Exploring geometry (Q2832141)

This is an undergraduate textbook of geometry, containing something of everything that one would expect in such a textbook. It encourages students to explore the truth of a conjectured state of affairs by means of the software \textit{Geometry Explorer}, that can be downloaded from \url{http://homepages.gac.edu/~hvidsten/gex/}, and which offers the possibility of testing hypotheses in plane Euclidean, hyperbolic, and elliptic geometry.NEWLINENEWLINEAfter a very thorough chapter on the nature and the origin of the axiomatic method, in which the author tries to make this approach palatable by presenting a fair number of axiom systems for non-mathematical situations, with a real world flavor, the next five chapters are devoted to aspects of Euclidean geometry, which is considered to be unknown to the reader (a realistic assumption in the present-day educational environment of the United States of America). Thus, the second chapter presents the essentials of plane Euclidean geometry, including inversion, from scratch, the third one is on analytic geometry (including Bézier curves), the fourth one on geometric constructions, including the impossibility of some of the classic construction problems, the fifth one on geometric transformations, and the sixth one on symmetry groups, including frieze and wallpaper groups, as well as tilings of the plane. Chapters 7 and 8 are on plane hyperbolic and plane elliptic geometry, whereas chapter 9 is an rather in-depth look at projective geometry, including conics. Chapter 10, the final one, is on fractal geometry. There are appendices on the nature of proofs, Book I of Euclid's \textit{Elements}, Birkhoff's and Hilbert's axiom systems, and wallpaper groups.NEWLINENEWLINEThere is a generous number of exercises in each section, of varying degrees of difficulty (the proof of Pappus implies Desargues, for example, is relegated to three exercises).