Exercises for Chaos Under Control

Chapter 3: A Dimension Primer

2. Can a fractal in the plane ever have a dimension greater than 2? Explain your answer.

3. Imagine a version of the Cantor MTS in three dimensions: start with a 1 x 1 x 1 cube and remove middle third slices so only eight little cubes (1/3 x 1/3 x 1/3) remain, one at each corner of the original cube. The first few stages are animated here. Try sketching the first stage in the construction. Repeat this process for each of these little cubes, and continue. What is the dimension of the resulting Cantor MTS in space? Answer

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