TITLE: Is it possible to construct a smooth curve with fractional Hausdorff dimension?
QUESTION [5 upvotes]: It is known that fractal curves have fractional Hausdorff dimension. These curves are not smooth and have undefined length. However, is the converse true? 

If a curve has a fractional Hausdorff dimension, then must it be
  non-smooth/not differentiable?

REPLY [4 votes]: A rectifiable curve has $\sigma$-finite and nonzero $1$-dimensional Hausdorff measure (essentially, the length is what $1$-dimensional Hausdorff measure is).  Therefore it must have Hausdorff dimension $1$.  Thus a curve whose Hausdorff dimension is not $1$ can't be rectifiable, and certainly can't be smooth.