By John Stillwell. I picked this up from the new arrivals counter at the library. The hyperbolic tessellation on the front cover was invitation enough for me.
The book is about teaching geometry to undergrads from four different perspectives: 1) axiomatic a la Euclid, 2) linear algebra, 3) Projective and 4) transformations. It’s quite interesting and comes with lots of illustrations, although personally, I would prefer a book that delves more deeply into each aspect.
One main focus of the book is the cross-ratio which was discussed at length in the second half of the book. There is this wonderful quote.
At this point I can hear someone asking, “What is the geometric significance of the cross-ratio?” Although I first encountered cross-ratios as a senior in high school, and have dealt with them many times since then, I must say frankly that I cannot visualize a cross-ratio geometrically. If you like, it is magic. Here is this algebraic quantity whose significance is impossible to understand, and yet it turns out to do something very useful. It works. You might say it was a triumph of algebra to invent this quantity that turns out to be so valuable and could not be imagined geometrically. Or if you are a geometer at heart, you may say it is an invention of the devil and hate it all your life.
– Robin Hartshorne, Geometry: Euclid and Beyond p 341.