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

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51
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4answers
2k views

Why does the Mandelbrot set contain (slightly deformed) copies of itself?

The Mandelbrot set is the set of points of the complex plane whos orbits do not diverge. An point $c$'s orbit is defined as the sequence $z_0 = c$, $z_{n+1} = z_n^2 + c$. The shape of this set is ...
27
votes
5answers
719 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 ...
18
votes
4answers
846 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 ...
4
votes
3answers
245 views

Discuss the convergence of $ \left \{ a_n \right\} $ where $ a_{n+1}=\frac{a_0}{2}+\frac{a_n^2}{2},n\geq 1 $

Let $$ a_{n+1} = \dfrac{a_0}{2} + \dfrac{a_n^2}{2} $$ where $ a_1 = \dfrac{a_0}{2} $ and $ n\geq 1 $ Discuss the convergence of $ \left\{a_n\right\} $
42
votes
1answer
938 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, ...
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 ...
8
votes
1answer
336 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 ...
6
votes
2answers
5k views

Continuous coloring of a Mandelbrot fractal

I've recently started making a small fractal app in Javascript using the famous Mandelbrot bulb $(z = z^2 + c)$. I've been trying to find the best method of coloring the points on the complex plane, ...
3
votes
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 $f$ ...
6
votes
1answer
172 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 ...
3
votes
3answers
131 views

Why is the Koch curve homeomorphic to $[0,1]$?

Henning Makholm has provided a nice proof that the limiting curve is a continuous function from $[0,1]$ to the plane. I was curios if the function is homeomorphism. A quick search gave me many sources ...
27
votes
5answers
3k views

What exactly are fractals

I have always been amazed by things like the Mandelbrot set. I share the view of most that it and the Koch snowflake are absolutely beautiful. I decided to get a deeper more mathematical knowledge of ...
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 ...
14
votes
2answers
389 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 ...
13
votes
2answers
373 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 ...
9
votes
1answer
400 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 ...
4
votes
3answers
495 views

Need good material on multifractal analysis

I'm searching for some good reading material on multifractal analysis. Preferably something accessible that doesn't put the stress too much on mathematical proofs but rather on applications. As long ...
2
votes
0answers
55 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 ...
7
votes
3answers
391 views

Mandelbrot boundary

Is there a sequence of parameterized expressions for the border of all the major bulbs of the mandelbrot set? By major meaning all bulbs with diameter greater than 0.01 for example. I am interested ...
7
votes
3answers
532 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 ...
6
votes
2answers
208 views

How to figure out the starting point for this Mandelbrot?

My answer for another math stackexchange question, asked by Gottfried, involved observing Mandelbrot bifurcation for the iterated function in question, $f(z)\mapsto z-\log_b(z)$. In particular, for ...
5
votes
3answers
234 views

Are there mini-mandelbrots inside the julia set?

I've seen a julia set zoom but it is not nearly as interesting as a mandelbrot zoom. I also have not seen corresponding julia sets for zooms in the mandelbrot deeper than the original image. I'm ...
4
votes
1answer
217 views

Sets of Constant Irrationality Measure

Let $\mu (r)>2$ be the irrationality measure of a transcendental number $r$, and consider the following set of points $P \in\mathbb{R}$: $P=\{r\in \mathbb{R}: \mu(r)=Constant\}$ Is this set a ...
7
votes
1answer
229 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 ...
6
votes
3answers
173 views

Does the Mandelbrot fractal contain countably or uncountably many copies of itself?

I've been working on a program that draws fractal images, and I was struck by a question that came to mind. It is clear that the Mandelbrot fractal contains infinitely many copies of itself, but I've ...
7
votes
3answers
591 views

variant on Sierpinski carpet: rescue the tablecloth!

I was playing around with Sierpinski carpets (see pretty GPU-produced picture here), and came up with a variation that I have been unable to find mentioned elsewhere. I'm wondering if anyone can tell ...
4
votes
2answers
98 views

Length of a Coastline

When B. Mandelbrot's typical example of measuring the length of a coastline is referenced, they mention how at every scale the length increases. In pure mathematics, I can imagine this quite well-- ...
4
votes
0answers
68 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 ...
4
votes
1answer
141 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 ...
3
votes
3answers
125 views

How do we solve $c_1^d+\ldots+c_n^d=1$ for $d$?

The question is motivated by the definition of self-similarity dimension for self-similar sets: Let $M \subset \mathbb R^d$ be self-similar. That is, there are $T_1, \ldots, T_m \subsetneqq M$ and ...
3
votes
1answer
103 views

Area of filled Julia set

This is a vague question, and I know nothing about this area. We fix some $c\in\mathbb C$ and iterate the map $z\mapsto z^2+c$. This gives some filled Julia set, i.e. the set of points $z\in\mathbb ...
2
votes
3answers
147 views

What is the topological dimension of the Peano curve?

The Hausdorff dimension of the Peano curve is know to be two. And I assume it to be a fractal since it's on the List of fractals by Hausdorff dimension. Moreover: According to Falconer, one of the ...
1
vote
1answer
26 views

How do you graph a fractal on a line, as a function of time and position?

I'm writing ANSI C code for work and just got working my LED Light Show Library. The first project it's going on is a sound bar with 10 volume indicator (1-color (white)) leds that are lined-up. ...
1
vote
1answer
139 views

What is the area of the apollonian gaskets?

I searched for the internet, but found nothing relavant to the area. The areas in each intermediate step form a bounded increasing sequence, so there is a limit. But wil it eventually fill in almost ...
0
votes
1answer
152 views

Hausdorff Measure and Hausdorff Dimension

Could someone explain the intuition behund the Hausdorff Measure and Hausdorff Dimension? The Hausdorff Measure is defined as the following: Let $(X,d)$ be a metric space. $\forall S \subset X$, ...
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 ...