A line or line segment has Hausdorff dimension 1, while a point has Hausdorff dimension 0. In general, ordinary n-dimensional space has Hausdorff dimension n. That's why I referred to the Hausdorff dimension of the Cantor set -- equal to log(2)/log(3), or approximately 0.63 -- as "intermediate".

1 for a line, 2 for a square, 3 for a cube. Hausdorff dimension agrees with the usual kind for normal shapes.