In the previous cases it is easy to find the dimension by simply reading the exponent. However it's not always so easy. Consider the Sierpinski Triangle - an example of a fractal.
Let's look at how it is generated: Begin with a triangle.
Draw the lines connecting the midpoints of the sides and cut out the center triangle.
Note that we have in our new triangle 3 "miniature" triangles. Each side = 1/2 the length of a side of the original triangle. Each "miniature" triangle looks exactly like the original triangle when magnified by a factor of 2 (magnification or scaling factor).
Take the result and repeat (iterate).
And again ...
Iterate this forever
Notice that the lower left portion of the triangle is exactly the same as the entire triangle when magnified by a factor of two. It is self-similar.
What is D?
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and 2018 by Mary