Modern Geometries: Non-Euclidean, Projective, and Discrete Geometry (2nd Edition)

Engaging, accessible, and extensively illustrated, this brief, but solid introduction to modern geometry describes geometry as it is understood and used by contemporary mathematicians and theoretical scientists. Basically non-Euclidean in approach, it relates geometry to familiar ideas from analytic geometry, staying firmly in the Cartesian plane. It uses the principle geometric concept of congruence or geometric transformation--introducing and using the Erlanger Program explicitly throughout. It features significant modern applications of geometry--e.g., the geometry of relativity, symmetry, art and crystallography, finite geometry and computation. Covers a full range of topics from plane geometry, projective geometry, solid geometry, discrete geometry, and axiom systems. For anyone interested in an introduction to geometry used by contemporary mathematicians and theoretical scientists.

The book also is too much of an overview--it makes a good introduction but a poor reference text. It is also very poorly indexed, which can make it hard to find things. The exercises are also poor--many new concepts are introduced in the exercises at the end of the chapters.

The writing is actually pretty good, for the most part. I think that the stuff that is explained in the book is explained well in most places, and the author does a very good job of tieing things together and bringing in historical background and significance of the topics being discussed.

I lastly might add that the name is very misleading--the geometries described in this book were mostly discovered over 100 years ago--there is nothing drastically "modern" about them.

Overall, this book was not prepared for being published--it needs a new edition to correct errors and tie up loose ends.