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

Take a line. Double all its lengths. Now it takes 2 times as much space, or 2^(1).  

 Take a square. Double all its lengths. Now it takes up 4 times as much space, or 2^(2).  

 Take a cube. Double all its lengths. Now it takes up 8 times as much space, or 2^(3).  

Those are examples of 1, 2, and 3 dimensional shapes.  

Now take Sierpinsky’s triangle (google it if you need). Double all its lengths. There are 3 of the original triangle inside this new doubled triangle. It takes up 3 times as much space. 2^(log2[3]). So this fractal has dimension log2[3]. 

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Here’s a good video on the topic: