A dust that almost every line misses
Keep the four corner squares of a square, then the four corners of each of those, and so on. What is left has a length, in the sense that measures length, and yet almost every straight line misses it entirely. Stage by stage the chance that a random line hits the dust falls — to 48% by the eighth stage — and how fast it falls to nothing is one of the few questions about a simple picture that is still open.
