**4**

votes

**4**answers

580 views

### Lecture on Fractals for Middle School Students

I'm going to have a one-hour lecture for middle school students next Monday. It will be about fractals. The students know virtually nothing about this subject.
I'll show some fractal images and a few ...

**4**

votes

**2**answers

372 views

### Algebraicity of the “outer” boundary of the Mandelbrot set

Let $M$ be the Mandelbrot set and let $\lambda\in M, \mu\in \mathbb C$ be algebraic numbers. Let $t_{\lambda,\mu}$ be defined as
$$
t_{\lambda,\mu} = \sup \lbrace t\in \mathbb R\colon \lambda +t\mu ...

**1**

vote

**1**answer

156 views

### Fractal dimension of 1D set, what if logN vs log(e) is a polygonal chain?

I have a finite set of points, and plot the graph log(N) vs. log(e). I see a polygonal chain (the final slope, starting at some size of e, is zero, of course). If the set represents some physical ...

**0**

votes

**1**answer

573 views

### What are integration on fractal? [closed]

Who can explain the proof of the formula (2.12) given here: J. Phys. A: Math. Gen. 20 (1987) 3861-3875. Printed in the UK ...

**8**

votes

**2**answers

981 views

### sequences with a fractal dimension

This is inspired by the self-similarity of the celebrated Golay-Rudin-Shapiro sequence, more exactly, of its alternating partial sums. (This latter one is oeis 020990). The pictures show the 550 first ...

**5**

votes

**2**answers

313 views

### Measures of full Hausdorff dimension for self-affine sets

Consider the iterated function system $T_{1}(x)=(\beta x,\tau y)$, $T_{2}(x,y)=(\beta x+(1-\beta),\tau y+ (1-\tau))$ for $\beta\in(1/2,1)$ and $\tau\in (0,1/2)$ with self affine set ...

**2**

votes

**2**answers

394 views

### Visualizing the l-adic fractal in the partition function p(n)

This page http://www.aimath.org/news/partition/ and this youtube lecture http://www.youtube.com/watch?v=aj4FozCSg8g speak of a fractal in the values of the partition function p(n).
"We prove that ...

**2**

votes

**1**answer

296 views

### Can we extend a continuous function with keeping Hausdorff dimension?

Let X be a compact subset of R^d, and K be a compact subset of X, such that Dim_H(X)=Dim_H(K). Let F be a continuous function on K, Can we extend F from K to X, with keeping the continuous and the ...

**8**

votes

**2**answers

722 views

### Fractal Tiling of Rhombic Dodecahedra

Hello, this is my first question on Math Overflow...
Rhombic dodecahedra can be tiled in 3-space, leaving no gaps. This tiling corresponds to the close-packing of spheres.
Consider a "nucleus" ...

**3**

votes

**2**answers

535 views

### Approximating fractal curves

Is there a known algorithm for approximating a fractal curve, say as specified by some iterative procedure e.g. a Koch snowflake, in terms of $f^{-1}(0)$ for some "simple" function $f$?
Specifically, ...

**12**

votes

**4**answers

948 views

### Fourier decay rate of Cantor measures

For $0<\theta<\frac{1}{2}$, denote by $C_\theta$ the Cantor set with dissection ratio $\theta$, i.e. the Cantor set obtained from dissection parttern $(\theta, 1-2\theta,\theta)$. It is known ...

**2**

votes

**4**answers

976 views

### Fractals as solution to optimization problem?

What's the scientific reason for fractals being present in nature at such a large scale? Is it perhaps the solution of an optimization problem? For example, would the fractal based shape of certain ...

**2**

votes

**0**answers

153 views

### A fractal object at origin but nowhere else: derived from Brownain motion

Hi,
Please consider this object: Start with a realization of Brownian motion in 2D, which I'll denote by rho(t) where -infinity < t < +infinity. Next, lets smooth rho. There are various ...

**2**

votes

**1**answer

443 views

### Hausdorff dimension of a subset of Cantor set

What is the Hausdorff dimension of the subset
$$F := \{ x = \sum^\infty_{n=1} \frac{2 x_n}{3^n} \in [0,1] : x_n \in \{ 0 , 1 \} , x_n = 1 \Rightarrow x_{n+1}=0 \}$$
of the Cantor set? Is it known ...

**0**

votes

**1**answer

328 views

### existence of fractal [closed]

I have a question about fractals;
Suppose $\alpha\in[0,1]$ is real number, is there any fractal $F_\alpha$, such that $Dimension(F_\alpha)=\alpha$?
If yes, do we have any method to construct such ...

**2**

votes

**0**answers

475 views

### Self-similarity of Riemann's “non-differentiable” function

I hope it doesn't seem inappropriate for me to raise on MO an unanswered question from MSE, indeed a question actually posed there by someone other than myself.
I want to ask the following:
1) ...

