Fractional Dimensions

A point has dimension 0, a line has dimension 1, and a plane has dimension 2. But did you know that some objects can be regarded to have “fractional” dimension?

You can think of dimension of an object X as the amount of information necessary to specify the position of a point in X. For instance, a block of wood is 3-dimensional because you need three coordinates to specify any point inside.

The  has fractional dimension! Why? Well it is at most 1-dimensional, because one coordinate would certainly specify where a point is. However, you can get away with “less”, because the object is self-similar. At each stage, you only need to specify which 2 out of 3 segments a point is in. Mathematicians have developed a notion of “dimension” which for the standard Cantor set works out to be: 

ln(2)/ln(3) = 0.6309… 

Most “fractals” have fractional dimension; for instance a curve whose boundary is very, very intricate can be expected to have dimension between 1 and 2 but closer to 2. This concept has been applied in other sciences to describe structures that appear to have some self-similarity, such as the coast of England or gaseous nebulae in interstellar space.

Presentation Suggestions:
It makes sense to do this Fun Fact after doing the one on the standard Cantor set.

The Math Behind the Fact:
Actually, the notion of “dimension” can be extended to crazy sets in many different ways. One notion isbox dimension, and another is Hausdorff dimension. These notions agree for the standard Cantor set and many other sets.

How to Cite this Page: 
Su, Francis E., et al. “Fractional Dimensions.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
K. Falconer, Fractal .

Fun Fact suggested by:
Francis Su

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