If I understand correctly, the initial appeal goes back to the early days of measure theory. Say, a 2-dimensional set is cut parallel to y-axis, and the cuts a

If I understand correctly, the initial appeal goes back to the early days of measure theory. Say, a 2-dimensional set is cut parallel to y-axis, and the cuts all have length 1. If the x-coordinates of the cuts themselves have length 1, you know that the 2-dimensional set must be "large". This is because you can integrate the cut length, and tell that it has area 1. (Think of a 1x1 square for an illustration). In Kakeya's problem, the cuts still have length 1, but are no longer parallel to any one axis. Besicovitch's construction shows, Kakeya set can have very small area, yet contain cuts of length 1 in many directions. This situation is quite different. This counterexample turned out to be useful in other areas of pure math, some discussed in the wiki entry for Kakeya problem.