Thomas Q. Sibley
American Mathematical Society
Thinking Geometrically: A Survey of Geometries (PDF) is a very well written and comprehensive survey of college geometry that would serve a wide variety of courses for both mathematics education majors and mathematics majors. Great attention and care is spent on developing visual insights and geometric intuition while stressing the historical development; logical structure; and deep interconnectedness of the ideas.
Students with less mathematical preparation than upper-division mathematics majors can successfully study the topics needed for the preparation of high school teachers. There is a multitude of projects and exercises in those chapters developing all aspects of geometric thinking for these schooll students as well as for more advanced students. These chapters include Axiomatic Systems and Models; Euclidean Geometry; Transformational Geometry; Analytic Geometry; and Symmetry. Topics in the other chapters; including Non-Euclidean Geometry; Projective Geometry; Differential Geometry; Finite Geometry; and Discrete Geometry; provide a broader view of geometry. The different chapters are as independent as possible; while the textbook still manages to highlight the many connections between topics.
The textbook is self-contained; including appendices with the material in Euclid’s 1st book and a high school axiomatic system as well as Hilbert’s axioms. Appendices give brief summaries of the parts of multivariable calculus and linear algebra needed for certain chapters. While some chapters use the language of groups; no prior experience with abstract algebra is presumed. The ebook Thinking Geometrically: A Survey of Geometries will support an approach emphasizing dynamical geometry software without being tied to any particular software.