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

learn more… | top users | synonyms

9
votes
0answers
208 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 ...
8
votes
0answers
231 views

Kakeya Needle problem video

I'm intruiged by the Kakeya Needle problem, described here on Wikipedia. Wikipedia has a nice animation of a needle turning through a hypo-cycloid: What I'm searching for is a visualisation of the ...
7
votes
0answers
199 views

About devil's staircases

We say that a function $f:\left[a,b\right] \to \mathbb{R}$ is a singular function or a devil's staircase if $f$ satisfies the following properties: $f$ is continuous; $f(a) < f(b)$; $f$ is ...
7
votes
0answers
219 views

Is The *Mona Lisa* in the complement of the Mandelbrot set.

Here is a description of how to color pictures of the Mandelbrot set, more accurately the complement of the Mandelbrot set. Suppose we have a rectangular array of points. Say the array is $m$ by $n$. ...
6
votes
0answers
242 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 ...
5
votes
0answers
108 views

Fractal dimension of Gaussian white noise is infinite?

I read in this paper that the fractal dimension of Gaussian white noise is infinite. The paper does not prove it nor give a reference to support it. I failed to find a reference from online searching. ...
5
votes
0answers
373 views

The Cantor Space as $\{0,1\}^{\mathbb{N}}$ and as $[0,1]$.

The Cantor-Space is defined as the space of all infinite binary sequences, i.e. the space $\{0,1\}^{\mathbb{N}}$. It has a natural metric, $$ d(x,y) = \inf\{ 2^{-|w|} : w \in pref(x) \cap pref(y) \} ...
5
votes
0answers
77 views

How difficult is it to impose a differential structure on a fractal?

Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply ...
4
votes
0answers
55 views

Why does the Mandelbrot shape show up in other fractals?

In the pictures below, the Collatz map fractal includes parts resembling the Mandelbrot set. Why? Do other fractals do so? The Mandelbrot set From Wikimedia Commons Part of the Collatz map fractal ...
4
votes
0answers
95 views

The Tribonacci constant and the Dragon

Let $x = \frac{\ln T}{\ln 2} = 0.879146\dots$ where $T$ is the tribonacci constant, then x solves the transcendental equation, $$4^x(2^x-1)=(2^x+1)$$ Let $x = \frac{\ln y}{\ln 2} = 1.523627\dots$ ...
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
0answers
142 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 ...
4
votes
0answers
104 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 ...
3
votes
0answers
109 views

Fractal derivative of complex order and beyond

Is there some precise definition of "complex (fractal) order derivative" for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would ...
3
votes
0answers
89 views

Help understanding this 'Fractal' I've just made?

I was messing around in C++, making an image where the pixels change depending on the the rectangle's dimensions and whether or not the space bar is down, and I formed this image: Could anyone ...
3
votes
0answers
172 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 ...
3
votes
0answers
157 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 ...
3
votes
0answers
52 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 ...
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 ...
3
votes
0answers
192 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 ...
3
votes
0answers
581 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 ...
2
votes
0answers
71 views

Points in a general Cantor set

We often look at the Cantor set with the construction that keeps removing the middle thirds of the remaining line segments at each iteration. Corresponding to this construction, we can determine ...
2
votes
0answers
53 views

Is this plot of Ford circles actually a fractal?

Is this plot of Ford circles actually a fractal?
2
votes
0answers
71 views

What is the name of this metric: Why is $(\mathcal{M}, L)$ complete

I am reading section 4 of this article about invariant measures: http://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf Let $(X,d)$ a complete metric space, ...
2
votes
0answers
49 views

what part of a m-set fractal showing spiral behaviour?

What part of a fractal Mandelbrot Set showing spiral behaviour like this one: what is it's direct equation?
2
votes
0answers
47 views

Show that Hausdorff measure is semifinite

I am currently reading a book about fractals and the author states the result that Hausdorff measure is semifinite. Can someone tell me how to prove or provide a hint for me?
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 ...
2
votes
0answers
68 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 ...
1
vote
0answers
52 views

Is the Mandelbrot set computable?

