We can show that this is self-similar by taking [ fig. 2 ] of the Koch curve and duplicating it to create another identical looking curve, [ fig. 3 ] Here you can see that if we take four copies of figure 2, and connect them at the same internal angle as the equilateral triangle, we create the same image as figure 2 at a larger scale. This shows its self-similarity, that one part reflects the whole pattern.
fig. 3: Demonstrating Self-Similarity
Characteristics of Fractals: Scaling and Dimension
The idea of Fractal Scaling is very similar to the idea of self-similarity. The scaling of a fractal refers to the extent to which you have to zoom or enlarge the observation scale, like a map, for a section of it to resemble the whole pattern. In the Koch Curve and in Sierpinski’s Triangle we have a one to one scale, where you would enlarge a section of the pattern by the same factor in all directions to reflect to the original pattern. They are like high quality zoom lenses that expand everything in the frame by the same degree in all directions. However, there are fractals that scale in one way more than another. For so-called self-affine fractals every horizontal enlargement has to be followed by a different number of vertical ones, depending on the fractal in question. They are more like a photocopier; shrinking an image more horizontally than vertically. The most complex type of fractal is called a Multi-Fractal; which scales in many different ways, at different points of the fractal. I will discuss this types of fractal when discussing the M-set One of the most important notions in fractal geometry is the idea of the fractal dimension. The fractal dimension can be described as a measure of how well a fractal fills the space in which it resides. For example, the Koch curve fills space better than a line, but worse than a square, so its fractal dimension lies between one and two. While there are many different definitions of a fractal dimension, each suited to varying situations; I will focus on two of them. For self-similar fractals, like the Koch curve, we can use the simplest “similarity dimension” , , whose formula is:
log() log()
=
Where r is the length of the previous iteration’s unit length as a multiple of the new unit length, see [ fig. 4 ]; and N is the total number of units per iteration. Therefore, for the Koch curve = 3 = 3 and N = 4. Thus for our Koch curve its similarity dimension is log(4)/log(3) = 1.2618…..
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