# Questions tagged [fractal-analysis]

Fractal analysis involves the study measures on fractals and their applications. This includes the study of self-similar integration and differential equations on fractals.

57 questions
Filter by
Sorted by
Tagged with
24 views

### How to determine the fractal dimension of a random walk in 2D?

I have to determine the fractal dimension of a random walk in 2D. How can I do that? I am first supposed to picture the random walk for 2D as in the following site. However, I am also asked to ...
• 113
1 vote
35 views

### Inequality on the uniform norm of a conformal differentiable transformation

I'm currently working on a question on chaotic dynamics on fractals. Particularly if we have an iterated function system of conformal differentiable contraction mappings {$f_1$,...,$f_n$} we denote ...
23 views

• 10.3k
59 views

### Does every point in the Koch curve belong to one of its approximating polygons?

So, pretty self-explanatory title: I was wondering wether if every point that the Koch curve has does necesarilly belong to one of its approximating polygons. Just in case, on the picture below, the ...
37 views

### Box-counting dimension: $\lim_{\delta \to 0} N_{\delta}(F)\delta^{s} = \infty$ when $s < \dim_{M}(F)$ and another similar limit.

I want to understand these two limits: $\lim_{\delta \to 0} N_{\delta}(F)\delta^{s} = \infty$ when $s < \dim_{M}(F)$ $\lim_{\delta \to 0} N_{\delta}(F)\delta^{s} = 0$ when $s > \dim_{M}(F)$ ...
• 1,676
96 views

### Do fractals necessarily have to be self-similar? What is the definition of a fractal? Is this figure a fractal? [duplicate]

I am doing a dissertation in Materials Engineering. I obtained the following images regarding crystal growth: In the literature they usually call these structures "fractals" and calculate ...
1 vote
192 views

### Integration of function using Hausdorff measure of Cantor Set

I am learning geometry on fractal shapes, along with how fractional calculus can relate to said geometry. At the moment I am trying to understand integration over a Hausdorff measure. According to ...
• 346
37 views

### How to calculate Assouad Dimension of sets

I am trying to work with sets $F$ as subsets of $\mathbb{R}^n$ and their Assouad dimension. I am a bit stuck on how to apply the definition as it asks that We find the infimum of possible $\alpha$ We ...
192 views

### Can the fractal dimension of a coastline be less than 1?

I am currently investigating the fractal dimension of the coastline of the island the Palm Jumeirah, including the crescent. Using the Hausdorff method I have reached an answer of 0.879. This is less ...
181 views

### Box-counting dimension (Minkowski Dimension) of {1, 1/4, 1/9, 1/16,...}

I want to determine the box-counting dimension (Minkowski Dimension) of the set $S=\{\frac{1}{n^2} \ | \ n \in \mathbb{Z}^+\}$. My first attempt was to first define my $\delta_n$. What I did was ...
130 views

### What fraction of a Sierpiński triangle is to the left of a given vertical line?

Let's say that we have a Sierpiński triangle of width 1 and height 1 I want to find the function $y(x)$ for $0<x<0.5$ that gives the fraction of the Sierpiński triangle contained in $[0,x]$ If ...
• 159
291 views

### Does the Mandelbrot set get smoother because of a rounding error

On many fractal explorer programs, after a certain level of zoom, the shapes begin to become smoother, as is shown. Is this a rounding error or an actual representation of the computation?
• 143
79 views

### what shape does this function $\delta_{x}(x)$ look like if it's a fractal what is it's fractal dimension?

what shape does this function $\delta_{x}(x)$ look like if it's a fractal what is its fractal dimension? so $\alpha$ is a real number and $\delta_{\alpha}(x)$ is a function based of $\alpha$ written ...
51 views

### Properties of the Cantor set constructed by removing just the middle point

Context: I am currently doing some exercises on the middle $\lambda$ Cantor set $C_\lambda$ (construction is similar to the usual Cantor set, but we remove the middle $\lambda$ proportion of the ...
• 1,185
159 views

• 1,185
1 vote
70 views

### Why the use of the term "snowflaking"?

I've seen a few places in the literature (in particular in fractal geometry) where we consider a metric space $(X,d)$, and then for $0 < \epsilon < 1$ define a new metric space $(X,d^{\epsilon})$...
• 4,614
105 views

### Hausdorff dimension for bounded set

Let $F$ be the set of of numbers $x\in [0,1]$ with base 3 expansions $0.a_1a_2...$ for which there exists an integer k such that $a_i\neq 1$ for all $i\geq k$. Find the Hausdorff dimension of $F$. ...
103 views

### How to find the box counting dimension of line segment [0,1]?

I uploaded this question and no one has answered because there were some flaws in the question but I have the text now. Please can some one explain how we arrive on example of line segment to have a ...
• 41
37 views

### Box counting dimension of line segment [0,1]?

I am not understanding this example from Kenneth Falconers book on Fractal geometry. can someone please explain how the inequalities follows from the definitions in the example given?
• 41
1 vote
90 views

### Hausdorff dimension of the relative complement of a set contained in a $G_\delta$

We have the following result: Every set is contained in a $G_\delta$ set of the same Hausdorff dimension I was wondering how tight can this inclusion be made, complement-wise. Is true that: Let ...
352 views

• 8,078
270 views

### Can the fractal dimension of a surface be less than 2?

I have two surfaces represented as raster images with heights as grayscale values. One is natural landscape elevations; the other is just distance from a line. I have computed Minkowsky D = 2 - H ...
• 193
641 views

### About moments of inertia, integrals and fractal dimensions.

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 ...
• 566