Axioms of Geometry?

Question

I have taken the generic low level undergraduate classes, such as Calculus 1-3, Differential Equations, and Linear algebra. Since I never learned Geometry past a basic high school level, I thought it would be cool for me to start from the axioms of Euclidean Geometry and try to prove/discover some geometry on my own. I did a little googling and found Euclid's $5$ postulates. However, other sources were talking about the existence additional axioms/postulates, such as the transitive property of equality, the partition axiom, etc. My question is, where is a good place for me to start? Should I just start with Euclid's $5$ postulates and assume the common rules I know from algebra (commutative property, etc.)? Are there more of these smaller axioms/postulates that are not generally talked about in "normal" math classes (calculus, etc.) that I should know about?

Answer 1

You shouldn't take Euclid's axioms too seriously. It was the first known TRY to axiomatise geometry, but as it turned out Euclid in his proofs derived some properties from diagrams and tacitly assumed some facts.

The first theory which is purely based on deduction from axioms was given by David Hilbert in Grundlagen der Geometrie in 1899. You can deduce all theorems one would want to have in geometry without referring to diagrams at all as Hilbert said in his famous quote

One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs

As for the matter of "common rules" (such as transitivity of equality). Hilbert's system is based on set theory so apart from geometry axioms we usually assume all ZFC axioms (and all logic rules). Besides, one would want to have the possibility to measure segments, angles, areas etc. (for instance measure of segments is needed in the formulation of Thales theorem) so we assume that we know the theory of real numbers. In fact, we may assume all results from all branches of mathematics.

I definetely recommend you to start with Hilbert axioms because they are the closest to Euclid's and the way I was taught to do synthetic geometry in secondary schools. You should also know that some of the axioms from the first original Hilbert's paper turned out to be redundant (because they were provable from other axioms) and also continuity axioms are now usually replaced with a different axiom.

Answer 2

I think one can learn a ton by simply reading and working through $\textit{The Elements}$. Seeing how Euclid built upon everything, from the original definitions to the five axioms, is astonishing when you think about everything that can be shown. The algebraic principles you know can be derived and shown to be true geometrically through these definitions and axioms as well.

Answer 3

It became clear a couple of hundred years ago that Euclid's axioms were flawed and incomplete. Even in his first theorem he implicitly assumes somthing that is not stated as an axiom (circles that "look like" they should intersect actually do have a point of intersection).

The two books I've heard are good are "Geometry: Euclid and Beyond" by Robin Hartshorne and "Euclidean and non-Euclidean geometries" by Marvin Greenberg. They both cover the more modern approaches to plane geometry.