The Shape of Everything

Geometry is an ancient science, rooted in the thought of Greek philosophers. The book recounts how the ideas and results of geometry were born and how they developed. Beginning with Euclid's Elements and continuing to present days, when geometry becomes abstract and helps us either to understand our multidimensional universe or to organize the multitude of data used in virtual space. The book describes "the new light in the darkness" found by Descartes to describe curves through an equation, which gives rise to modern analytic or algebraic geometry. Then it considers the revolution of Leibniz, Newton and others who, with the invention of Calculus (differential and integral), solved problems arising in physics constructing new curves. The central section is dedicated to Frederic Gauss and Bernard Riemann, who, with incredible imagination and rigor, paved the way for contemporary geometry, a path that would lead to the study of non-Euclidean Geometries, General Relativity, the shape of biological structures (Protein Folding), and much more. The conclusion is dedicated to present various Contemporary Research Programs, which organize many aspects of geometry, according to a celebrated Manifesto by Felix Klein. These include The Program of Minimal Models, by S. Mori, and The Program for the Classification of Topological Manifolds, by W. Thurston and G. Perelmann. This book will be of interest not just for mathematicians but for anyone who has curiosity in the field.