Non-Euclidean Geometries: A Modern Approach
Description
Non-Euclidean Geometries: A Modern Approach offers a comprehensive introduction to the fascinating world beyond Euclid's fifth postulate. This masterful text by Dr. Anna Nowak, a leading Polish mathematician, presents hyperbolic and elliptic geometries through both classical and contemporary perspectives.
The book bridges the gap between abstract theory and practical understanding, making non-Euclidean geometries accessible to readers with a basic background in mathematics. Dr. Nowak's approach emphasizes intuitive understanding through carefully developed visualizations and models, complemented by rigorous mathematical formalism.
Key Features:
- Historical context that traces the development of non-Euclidean geometries from antiquity to modern times
- Clear explanations of the fundamental differences between Euclidean, hyperbolic, and elliptic geometries
- Innovative visualizations and models to build geometric intuition
- Applications to modern physics, cosmology, and computer graphics
- Numerous exercises ranging from straightforward applications to challenging explorations
- Digital supplements including interactive models and visualizations
This English translation of Dr. Nowak's acclaimed work brings her unique pedagogical approach to an international audience. The book is ideal for undergraduate mathematics students, physics students, teachers, and anyone fascinated by the structure of geometric space.
Table of Contents
- Foundations and Historical Development
- Euclid's Elements and the Parallel Postulate
- Early Attempts to Prove the Fifth Postulate
- The Revolutionary Work of Bolyai, Lobachevsky, and Gauss
- Riemann's Contribution
- The Structure of Geometric Space
- Axiomatic Systems
- Consistency and Independence
- Models of Non-Euclidean Geometries
- Hyperbolic Geometry
- The Poincaré Disk Model
- The Klein Model
- The Upper Half-Plane Model
- Triangles in Hyperbolic Space
- Area and Defect
- Elliptic Geometry
- The Projective Plane
- Spherical Geometry
- Triangles in Elliptic Space
- Great Circles and Geodesics
- Differential Geometry Perspective
- Curvature and Metrics
- Geodesic Equations
- The Gauss-Bonnet Theorem
- Modern Applications
- Einstein's General Relativity
- Cosmological Models
- Computer Graphics and Hyperbolic Tessellations
- Network Visualization
- Appendices
- Historical Timeline
- Construction of Models
- Solutions to Selected Exercises
About the Author
Dr. Anna Nowak is a distinguished mathematician specializing in non-Euclidean geometries and their applications. Born in Wrocław, Poland in 1968, she completed her doctoral studies at the University of Warsaw, where her dissertation on "Visualization Methods in Hyperbolic Geometry" received exceptional recognition.
Currently serving as a Professor of Mathematics at Jagiellonian University in Kraków, Dr. Nowak has dedicated her career to making complex geometric concepts accessible through innovative teaching methods. Her research spans theoretical aspects of non-Euclidean geometries and practical applications in data visualization and network analysis.
Dr. Nowak has received numerous awards for her contributions to mathematics education, including the Polish Ministry of Education's Excellence in Teaching Award (2015) and the European Mathematical Society's Education Prize (2018). She has published over 40 research papers and three textbooks that are widely used throughout Eastern Europe.
As a visiting professor at universities in Germany, France, and the United States, Dr. Nowak has brought her unique perspective on geometric education to an international audience. This book represents the culmination of her twenty years of experience in teaching non-Euclidean geometries to students at various levels.
