Bridging Worlds: Non-Euclidean Geometry across Disciplines

How did one little sentence in one of the most published books ever, other than the Bible, give rise to thousands of years of dedicated research, a completely new field of study, and even shape our understanding of the universe? This paper traces how questioning Euclid's 5th postulate led Gauss, Lobachevsky and Bolyai to non-Euclidean geometry, and explores its applications across disciplines: hyperbolic and mixed-curvature spaces for graph embeddings and data visualization in computer science, geometry-based methods in modern cryptography, and the role of curved space-time in General Relativity and in measuring the shape of the universe. About this publication: This paper is original research by a young researcher, published through the International Scientific Youth Community (ISYC), a youth-led, non-profit scientific community of students and young researchers from around the world. ISYC gives young people the opportunity to carry out independent research and publish it under their own names. Each paper is reviewed and edited by ISYC for integrity, sourcing and formatting before publication. The author retains full copyright of this work.