I would not have thought it possible to produce a 'popular' explanation of the Banach-Tarski Theorem, the most counter-intuitive result in mathematics. Yet Wapner succeeds with flying colours.
Just one caveat: although the book presumes some mathematical background (all school stuff: mappings, trig functions, radians..), and Wapner skillfully, and apparently effortlessly, guides the reader through the first 3 chapters (by which time the reader will have realised that he or she is on to something truly astonishing), chapter 4 does make some demands (already mentioned; plus the Axiom of Choice, which is explained); and chapter 5 would not be found at all easy by school pupils. But it's so enticingly written that it will have them re-reading in determination to grasp the argument.
The final 3 chapters are accessible and non-technical, but highly thought-provoking, and there's a very good bibliography.
I don't understand how I missed this book until now. If you like popular paradoxes, optical illusions, etc (which look pedestrian and silly by comparison) get hold of this book. It's the real thing!