Erlangen program

Abstract: In the Ashes of the Ether: Differential Topology.- Looking for the Forest in the Leaves: Folations.- The Fundamental Theorem of Calculus.- Shapes Fantastic: Klein Geometries.- Shapes High Fantastical: Cartan Geometries.- Riemannian Geometry.- Mobius Geometry.- Projective Geometry.- Appendix A - E.

Abstract: PART A: Introduction Elementary Geometry and Trigonometry Solid Geometry and Spherical Trigonometry Curve Sketching Convexity Topology Graphs and Networks Tensors Catastrophe Theory Finite Spaces and Combinatorial Design Projective Geometry Symmetry PART B: Introduction Differential Geometry Vectors Analytic Manifolds and Lie Groups Enumeration Coding Theory Patterns Bibliography Index.

Abstract: Throughout most of this book, non-Euclidean geometries in spaces of two or three dimensions are treated as specializations of real projective geometry in terms of a simple set of axioms concerning points, lines, planes, incidence, order and continuity, with no mention of the measurement of distances or angles. This synthetic development is followed by the introduction of homogeneous coordinates, beginning with Von Staudt's idea of regarding points as entities that can be added or multiplied. Tranformations that preserve incidence are called collineations. They lead in a natural way to isometries or 'congruent transformations'. Following a recommendation by Bertrand Russell, continuity is described in terms of order. Elliptic and hyperbolic geometries are derived from real projective geometry by specializing an elliptic or hyperbolic polarity which transforms points into lines (in two dimensions) or planes (in three dimensions) and vice versa.