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S: 1,0.3,0.22,0.12,0.06 N: 7,201,478,2595,17950 (no idea how to put this in a tally)

I've got a question here where S is not shrunk by the same fraction throughout, I know how to work out the dimension Fractal when the Shrink is continuous E.G 1/3 each time but not when it's shrunk different each time. Can anyone help?

thanks

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  • $\begingroup$ It would help if you give some context to your question. I have no idea what you're talking about. For example, the word "Shrink" is used as a noun and not defined, the expression "E.G 1/3" seems code for something, "S" is used as a noun and not defined, "S:1,0.3,0.22,0.12,0.16" seems code for something, etc. $\endgroup$ – Dave L. Renfro Feb 7 '14 at 15:38
  • $\begingroup$ where N(s) is the number of copies of the cauliflower of linear size s, which make up the original. $\endgroup$ – Unknownstarz Feb 7 '14 at 15:40
  • $\begingroup$ Ah, cauliflowers. $\endgroup$ – TonyK Feb 9 '14 at 19:43
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In the definition of box-counting dimension, the dimension $d$ is given by $$d = \lim_{\varepsilon \rightarrow 0} \frac{\log N(\varepsilon)}{\log 1/\varepsilon},$$ where $N(\varepsilon)$ is the smallest number of sets of diameter $\varepsilon$ required to cover the set. In numerical applications, $N(\varepsilon)$ is often estimated using any of a number of equivalent formulations. It sounds like your assignment presents the following estimates:

\begin{align} N(1) &= 7 \\ N(0.3) &= 201 \\ N(0.22) &= 478 \\ N(0.12) &= 2595 \\ N(0.06) &= 17950 \end{align}

It might be tempting to simply use the smallest given value of $\varepsilon$ to conclude that your dimension is best approximated by $$\frac{\log(17950)}{\log(1/0.06)} \approx 3.48.$$ One problem with this is that we would typically like to take this sparse data, given over a broad range of $\varepsilon$ values and extrapolate a limit. In this example, if we compute $\log(N(\varepsilon))/\log(1/\varepsilon))$ for the other given values of $\varepsilon$, we see that the computations are decreasing sharply. Thus, the usual approach is to plot $\log(N(\varepsilon))$ versus $\log(1/\varepsilon)$ like so:

enter image description here

Thus, the $x$ coordinates are \begin{align} \log(1) &= 0 \\ \log(1/0.3) &\approx 1.2 \\ \log(1/0.22) &\approx 1.5 \\ \log(1/0.12) &\approx 2.1 \\ \log(1/0.06) &\approx 2.8 \end{align} The $y$ coordinates are $\log(N(1)), \log(N(0.3)),\ldots,\log(N(0.06))$. The line is a least squares fit to this data. The ridiculously good fit suggests that the problem has been manufactured. Since we're interested in how $\log(N(\varepsilon))$ changes in response to a change of $\log(1/\varepsilon)$, the slope of that line represents the fractal dimension. Any number of techniques can be used to compute the least squares fit to be $y\approx 1.945 + 2.79 x$.

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