
3
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0answers
525 views

How can I generate grid-based Fractals?

Please let me know if there's a better site to ask a question like this. I play a little indie game called Dwarf Fortress and a major part of the game involves building the titular Fortress for your ...
6
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0answers
228 views

Do fractals contain solutions for problems?

I notice that fractals resemble natural shapes such as leaves or rivers. Leaves and rivers are solutions to problems in themselves. A leaf is trying to distribute the water to the leaf while the leaf ...
4
votes
2answers
319 views

Independent High School Research

StackExchange Gurus! I'll keep this question short and to the point. I'm going into my senior year of high school, will have an independent research period, and have few ides of what to do during ...
9
votes
3answers
307 views

Hilbert curve, understanding the original article

I'm trying to read and understand the article in which Hilbert gave an illustration of a space filling curve, namely "Ueber die stetige Abbildung einer Linie auf ein Flächenstück". It's only a short 2 ...
4
votes
1answer
288 views

Hausdorff dimension of graphs of one-dimensional Brownian motion

First question here, my apologies if it is a duplicate or inappropriate. There is a page on Wikipedia listing fractals by Hausdorff dimension and it includes the graph of a "regular Brownian ...
13
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2answers
344 views

Mini Mandelbrots, are they exact copies?

(This one was found by magnifying 280,000,000 times.) In popular "zoom movies" of the Mandelbrot set the last image is often what appears to be an exact copy of the original set. This is always ...
6
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1answer
153 views

Fractal dimension after nonlinear transformation

Let's assume X(s) is a fractal surface with Hausdorff dimension D. Now we take a nonlinear transformation f which transforms X(s) to f(X(s)). In this case, what will be the Hausdorff dimension of the ...
28
votes
2answers
541 views

Why does this distribution of polynomial roots resemble a collection of affine IFS fractals?

Consider the following spectacular image, created by Sam Derbyshire and described in John Baez's article "The Beauty of Roots": In this image are plotted all the complex roots of all polynomials of ...
7
votes
3answers
463 views

Given a Pattern, find the fractal

Is it possible, given a pattern or image, to calculate the equation of the fractal for that given pattern? For example, many plants express definite fractal patterns in their growth. Is there a ...
7
votes
1answer
218 views

What is known about nice automorphisms of the Mandelbrot set?

It is often stated that fractals, such as the Mandelbrot set M, are self-similar, although I've never heard of any functions to formally model this perspective. I'm curious to learn about any ...
5
votes
1answer
232 views

What is the mathematical principle that describes a series of dots on concentric circles that form a spiral pattern?

Apologies for the vagueness of the question, I'll clean it up once an answer helps me describe it better. I'm fascinated by the pattern demonstrated in this image. It's made up of dots on a series ...
3
votes
2answers
280 views

Dimension of fractals

I would like to know is it possible to generate a fractal in the plane with dimension higher than 2? If that is possible, please could you explain the intuition behind that? If it is not possible, is ...
0
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1answer
137 views

What is the tangent point of any given co-prime on the Mandelbrot Set in pseudocode?

Given a computer program generating the Mandelbrot Set - using this one for example, which uses a module called mandel.js - what would be the pseudocode to find the complex coordinates, capable of ...
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1answer
166 views

Interesting non-stem questions about Koch/Sierpinski fractals

Exam time and I am having a hard time finding any inspiring questions about fractals for our "contemporary math" course. We found the perimeter and area of various Koch snowflakes and Sierpinski ...
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4answers
797 views

Mandelbrot fractal: How is it possible?

I'm a programmer and have recently played around a bit with rendering Mandelbrot fractals / zooming into them. What I can't grasp: How can such infinite, complex shapes come out of somewhat 10 lines ...
6
votes
2answers
388 views

Odd fractal-looking illusion with $x,y,z \in [0,1]$ such that $x+y+z=1$, what is wrong?

Thanks to comments, it should be a plane but why does it look a bit like a fractal? Does my code overlook something or some err in plotting tool? I used Python and GNUplot. Apparently an animated ...
3
votes
2answers
346 views

quasiconformal “automorphism” groups of julia sets

To motivate this question, let me begin with a picture: Each letter labels a "blob" of this quartic julia set. (is there a technical term for these parts?). Because of resolution limitations I ...
3
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0answers
100 views

The Hausdorff dimension of the set of solutions of a system of coupled differential equations

I am interested in the relationship between non-linear differential equations and the Hausdorff, or fractal, dimension of the set of solutions. For example, the Lorenz Attractor, which is a standard ...
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0answers
201 views

Visualizing the Partition numbers (suggestions for visualization techniques)

So Ken Ono says that the partition numbers behave like fractals, in which case I'd like to try to find an appropriately illuminating way of visualizing them. But I'm sort of stuck at the moment, so ...