**3**

votes

**0**answers

237 views

### Picture of the set of discontinuity of degree 2 rational Julia sets

Let $Rat_d$ be the set of all rational fraction of degree $d$ and $X_d \subset Rat_d$ be the bifurcation locus of rational fractions of degree $d$, i.e. the closure of the set of discontinuity of the ...

**13**

votes

**3**answers

540 views

### Julia sets using other fields

I hope I am forgiven for my noob question. But, does it make sense to think of Julia sets using other fields? More precisely I would like to think of fields in which closed and bounded isn't ...

**1**

vote

**2**answers

323 views

### Terrain Generation: Infinite 2D space filled with Diffusion-limited aggregation clusters?

Disclaimer: I don't have a deep understanding of fractals or any higher math, I'm just personally interested in it, so please excuse me if I'm using wrong terms or if I'm being inaccurate. Making ...

**1**

vote

**0**answers

189 views

### Classification of Self similar sets

I am looking at self similar sets in $\mathbb{C}$ defined as the fixed set or a sequence of contractions or an iterated function system. I am currently trying to classify these sets by how they are ...

**12**

votes

**2**answers

1k views

### Is there some known way to create the Mandelbrot set (the boundary), with an iterated function system?

Is there some known way to create the Mandelbrot set (the boundary),
with an iterated function system (IFS)?
Julia sets can be formed by iterating the two functions $z \mapsto \pm \sqrt{z-c},$
and ...

**3**

votes

**1**answer

663 views

### Hausdorff dimension of graphs .

Is there an easy way to calculate the Hausdorff dimension of the graph of a real "elementary" function, like $f(x)=\sin(1/x)$ ?

**4**

votes

**3**answers

778 views

### Estimating the fractal dimension of a point cloud

I have finite set of geolocation point data, and I'd like to estimate the fractal dimension. I know there are several ways to do this, and some of them give different numbers. What is the most ...

**1**

vote

**2**answers

883 views

### Lyapunov Exponent and degree of chaos

I am aware that having positive Lyapunov exponents in a system signifies that a system is chaotic. However, I would like to know if there is a means to know the degree of chaos in the system from the ...

**3**

votes

**2**answers

246 views

### Hausdorff dimension of inverse images.

Let $f: \mathbb{R}^d \to \mathbb{R}$ be a continuous function. Let $t \in (\inf(f), \sup(f))$ and define $C = f^{-1} (t)$. Is it true that the Hausdorff dimension of C is $\geq d -1$? If no how does ...

**2**

votes

**4**answers

922 views

### Fractal questions: Weierstraß-Mandelbrot

Hi,
Coming from a specific field in algebraic geometry I am a total noob in Fractal Theory and I'd like to learn it a bit. I hope I am tolerated for my maybe-trivial questions. I just read about the ...

**0**

votes

**1**answer

2k views

### PhD at 30 vs 33 [closed]

I'm almost 28. I have two bachelors degrees, all from the UK. One in Computing and the other in Electrical & Electronics Engineering. My area of interest is nonlinear dynamics/chaos and complex ...

**0**

votes

**1**answer

481 views

### Is the notion of fractional dimension compatible with considering a dimension a set of n-tuples?

Hi there,
I am trying to teach a friend about higher-dimensions and I have explained them in the following two manners:
A higher dimension, e.g. the 56th is the set of all 56-tuples.
A fractional ...

**5**

votes

**5**answers

1k views

### Hausdorff dimension for invariant measure?

A fractal set has a Hausdorff dimension.
In some cases, we may generate a fractal by iterating $f,$
and let the fractal be the set of starting points $x$ such
that $|f^{\circ n}(x)|$ is bounded as ...

**2**

votes

**1**answer

432 views

### Mandelbrot and “log-derivative”

I am reading Mandelbrot, and stubling upon his use of the limit ("almost a Hölder exponent")
\lim_{\epsilon -> 0} log(f(x+\epsilon) - f(x))/log(\epsilon).
To simplify, lets assume that f is ...

**3**

votes

**2**answers

303 views

### Hausdorff dimension of non-recurrent walks

Preface: I am fairly new to the concept of Hausdorff dimension, so I don't know how interesting a question this is.
Identify walks on $\mathbb{Z}$ with infinite binary sequences (say $0$ means moving ...

**21**

votes

**6**answers

5k views

### How might M.C. Escher have designed his patterns?

I realize this question isn't strictly mathematical, and if it doesn't fit with the content on this site then feel free (moderators/high-rep users) to close it. But when I thought up the question it ...

**15**

votes

**6**answers

2k views

### Parametrization of the boundary of the Mandelbrot set

Does anyone know how to parametrize the boundary of the Mandelbrot set? I am not a fractal-geometer or a dynamical systems person. I just have some idle curiosity about this question.
The ...

