Jerzy Dydak sent me an interesting paper a couple of months ago, which you can find on his website here. Entitled “A Topological Approach to the Axiomatization of Geometry”, it proposes a new way of building Euclidean plane geometry from he ground up.
Readers will know that although Euclid was the first to propose an axiomatic foundation for geometry, his axiom system is not precise enough by modern standards. In the 19th century Hilbert gave a complete system of axioms for geometry, and other mathematicians followed in his steps. In 1932 G.D.Birkhoff published a paper called A system of axioms for plane geometry based on scale and protractor. This appeared in the Annals of Mathematics! Birkhoff’s innovation was to assume the real numbers as given: his axioms stated that certain geometric quantities could be “measured” by real numbers. This was of course quite different from Euclid’s approach, where the “theory of proportion”, equivalent to what we would call today the theory of the real numbers, was developed as part of geometry (Elements, book 5). Birkhoff’s approach has been followed by many later writers of textbooks (such as my own Elementary Geometry for example).
Dydak wants to get back to the Euclidean order of business where the real numbers are developed concurrently with the foundations of geometry. The fundamental undefined notion in his theory is the ternary relation of betweenness (as in “C is between A and B”). This allows one to define line segments, rays, and so on. What would conventionally be called “the completeness of the real numbers” is expressed in terms of connectedness and compactness properties of line segments (themselves defined in terms of betweenness, of course); and the “arithmetic” of real numbers appears as the result of adjoining and subdividing line segments, very much in the Euclidean manner.
It would be interesting to try teaching geometry this way. You spend a lot of time thinking about one-dimensional geometry in this development; but, as a bonus, you understand the foundations of the real number system in an intuitively appealing way. Dydak also suggests that the concept of lines as defined by an abstract “betweenness” might help make models of other geometries, such as the Klein model or Poincare model of hyperbolic geometry, less mysterious: no more “we are going to call this a ‘line’ even though it doesn’t look like one.”