Fourth Homework Set Answers

2. (a) From the similarity dimension we see this fractal is made of N = 5 pieces, each scaled by a factgor of r = 1/3. Five squares of side length 1/3, placed inside the unit square and arranged so they overlap at most along edges, suffice. Within these restrictions, the placement of the squares does not affect the dimension. The left image below shows one such placement.

(b) This fractal is made of N = 5 pieces, each scaled by a factor of r = 1/2. Note this dimension is larger than 2, so the fractal cannot be a subset of the plane. Five cubes of side length 1/2, placed inside the unit cube and arranged so they overlap at most along faces, suffice. Within these restrictions, the placement of the cubes does not affect the dimension. The right image below shows one such placement.

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