Geometry by Felix Klein

Part Three: Systematic Discussion of Geometry and its Foundations

There are many ways to approach this book, but I have enjoyed starting from Part Three and jumping back. A play with necessary flashbacks for context.

What I find most enjoyable about Klein is his development of geometry primitives through the lens of transformations. In particular, he builds up intuition for four primary “special linear substitutions of x, y, and z” - parallel displacements, rotations about the origin O, reflections in O, and similarity transformations with O as center. And yet, he reminds us that “geometry is concerned only with those relations between the coordinates which remain unchanged by the linear substitutions… geometry is the invariant theory of those linear substitutions.” The properties of geometry that define the relations that remain unchanged get bundled up in an umbrella subset of geometry Klein calls metric geometry. This term is used to against other geometries, such as affine geometry and projective geometry, but from the onset, it’s not clear why he bothers to make this distinction. It seems the main purpose is to separate affine transformations from projective transformations.

But the list doesn’t stop there, he goes onto introduce the “geometry of reciprocoal radii”, where lines and planes don’t have independent meanings, but rather are seen as special cases in circles and spheres, respectively. Once again, this blurring of the lines of those concepts is not set up, so we’ll have to see why we need these tools.

Part 2: Geometric Transformations