Pillars of Geometry

The Four Pillars of Geometry

For over 2000 years Euclid and geometry were synonymous.

Euclid's geometry, now called Euclidean geometry, was based on a set of axioms. The most famous of these is the so-called "Parallel Postulate", which states that given any line and a point not on that line, there is exactly one line through the point that does not intersect the line.

( For many centuries it was assumed that this axiom could be derived from the others.)

The 18th century, though, saw a description of a "hyperbolic geometry" which satisfied all of Euclid's axioms except the parallel postulate. (hyperbolic spaces have too many parallel lines.) Later developments included projective geometries, which have too few parallel lines.

We discuss three approaches (pillars) to understanding geometry. The first is constructive. These straightedge and compass constructions will likely be the most intuitive to most students. The second is algebraic. Introducing coordinates to a Euclidean plane can reduce complicated geometric arguments to simple calculation. The third approach involves "invariants" of "transformations". For instance, length and angle are invariants of the "rigid motions" of the plane. The fourth pillar is projective geometry, which describes why things look the way they do, and points toward some deeper connections between algebra and geometry.