Beauty plays an important role in the domain of science and mathematics, and has often served as a "pointer" to new discoveries. Although we cannot fully explain why this approach has been so effective, we can say a great deal about how it works in practice. This book provides the technical background that will allow readers to explore this topic, and gain a better understanding of how aesthetic criteria are used in science. It will also help them see how our sense of beauty relates to natural forms and processes, and contributes to our ability to analyze them. These questions are examined from the perspective of information theory, fractal geometry, complexity theory and nonlinear dynamics, each of which provides us with a different way to investigate the connection between our mathematical models and the physical reality that they represent. Specific topics that are discussed in this context include information entropy, quantum computing, chaos theory, catastrophes, cellular automata and random Boolean networks. The book also contains a separate section on infinite sets, whose properties defy our intuition and stretch the human imagination far beyond anything that we can experience. Our capacity to engage in such speculations is a unique gift, and can often be the source of profound aesthetic experiences (as Kant rightly recognized many years ago).