1. Editor’s note: After I wrote this introduction, some of my early readers had questions about the Banach-Tarski paradox and the axiom of choice. I did a web search and found a number of sites that talk about the concepts. The one I liked best was an essay in the blog Good Math, Bad Math, written by Mark Chu-Carroll. If you’d like a look, see tinyurl.com/AxiomChoiceBlog.
2. For the video of the Mandelbrot fractal, see “Mandelbrot Zoom,” YouTube video, 1:18, posted on July 8, 2009, by “Kevin O’Toole,” www.youtube.com/watch?v=rajXu9E_Ry0 (accessed March 11, 2013).