Geometries and Transformations

Euclidean and other geometries are distinguished by the transformations that preserve their essential properties. Using linear algebra and transformation groups, this book provides a readable exposition of how these classical geometries are both differentiated and connected. Following Cayley and Klein, the book builds on projective and inversive geometry to construct 'linear' and 'circular' geometries, including classical real metric spaces like Euclidean, hyperbolic, elliptic, and spherical, as well as their unitary counterparts. The first part of the book deals with the foundations and general properties of the various kinds of geometries. The latter part studies discrete-geometric structures and their symmetries in various spaces. Written for graduate students, the book includes numerous exercises and covers both classical results and new research in the field. An understanding of analytic geometry, linear algebra, and elementary group theory is assumed.

Introduction; 1. Homogenous spaces; 2. Linear geometries; 3. Circular geometries; 4. Real collineation groups; 5. Equiareal collineations; 6. Real isometry groups; 7. Complex spaces; 8. Complex collineation groups; 9. Circularities and concatenations; 10. Unitary isometry groups; 11. Finite symmetry groups; 12. Euclidean symmetry groups; 13. Hyperbolic coxeter groups; 14. Modular transformations; 15. Quaternionic modular groups.

Norman W. Johnson was Professor Emeritus of Mathematics at Wheaton College, Massachusetts. Johnson authored and co-authored numerous journal articles on geometry and algebra, and his 1966 paper 'Convex Polyhedra with Regular Faces' enumerated what have come to be called the Johnson solids. He was a frequent participant in international conferences and a member of the American Mathematical Society and the Mathematical Association of America.