"Worlds Out of Nothing" is a comprehensive academic book based on the latest historical research, offering a unique insight into the history of geometry in the 19th century. It covers central themes such as projective geometry and non-Euclidean geometry, with a focus on the concept of duality. The book analyzes the singularities of algebraic curves and their significance in resolving paradoxes within duality theory. It also discusses Riemann's contribution to differential geometry and Beltrami's role in establishing non-Euclidean geometry as a rigorous mathematical discipline. In the concluding section, the importance of projective geometry and Poincaré's ideas on non-Euclidean geometry are explored from both physical and philosophical perspectives. Additionally, the book provides valuable chapters on engaging with the history of mathematics in writing, including examples and advice for essay writing.