**8**

votes

**2**answers

517 views

### local behavior of a finite Borel measure

Let $\mu$ be a finite Borel measure on $\mathbb{R}^n$. I am interested in how does $\mu(B(x,r))$ behave, where $B(x,r)$ is the open ball of radius $r$ centered at $x$. For instance, as far as I ...

**1**

vote

**2**answers

336 views

### Fractional Gaussian noise in higher dimensions

I'm having difficulty imagining (or explaining to myself) how the partial sums a two dimensional Gaussian noise can produce a surface. According to equation (20) of the paper, "On two-dimensional ...

**2**

votes

**1**answer

452 views

### Attractive Basins and Loops in Julia Sets

I recently learned about the Mandelbrot set for the first time from a presentation by some undergraduates in honor of Mandelbrot's death. The presentation was short and by non-experts so I left with ...

**-1**

votes

**5**answers

805 views

### finding cutting edge papers and books

Hi all,
what are the best strategies to find cutting edge papers and books on a field of mathematics?
..
Example:
2-3 years ago I had to analyze a time series. I found a paper and showed that to ...

**1**

vote

**1**answer

717 views

### Need help understanding Mandelbrot and Van Ness Fractional Brownian Motion

I need help understanding the Mandelbot and Van Ness' definition of Fractional Brownian motion
$
B_H( t , \omega ) - B_H( 0 , \omega ) = \frac{1}{\Gamma(H + \frac{1}{2})} \left\( \int_{-\infty}^0 ...

**2**

votes

**0**answers

248 views

### Complexity of a variant of the Mandelbrot set decision problem?

This is a modified version of a question posted on StackExchange TCS.
Mandelbrot set is defined using the complex equation $P_c (z)=z^2 +c$ where $c$ is a complex number. Let us define
...

**1**

vote

**2**answers

355 views

### Topologizing a free product G*H of discrete groups?

My -possibly flawed- mental picture of free products of groups certainly comes from the special case usually performed to illustrate the construction that proves the Banach-Tarski paradox. Thus I'm ...

**6**

votes

**2**answers

606 views

### Symmetries of the Julia sets for $z^2+c$

The julia set seems to have symmetries roughly corresponding to translation, rotation and scaling.
In the following image
You can see the horizontal translation, which leaves the extremal left and ...

**1**

vote

**1**answer

194 views

### Hausdorff dimension of higher powers of the Mandebrot set ?

My third question about Shishikura's result :
Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper1. The Mandelbrot set is defined by ...

**13**

votes

**2**answers

1k views

### Area of the boundary of the Mandelbrot set ?

My second question about Shishikura's result :
Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper 1. In a sense, could we consider it ...

**2**

votes

**1**answer

257 views

### Hausdorff dimension of subsets of the Mandelbot set.

Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper, but I can't figure out one thing : can we say all open subsets of this boundary has ...

**6**

votes

**4**answers

1k views

### Measure 0 sets on the line with Hausdorff dimension 1

I use $\dim_H(E)$ to denote the Hausdorff dimension of a set $E \subseteq \mathbb{R}$ and $|E|$ to denote its Lebesgue measure. It is easy to see from the definition of Hausdorff dimension that if ...

**2**

votes

**3**answers

598 views

### Hausdorff dimension: subset of $\mathbb{R}^n$ vs. boundary of this subset

Let $n$ be a positive integer.
Let $S \subseteq \mathbb{R}^n$. Is the Hausdorff dimension of the boundary of $S$ always smaller than the Hausdorff dimension of $S$?
I have not found anything ...

**3**

votes

**5**answers

1k views

### Reference for the iterated function system of the Koch snowflake

Let $KS$ be the Koch snowflake. This fractal has an iterated function system (IFS) of the form
$$ KS = \bigcup_{0 \leq k \leq 6} f_k(KS) $$
with
$$ f_0(z)=\frac{1}{\sqrt{3}} e^{i\pi/2} z $$
and for $0 ...

**5**

votes

**2**answers

572 views

### A calculus question related to quantization dimension

Going through some old papers, I came up with a simple-looking problem I thought about 5 years ago or so.
MO wants motivation ... Associated to a probability measure on a metric space is something ...

**0**

votes

**3**answers

579 views

### Is there a way to find regions of depth in the Mandelbrot set other than simply poking around?

Ie, is there a way to probe it for regions of depth that involves a function, the domain of which is the Mandelbrot set itself, or a part of that set?

**7**

votes

**2**answers

311 views

### Minimum number of contractions needed to obtain a particular invariant set

Consider the Koch curve $G \subseteq \mathbb{R}^2$. Clearly $G$ is the invariant set (IS) of the iterated function system (IFS) $\lbrace \phi_1, \phi_2, \phi_3, \phi_4 \rbrace$. Where (not wanting to ...