**4**

votes

**0**answers

683 views

### What is known about the area of the symmetric Pythagorean tree?

What is known about the area of the symmetric Pythagorean tree? (Closed unit square as base, area of enclosed triangles not included.) The problem in calculating the area is that squares start to ...

**6**

votes

**1**answer

149 views

### Random Cantor sets on the unit interval

Denote $A=\{0\}, B=\{0,1\}$. Then any subset of $\Omega:=\{A,B\}^{\mathbb N}$ is a continuum provided the number of $B$'s is infinite. We treat these as binary expansions of numbers in $[0,1]$.
For ...

**4**

votes

**1**answer

66 views

### Usable Change-of-Variables Formula for Hausdorff Measure

Let $H^{s}$ be the $s$-dimensional Hausdorff measure, let $D$ be a nonsingular matrix. Consider the change of measure formula:
$$
\int\limits_{A} f(Dx) \; \mathrm{d}H^{s}(x) = \int\limits_{ D A} f(y)...

**3**

votes

**1**answer

497 views

### Mandelbrot and “log-derivative”

I am reading Mandelbrot, and stubling upon his use of the limit ("almost a Hölder exponent")
$$\lim_{\varepsilon\to0}\frac{\log(f(x+\varepsilon) - f(x))}{\log(\varepsilon)}.$$
To simplify, lets ...

**0**

votes

**0**answers

111 views

### Can infinitely many orbifolds be “added up” to form a fractal space? [on hold]

Disclaimer: this question is rather vague and thus might not be suitable for this site. If so, feel free to tell me and I'll delete it.
Intuitively, an orbifold, from what I understand, is a "...

**5**

votes

**2**answers

141 views

### Integral over the Cantor's set Hausdorff dimension

As can be seen in the David Morin's Classical Mechanics, there are some scaling strategies in order to calculate the moments of inertia of certain fractals, for example, the Cantor's set has a moment ...

**14**

votes

**7**answers

1k views

### Unexpected Occurences of the Sierpinski Triangle

The probably most wellknown occurance of the Sierpinski Triangle is as the odd entries of the Pascal triangle
Some month ago however, there was an article about mathematical models of sandpiles along ...

**1**

vote

**2**answers

404 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 ...

**3**

votes

**0**answers

149 views

### Product Fractals

Here is a theorem found in the Falconer's book on fractal geometry:
Theorem: For any sets $E\subset \mathbb{R}^n$ and $F\subset \mathbb{R}^m$
$$
\dim_HF+\dim_HE\leq \dim_H(E\times F)\leq \dim_HE+\...

**2**

votes

**0**answers

55 views

### Critical set of a self similar structure

A self similar structure is a triple $L=(K, S, \{F_i\})$ where $K$ is a compact metric space, $S$ is a finite set and for every $i\in S$ the functions $F_i:K\rightarrow K $ are injective and ...

**1**

vote

**0**answers

43 views

### Fundamental system of neighborhoods. Self similar set

I have been reading the text Analysis on Fractals of Jun Kigami. There is a theorem about the fundamental system of neighborhoods of a point in a self similar set. It is stated as follows
Let $\...

**19**

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 ...

**16**

votes

**1**answer

669 views

### Are the algebraic numbers dense everywhere on the boundary of the Mandelbrot set?

Let $\mathcal{B}$ denote the boundary of the Mandelbrot set, and let
$\overline{\mathbb{Q}}$ denote the algebraic closure of the rationals.
Further put $\mathcal{B}_{\overline{\mathbb{Q}}} := \mathcal{...

**7**

votes

**0**answers

147 views

### Does the Mandelbrot set have infinite VC dimension?

Define a binary classifier for points in the complex plane, whose parameter $\theta$ is an isometry of $\mathbb{C}$, and which classifies $z \in \mathbb{C}$ based on whether or not $\theta(z)$ is in ...

**27**

votes

**3**answers

816 views

### A point set of power series with coefficients in {-1, 1}. Connected or not?

Let $z$ be a fixed complex number with $|z|<1$ and consider the set
$$X_z := \Big\{\sum\limits_{i=1}^{\infty} a_i z^i \ \Big|\ a_i\in \{-1,1\} \forall i\Big\}.$$
What can be said about the set $M$ ...

**1**

vote

**0**answers

78 views

### Quantification of the extent of periodicity in a time series using fractal analyses

I need metrics to quantify and compare the extent of periodicity between any two given time series, considering the time series were "almost periodic". By "almost periodic" I mean: if I were to take ...

**0**

votes

**0**answers

55 views

### What results exist for functions with regionally fluctuant fractal dimension?

I'm interested in functions that have a varying fractal dimension at different scales and/or regions. Has this been investigated in detail? I'd be interested in results and references in this area of ...