This is a weakened version of Is the measure induced by the Mandelbrot set computable on rational rectangles? ; Given a (computable, or rational) rectangle in the complex plane, is it computable ...
1
vote
0answers
22 views

Intersections of fractal sets with connected sets

Let $\beta \geq \alpha > 0$. Let $A\subset\mathbb R^n$ be a measurable set with Hausdorff dimension $\alpha$. Must there exist a closed connected set $B$ with Hausdorff dimension $\leq \beta$ ...
1
vote
0answers
118 views

Sequential Algorithm to generate Fractal (Koch's snowflake)

As part of an assignment I had developed a sequential algorithm to generate a Koch's snowflake. Algorithm I have encountered so far have been recursive and iterations generate closer approximations. ...
1
vote
0answers
105 views

Mandelbrot set and riemann hypothesis

Has anyone tried to make a connection between the Mandelbrot set and the non-trivial zeros the zeta function? Looking at the Mandelbrot set, it appears that all points are to the left of the line 0.5 ...
1
vote
0answers
145 views

Fractal Analysis

Is there any way to compare two fractals and analyse the difference between the two. I'm doing a project on fractals and It'll be very easy if there is a module which can be used to analyse and ...
1
vote
0answers
105 views

L-systems and Sierpinski Triangle

I was just shocked when I saw these consecutive outcomes of an L-system converging to the Sierpinski triangle (shown in this picture). I'm interested to know how can one arrange the rules of an ...
1
vote
0answers
66 views

what is the fractal dimension of the henon map?

I have some questions about the Henon map that are not clear for me. I have seen that the correlation dimension of the Henon map is approximately 1,21, is that measure similar to its fractal ...
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...
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 ...
1
vote
0answers
236 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
votes
0answers
12 views

Can Wiener process on a fractal random graph be reduced to a levy flight?

Weiner process on small-world graphs is a Levy flight. But does the condition still hold for a random graph that connects the edges of a fractal?
0
votes
0answers
14 views

How do you detect when an iterated function converges / diverges and calculate limit accurate-enough?

See this post for a background on what I'm doing. So how many iterations $N$ of $P_c(z)$ does it take so that if $f(c) = P_c^N(z)$, and $g(c) = $ infinite iterations of $P_c(z)$, then $h \circ ...
0
votes
0answers
10 views

Singular distributions: Applications and Instances

This is the duplication of the question I asked here. I repeat it here with hope of getting new answers. Singular distributions are special mathematical objects. They have an interesting property ...
0
votes
0answers
15 views

algorithm for traversing a fractal in a “maximally ordered” way

consider a multidimensional fractal that can be "traversed" in an arbitrary order. is there an algorithm for traversing a fractal in a "maximally ordered" way? in other words the algorithm has ...
0
votes
0answers
64 views

Hausdorff dimension is less than box counting dimension?

I have been asked to prove that for a bounded set $F\subset\mathbb{R}^n$, $dim_H F\le \underline{dim}_B F \le \overline{dim}_B F$ where $dim_H F$ is the Hausdorff dimension, $\underline{dim}_B ...
0
votes
0answers
17 views

How to… quotient set on a fractal continuous curve.

I'm really not good at math so I can't really formulate my problem in a closed form :) There is a curve $C$ in $R^2$. There are some rulers of length ${L1,L2,L3,L4,L5,....}$ I need to find a way to ...
0
votes
0answers
21 views

information dimension and correlation dimension, what do they really mean?

If I have measure the information dimension and correlation dimension of a couple of fractals, I would like to know what these measures really stands for. For example, lets suppose: fractal 1, inf ...
0
votes
0answers
74 views

Fractal geometry of literature: First attempt to Shakespeare's works

I found this article on arxiv.org. It has been written by some unknown guy named Ali Eftekhari. Apparently, he is a chemist with original works and publications in chemistry and nano-technology. ...
0
votes
0answers
34 views

Addressing/traversing an infinite 2D grid using a Z-line?

I'm looking for a method to map an infinite 2D grid using a line, so that I would have just one integer from which I would compute the X and Y. I know something like that exists, but can't recall the ...
0
votes
0answers
92 views

doubt in a book proof from 'The Geometry of Fractal Sets'

I am reading the proof of existence of positive finite $H^s$-measure (Theorem 5.4) on page 67-68 of The Geometry of Fractal Sets.I am not quite convinced that $E_k$ are closed set by the construction ...
0
votes
0answers
75 views

Area fractal pentagrams III

how can I find the area of these two fractal? I've been try to solve some geometry exercises here but this and some other are so much difficult.
0
votes
0answers
121 views

Area fractal pentagrams II

A simple fractal. How to find the area of it? (only the arms of the star) Working with pentagrams is quite complicated, I can not solve this.