This book explores the history of mathematics from the perspective of the creative tension between common sense and the "impossible" as the author follows the discovery or invention of new concepts that have marked mathematical progress: irrational and imaginary numbers, the fourth dimension, curved space, infinity, and others. The authors puts these creations in to a broader context involving related "impossibilities" from art, literature, philosophy, and physics. By embedding mathematics into a broader cultural context and through his clever and enthusiastic explication of mathematical ideas the author broadens the horizon of students beyond the narrow confines of rote memorization and engages those who are curious about the place of mathematics in our intellectual landscape.