**10**

votes

**3**answers

589 views

### Self-Similar Graphs

Many fractals can be generated using and infinite sequence of graphs. For example, Sierpinski's Gasket could be generated by the following sequence of graphs.
Many definitions of fractal dimensions ...

**7**

votes

**2**answers

201 views

### Angles and proportions occurring in L-system fractals

This is about properties of certain fractals defined by Lindenmayer systems, a.k.a. L-systems. Unlike “classical” fractals like Julia sets or the Mandelbrot set (the name “set” says it all), these ...

**0**

votes

**1**answer

188 views

### Lightning strike fractal formula [closed]

I need to generate random gold ore channels for a game, I was thinking they would look kinda like lightning strikes. Anyone know any good fractals (recursive functions) that looks like it? Or ...

**10**

votes

**2**answers

389 views

### Nim and the Sierpinski Gasket

(I discovered this in high school, sent it off to a journal, never heard back, and forgot about it. I've never found anyone else who appeared to know about it; the combinatorial game theorists I've ...

**13**

votes

**3**answers

917 views

### Is there an intrinsic definition of fractal (i.e. not embedded in euclidean space)?

Long ago, manifolds were embedded subsets of euclidean space defined by polynomials. Later, using the gluing of open sets, people realized they could define manifolds intrinsically. And in certain ...

**10**

votes

**1**answer

364 views

### Are there any exact results for Hausdorff Measure?

The computation of the Hausdorff measure is extremely difficult due to the infinum appearing in its definition. This has made the calculation of the Hausdorff measure for nearly all fractals difficult ...

**0**

votes

**0**answers

65 views

### Fractal in discrete time series/discrete time sequence

Consider a time series of real number $x_1, x_2,\dots,...x_n$. How one can define fractal dimension of this series?
I would like to know famous formula $F+H=2$ where H is Hurst exponent and F is ...

**13**

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 ...

**12**

votes

**2**answers

631 views

### What is the Hausdorff dimension of this fractal?

Let $\sum_{i=h}^\infty d_i/b^i $ be the base $b$ representation of $x \geq 0,$ where $b>1$ and the $d_i$ are uniquely determined by the greedy algorithm. For fixed $c>1,$ let $f(x)= \sum_{i=h}^\...

**3**

votes

**1**answer

72 views

### Subsets of sets of positive Hausdorff dimension with controlled upper Minkowski dimension

Call a Borel set $A \subseteq [0,1]$ good if $$0 < \dim(A) \leq \overline{\dim_\text{M}}(A) < 2 \dim(A),$$ where $\dim(A)$ is the Hausdorff dimension of $A$ and $\overline{\dim_\text{M}}(A)$ is ...

**7**

votes

**1**answer

288 views

### Fractal dimension of scaling limits of discrete structures

Let $S$ be the set of positive integers whose base-three expansion contains only the digits 0 and 2. The discrete set $S$ in a sense has (negative) fractal dimension $(\log 1/2)/(\log 3)$, since if ...

**11**

votes

**2**answers

524 views

### On the boundary of the twindragon

Let $\mathcal T$ be the famous twindragon, i.e.,
$$
\mathcal T=\left\{\sum_{n=0}^\infty a_n\left(\frac{1+i}2\right)^n : a_n\in\{0,1\}\right\}.
$$
Then, as is well known, $\mathcal T$ has a non-...

**2**

votes

**2**answers

227 views

### Is there an (almost) dense set of quadratic polynomials which is not in the interior of the Mandelbrot set?

For the parameter plane of complex quadratic polynomials, $(z\mapsto z^2+c)_{c\in\mathbb{C}}$ :
Is it possible to find a part of the parameter plane, scanned with a given limited precision (...

**6**

votes

**1**answer

291 views

### Precise density estimates for Cantor sets

Let $C_\lambda$ be the classical Cantor set associated to a real number $0<\lambda<\frac{1}{2}$, as defined for example in the book of K. J. Falconer The geometry of fractal sets. I recall ...

**1**

vote

**1**answer

163 views

### What is “graph-directed iterated function”?

Im translating an article about Rauzy fractal and I ran into this sentence:
...

**6**

votes

**1**answer

218 views

### Isotropy of Apollonian disk-packing

Is there any sense in which the "epsilon-tail" of an Apollonian disk-packing (by which I mean the union of the disks of radius less than epsilon) exhibits more and more statistical isotropy as epsilon ...

**6**

votes

**1**answer

188 views

### Spirals in Apollonian circle-packings

Given mutually (externally) tangent circles $C_1,C_2,C_3$,
let $C_n$ be the unique circle externally tangent to
$C_{n-1}$, $C_{n-2}$, and $C_{n-3}$ for $n \geq 4$.
Let $P_{\infty}$ be the point toward ...

