This book aims at introducing the reader with some high school mathematics, to both the higher and the more fundamental developments of the basic themes of elementary mathematics. Most chapters begin with a series of elementary problems, behind whose diverting formulation more advanced mathematical ideas lie hidden. These are then made explicit and further developemnts of them explored, thereby deepending and broadening the reader's understanding of mathematics — enabling him or her to see mathematics as a hologram. The book arose from a course for potential high school teachers of mathematics taught for several years at St. Petersburg University, and nearly every chapter ends with an interesting commentary on the relevance of its subject matter to the actual classroom setting. However, it can be recommended to a much wider readership; even the professional mathematician will derive much pleasureable instruction from reading it.