Real Analytic Plane Geometry

Abstract

Among the few choices of systems of axioms to construct a geometric model of the plane (for example, via Euclid or Hilbert) we take the least strenuous path; and, in making use of the real number system already in place, we develop real analytic plane geometry using Birkhoff’s axioms of metric geometry. One of the main purposes of this chapter is to explain what is classically known as the Cantor–Dedekind Axiom: The real number system is order-isomorphic to the linear continuum of geometry. Unlike the original approaches of Hilbert and Birkhoff, we are working here in a concrete model of the plane, built from the real number system of Chapter 2. Verifying that the Birkhoff postulates hold in our concrete model is much less demanding than the synthetic (purely axiomatic) approach. Nevertheless, our model oriented exposition still encounters some struggle, as in Sections 5.6–5.7, where the existence and properties of the circular arc length are shown using purely metric tools, and paving the way to trigonometry (Chapter 11). This also gives a precise answer to the question: “What is π?” Once again, this relies on the Least Upper Bound Property of the real number system, the main common thread with the first two chapters. A natural offspring of this technical passage is concluded with an optional section on the (often neglected) Principle of Shortest Distance, given here in full details. To ease up the complexity of the material, we make frequent side tours to develop metric properties of many geometric configurations. We determine all Pythagorean triples not by elementary number theory, but via analytic geometry: the method of rational slopes. We introduce here additional important tools that will play pivotal roles in the sequel: the Cauchy–Schwarz inequality, the AM–GM inequality, and their offsprings. Finally, still in this chapter, we present Archimedes’ duplication method to approximate π, once again with a view to algebraic formulas for many special angles given subsequently in trigonometry in Chapter 11.