It's a small world after all...

In school, one of our first introductions to theorem-proving came from geometry. Sometime in high school, we learnt about Euclid's axioms and several geometric theorems that were proved from the axioms.  One of the main techniques for proving geometric theorems was by the use of a compass and a (uncalibrated) ruler. For instance, we learnt how to bisect a line segment using a compass and marking intersecting arcs from either ends and connecting the arcs together to cross the line.  Every method of inquiry has its own advantages and limitations, and embeds within it some deep assumptions about the underlying worldview. So too it is with the way in which Euclid's methods developed. Geometry literally means "earth measurements"-- and Euclid's geometry is deeply rooted in this application area.  One of the vexing problems with the compass and ruler approach is to make arbitrary divisions of geometric elements. For instance, we used to be challenged by "Maths Olym...