Starting with numerical algorithms resulting in new kinds of amazing fractal patterns on the sphere, this book describes the theory underlying these phenomena and indicates possible future applications. The book also explores the following questions: What are fractals? How do fractal patterns emerge from quantum observations and relativistic light aberration effects? What are the open problems with iterated function systems based on Mobius transformations? Can quantum fractals be experimentally detected? What are quantum jumps? Is quantum theory complete and/or universal? Is the standard interpretation of Heisenberg's uncertainty relations accurate? What is Event Enhanced Quantum Theory and how does it differ from spontaneous localization theories? What are the possible applications of quantum fractals?