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  1. Suppose we have two given fractals $K_1$, $K_2$ of dimension $d_1$,$d_2$ respectively. What can we say about the dimension of $K_1 \cap K_2$, and $K_1 \cup K_2$?

  2. Is there any technique to describe a fractal with given dimension $d$?

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What is your definition of the term "fractal"? – KCd Dec 12 '11 at 1:41
up vote 4 down vote accepted

For 2 using Hausdorff dimension, if you review how the Koch snowflake dimension is shown to be $\frac{\log 4}{\log 3}$ you can see how to make a fractal with any dimension of the form $1<\frac{\log m}{\log n}<2$-just divide the side into $n$ pieces and replace each with $m$ segments each time. Similarly the Cantor set has dimension $\frac{\log 2}{\log 3}$ and shows how to get other ratios

For 1, the same Wikipedia article says the union has the higher dimension. The intersection can clearly have dimension $0$ if they are disjoint and no higher than the lower.

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