# Finding base number with Hausdorff Dimension Approximation methods?

## Main Question or Discussion Point

According to the link below, fractal dimension is an exponent of some sort:
http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html

The Hausdorff Dimension (aka fractal dimension) is denoted as D in the website above. And r is the base number.

If we were to look at any image and use Hausdorff Dimension approximation methods such as the box counting method (http://classes.yale.edu/fractals/fracanddim/boxdim/BoxDim.html) for approximating the Hausdorff Dimension which is D in N=r^D. The link describes how to find D using the box-counting method, but it doesn't explain how to derive at r. Is there a way in how we get r using the box counting method or any other Hausdorff Dimension approximation methods?

The reason I ask is because in the case of the Koch Snowflake, we know the initiator and generator (refer to first link if you're not familiar with these two terms) because it is something created by man; in other words, we already know its D and r because these values are chosen by man (aka man-made). However, if we were to take a picture of a real tree in my backyard for example (trees in general have a wonderful fractal dimensional branching pattern), we can use the box counting method to approximate at D without knowing r. So I wanted to know if we can derive at r using the box counting method or any other Hausdorff Dimension approximation methods.

you don't 'derive' r- it has to be given or chosen. The idea is simply that area has units of "length squared" so the area of a two dimensional figure is proportional to $r^2$ where r is a 'typical' length. Volume of a three dimensional figure is proportional to $r^3$- again r can be any 'typical' length in the figure. Changing to a different length would just give a different proportion.
A figure has hausdorff dimension 'd' if and only if its measure can b written as proportiona to $r^d$ where r can be any length in the figure. If you change to a different length you will have a different 'coefficient of proportionality' but the measure will still be proportional to that length to the d power.