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 standard cantor set 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 other “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 is*box 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 geometry*.

**Fun Fact suggested by: **

Francis Su