How can fractals have fractional dimensionality?

Question

I grasp how fractals can be self-similar and have other weird properties. But I don’t quite get how they can have fractional dimensionality, even though that’s the property they’re named after.

How can a shape have a dimensionality *between*, say, two and three?

In: Mathematics

Answer 1

I had a professor crumple a paper ball and ask if this is a 2-dimensional object. Yes, but it occupies 3-D space. If you lightly crumple it and spread it out, that wrinkled plane is slightly more than 2. If you wad it into the tightest ball you can, it’s almost 3. Similarly for lines that fill a plane. Etc

Edit: spelling