## Book Review: Experiencing Geometry

Experiencing Geometry In Euclidian, Spherical, And Hyperbolic Spaces, by David W. Henderson

The increasing popularity of flat earth thinking in certain circles has led me to ponder about the deficiency of my own education in geometry in spherical spaces, and seeking a better way to understand this sort of issue and how it is that a failure to know spherical geometry can lead to negative consequences in how one sees the world, I thought it would be worthwhile to bone up on some geometry, since the last time I did coursework on it was in high school and that was focused on plane geometry.  This book is written for students in Cornell, and the author clearly is dealing with an intended audience that is both intensely practical when it comes to modeling more complicated geometries but also familiar with the basics of geometry.  This is the sort of book that I can well recommend to someone who wants to be more familiar with advanced geometry, at least advanced undergraduate-level geometry, so as to be able to develop a better informed intuition on geometry based on reading and modeling.  Not everyone is going to find a book like this interesting, of course, but at the same time those who do will get a lot out of it.

This book is more than 300 pages long and has twenty-two chapters.  The book begins with a preface and some suggestions on how to read the book depending on the interests of the reader as well as a discussion on the nature of proofs.  After that the author discusses the meaning of straight (1), before looking at what it means in spherical geometry (2), as well as cylinders and cones (4), and hyperbolic planes (5).  The author also examines the question of what an angle is (3), discussing after that triangles and congruencies (6), and problems on SSS, ASS, SAA, and AAA (9).  The author also discusses area and holonomy (7), parallel transport (8), and some postulates that relate to parallel lines (10).  He shows himself interested in isometries and patterns (11), dissection theory (12), and the geometry of square roots, Pythagoras, and similar triangles (13).  There are chapters on circles in the plane (14), projections of a sphere onto a plane, useful for maps (15), projections of hyperbolic planes (16), geometric 2-manifolds and coverings (17), geometric solutions of quadratic and cubic equations (18), trigonometry and duality (19), 3 spheres and hyperbolic 3-spaces (20), polyhedra (21), 3-manifolds (22), as well as appendices on Euclid’s definitions, postulates, and common notions (i) and square roots in the Sulbasutram (ii), an annotated bibliography, and an index.

There are many insights that this book provides, but space and time constraints prevent me from mentioning more than a few of them, so here goes.  The author’s focus is on informing readers (and his students) by developing an intuitive grasp of geometry and its importance in algebra and trigonometry, for example, rather than focusing on the formalist approach that is most common in schools.  Among the useful tips that the author encourages is recognizing that when it comes to spherical geometry we are surface dwellers whose intrinsic experience of being on the surface of an oblong spheroid (which we call earth) is different than the extrinsic knowledge we have about spheres from a formal perspective.  The author also suggests better understanding hyperbolic space through crocheting a surface and then working with it to model the space and see its quirks, which is an inventive solution for mathematical matters.  In reading this book I was struck by the melancholy thought that an approach to algebra and trigonometry that utilized geometry and that defended the essential value of standard measurements (as opposed to metric ones) for their geometric value would have led my father to have been far better at math because of his inability to deal with the abstractions of higher math.  Alas, math is full of melancholy thinking in such ways.