The approach to geometry is that of the Erlangen Program. This approach draws on content and uses modes of thought that appear in advanced courses in the undergraduate mathematics curriculum. 

The use of constructions to prove theorems and motivate definitions gives a tangible and intuitive approach to the subject.  The constructive approach adopted in the text enables the students to connect more deeply with the material. 

Includes 164 illustrations to aid visualization, and several applets for manipulating figures in non-Euclidean geometry.

The exercises come in many varieties to help students practice different skills: intense computations (e.g., computing hyperbolic distances, the hyperbolic law of cosines), constructions (e.g., symmetry points, hyperbolic circles), proofs both big (e.g., proof of arc length invariance) and small (that a collection of transformations forms a group), open-ended exercises (do you think the pair separation histogram came from a toroidal universe or a Klein bottle universe?), applications of geometry and topology to cosmology (Sections 7.4, 7.6, 8.2), and investigative.

Chapter 8 includes a general discussion of two current research programs in cosmic topology: cosmic crystallography and circles-in-the-sky. Chapter 8 also includes a review of our understanding of the current state of the universe, a view that is based on significant observational discoveries over the last 10 years.