The shape can be decomposed into
N = 4 pieces, each scaled by a factor of
Consequently,
ds = Log(4)/Log(2) = 2.
This calculation suggests a bad joke.
Though hardly a surprise after the interesting variation of example (d), this shows 2-dimensional shapes need not be smooth surfaces. This shape could be called the Sierpinski tetrahedron.
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Plane projections of complicated shapes in three dimensions can be difficult to parse.
To help with this, here are the three stages leading to the decomposition on the right.
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Finally, here are Gayla Chandler's site of Sierpinski tetrahedron models, and some photos of of a lab exercise.
Return to Similarity Dimension Exercises.