Geometry

In recent years, I have been teaching a junior-senior-level course on the classi­ cal geometries. This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid’s Elements. Students are expected to read concurrently Books I-IV of Euclid’s text, which must be obtained sepa­ rately. The remainder of the book is an exploration of questions that arise natu­ rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert’s axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And in the last chapter we provide what is missing from Euclid’s treatment of the five Platonic solids in Book XIII of the Elements. For a one-semester course such as I teach, Chapters 1 and 2 form the core material, which takes six to eight weeks.

Table of Contents

Chapter 1. Euclid’s Geometry

Chapter 2. Hilbert’s Axioms

Chapter 3. Geometry over Fields

Chapter 4. Segment Arithmetic

Chapter 5. Area

Chapter 6. Construction Problems and Field Extensions

Chapter 7. Non-Euclidean Geometry

Chapter 8. Polyhedra

Appendix: Brief Euclid

About the Author

Robin Hartshorne is an influential American mathematician known for his contributions to algebraic topology and differential topology. Born on April 4, 1934, he has made significant impacts in areas such as the study of manifolds and complex structures.