**3**

votes

**1**answer

241 views

### “Nice” functions on infinite-dimensional space of germs of continuous functions at a point

Consider set of all germs of continuous functions at some point.
Question: What are some functions ("any/nice/constructive/whatever") from this set to R (reals) ? (Except evalution at point and made ...

**7**

votes

**1**answer

296 views

### For a 3D Apollonian packing, do we really know that the Hausdorff dimension of the complement is approximated by the growth rate of curvature?

The fractal dimension of the 3D Apollonian packing is computed in this paper.
In the introduction, the authors cite three of Boyd's paper (Ref 2, 5, 6) to support that the fractal dimension (...

**3**

votes

**1**answer

149 views

### boundary density of the Von Koch flake

Given a measurable set $K\subset \mathbb{R}^d$ we consider the occupation ratio $$f_r(x)=vol(K\cap B(x,r))/r^d$$ and especially the asymptotics when $r\to 0$. When $K$ has a fractal boundary and $x$ ...

**5**

votes

**3**answers

987 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 ...

**25**

votes

**2**answers

1k views

### Fractal-like structures arising from the action of a group on $\mathbb{Z}^2$

Let $G := \langle a, b, c \rangle < {\rm Sym}(\mathbb{Z}^2)$ be the group
generated by the permutation
$$
a: \ (m,n) \ \mapsto \ (m-n,m)
$$
of order $6$ and the involutions
$$
b: \ (m,n) \ \...

**11**

votes

**1**answer

286 views

### Cantor set intersecting a geometric sequence

I was working on a problem involving finding all points in the intersection of the Cantor set $C$ and the geometric sequence $\{ (2/3)^i \}_{i=1}^\infty$. The only points I have in this intersection ...

**-4**

votes

**1**answer

295 views

### Proof of im/possibility of constructing any fractal by iterated function systems? [closed]

Well the question is as simple as that and what I really want to see is if there is a mathematical proof that can tell whether every fractal(object of any non-integer dimension) can be constructed by ...

**12**

votes

**3**answers

513 views

### Dimensions of self-affine sets

Let $A$ be a $2\times 2$ matrix which we assume to be contracting, i.e., the exists $\alpha\in(0,1)$ such that
$$
\|A {\mathbf x}\|_2\le \alpha\|{\mathbf x}\|_2,\quad \forall {\mathbf x}\in\mathbb R^...

**4**

votes

**3**answers

521 views

### Precise location of the Mandelbrot Bulb Attachment to the main Cardioid

Is there an analytical formula for determining the location of the attachment points of the bulbs on the main cardioid? I was told there is an exact parametrization of the boundary of the main ...

**1**

vote

**1**answer

218 views

### notable inductive proofs relating to fractals

what are notable/ prominent inductive proofs relating to fractals?
the motivation for this question is:
fractals are very difficult mathematical objects to work with, and many problems/questions ...

**1**

vote

**1**answer

329 views

### Singular distributions: Applications and Instances

Singular distributions are special mathematical objects. They have an interesting property of not having a density function, defined on a set with Lebesgue measure zero. Cantor distribution is the ...

**2**

votes

**1**answer

450 views

### Level sets of a Weierstrass nowhere-differentiable function

Can anyone describe level sets of a Weierstrass nowhere-differentiable function? For example, let $f(x) = \sum_{n=0}^\infty \frac{1}{2^n} \cos( 4^n \pi x)$. For some $c \in (-2,2)$, what is known ...

**5**

votes

**1**answer

371 views

### Smallest positive zero of Weierstrass nowhere differentiable function

Consider the Weierstrass nowhere differentiable function $f(x) = \sum_{n=0}^\infty \frac{1}{2^n} \cos(4^n \pi x)$. It seems that the smallest positive zero of $f(x)$ occurs at $x=\frac{1}{5}$, but I ...

**9**

votes

**2**answers

1k 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 ...

**6**

votes

**1**answer

193 views

### Escape Time of Fractional Brownian Motion

Let $B(t)$ be Brownian motion with $B(0)=x>0$ and let $A>x$. It is well known that the expected time for $B(t)$ to escape the interval $[0,A]$ is equal to $x(A-x)$.
Is the expected time known ...

**3**

votes

**1**answer

135 views

### How is the Fractal Dimension of a Parametric Curve Related to the Fractal Dimensions of its Coordinate Functions?

Question:
Suppose the fractal dimension $1\le d_c\le2$ of a planar parametric curve $c(t) := (x(t),y(t))$ is given;
can any nontrivial estimates for the fractal dimensions $d_x$ of $x(t)$ and $d_y$ of ...