Non-Euclidean Geometry

Non-Euclidean Geometry is a mathematics textbook that covers geometrical systems beyond traditional Euclidean geometry. The book presents both hyperbolic and elliptic geometries through rigorous mathematical treatments and proofs. Coxeter builds the concepts progressively, starting from basic axioms and definitions before moving to more complex theorems and applications. The text includes historical context about the development of these geometric systems and the mathematicians who contributed to the field. The work features illustrations, diagrams, and exercises that help readers grasp abstract geometric concepts. Technical discussions cover topics like parallel lines, triangles, circles, and transformations in non-Euclidean spaces. This book demonstrates how questioning fundamental assumptions can lead to new mathematical frameworks and ways of understanding space. The text serves as both a scholarly examination of geometric theory and a testament to mathematical innovation.

Readers report this book requires significant mathematical background, with many noting it's not suitable for beginners. Advanced math students and professors favor it for its rigorous treatment and comprehensive proofs. Liked: - Clear progression from basic concepts to complex theorems - Thorough historical context for each development - High quality diagrams and illustrations - Strong focus on foundations and first principles Disliked: - Dense notation that can be hard to follow - Assumes prior knowledge of advanced calculus - Limited worked examples - Some sections feel rushed or compressed Ratings: Goodreads: 4.0/5 (21 ratings) Amazon: 4.2/5 (8 ratings) One reviewer noted: "Not for the mathematically faint of heart, but rewarding for those willing to work through it carefully." Another stated: "The proofs are elegant but require close attention - I had to read some sections multiple times to grasp the concepts."

Euclidean and Non-Euclidean Geometries by Richard Silverman This text explores the development and comparison of various geometric systems through an axiomatic approach similar to Coxeter's methodology.

The Foundations of Geometry by David Hilbert Hilbert's formalization of geometric axioms provides the mathematical groundwork that underpins many concepts found in Coxeter's work.

Regular Polytopes by H.S.M. Coxeter This companion work delves deeper into the specific geometric structures and symmetries that extend from the principles discussed in Non-Euclidean Geometry.

Geometry: Euclid and Beyond by Robin Hartshorne The text builds from Euclidean foundations to modern geometric theories using the same rigorous mathematical approach found in Coxeter's work.

Introduction to Hyperbolic Spaces by Arlan Ramsay and Robert D. Richtmyer This book focuses on one of the key areas of non-Euclidean geometry through concrete models and analytical methods that complement Coxeter's treatment.

🔹 H.S.M. Coxeter was considered the greatest classical geometer of the 20th century, and his work influenced not just mathematics but also inspired artist M.C. Escher's famous tessellation artworks. 🔹 Non-Euclidean geometry, the book's subject, arose from mathematicians questioning Euclid's parallel postulate for over 2000 years, eventually leading to revolutionary new geometries that would later prove essential for Einstein's theory of relativity. 🔹 The book was first published in 1942 and has remained a standard text for over 80 years, helping generations of students understand concepts like hyperbolic and elliptic geometry. 🔹 Coxeter wrote this book while at the University of Toronto, where he taught for more than 60 years and continued publishing mathematical papers well into his 90s. 🔹 The principles discussed in this book have practical applications in modern GPS systems, which must account for the Earth's curved surface rather than treating it as a flat, Euclidean plane.