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(I do not speak English. Please, excuse my bad grammar.)

This is about fractals over the real line, composed of points or segments inside the real line.

If the integers set is a fractal with dimension D=0, and the real line is a fractal with dimension D=1, then is possible to continuously vary the dimension D from the integer set to the real line set? (making some continuous figure in the plane)

There is a drawing of all the fractals side by side? (something like this image)

I do not mean that all the fractals make one larger fractal, but just that for each value D there is one fractal over the real line with that dimension, and all the fractals "touch each neighbor" continuously.

In other words, I want to continuously transform a fractal, from the integer set to the real line, by continuously changing his dimension D from 0 to 1.

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This can be done by generalizing the Cantor set. In particular, fix some $\alpha$ to be the dimension and let $\beta=2^{-1/\alpha}$. Define two functions $f_1(x)=\beta x$ and $f_2(x)=1-\beta+\beta x$, which are contractions towards $0$ and $1$ respectively. One may prove that there is a unique compact set $S_{\alpha}$ such that $S=f_1(S_{\alpha})\cup f_2(S_{\alpha})$. We have that $S_{1/ \log_2(3)}$ is the Cantor set and $S_1=[0,1]$ and in the limit as $\alpha$ goes to $0$, we get the set $\{0,1\}$. Additionally, it is fairly clear that $S_{\alpha}$ has dimension $\alpha$, since, being composed of two copies of it scaled by $\beta$, its dimension satisfies $\beta^{d}=1/2$.

One may prove that $S_{\alpha}$ consists exactly of points $x$ of the form, where $s_i$ is a sequence in $\{0,1\}$: $$x=\sum_{i=0}^{\infty}s_i(1-\beta)\beta^i$$ which clearly varies continuously when $s_i$ is fixed. A plot of these fractals follows, where the cross-section of the graph at $x=d$ is a fractal of dimension $d$: enter image description here

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  • $\begingroup$ Of course, if you actually want to continuously move from $\mathbb Z$ to $\mathbb R$, you can just union together the translates of $S_{\alpha}$ by every integer. That's harder to plot. $\endgroup$ Dec 28, 2016 at 4:48
  • $\begingroup$ Thanks Milo. I didn't knew that the cantor set could be modified to have any dimension. $\endgroup$
    – Nuwo Pagog
    Dec 29, 2016 at 1:40
  • $\begingroup$ Thanks Milo, and thanks for making the graphic! You are right. I'm looking for a grasp of what are natural sets between $\mathbb{Z}$ and $\mathbb{R}$, and I'm still lost... :( I imagine that there are many ways to "interpolate" between $\mathbb{Z}$ and $\mathbb{R}$ with fractals. I don't even know where to look. $\endgroup$
    – Nuwo Pagog
    Dec 29, 2016 at 2:27

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