The two-dimensional Sierpinski gasket can be created as follows:
In three dimensions the process is essentially the same except that you start with a tetrahedron. In both cases, the result is a successive subdivision of the original shape.
Letting n denote the the "subdivision factor" and k denote the number of similar objects created by the subdivision process, he fractal dimension is usually defined as:
d = (ln k)/(ln n)
Intuitively, the fractal dimension of an object is how many similar objects we create when we subdivide it.
In the case of the two-dimensional Sierpinski gasket, each time we subdivide a side by a factor of 2 we keep 3 of the 4 triangles created. Thus, d = (ln 3)/(ln 2) = 1.58496. In the case of the three-dimensional Sierpinski gasket, each time we subdivide a side by a factor of 2 we keep 4 of the tetrahedra. Thus, d = (ln 4)/(ln 2) = 2.
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