
Description
For generations of high school students, the word "geometry" conjures up images
of straightedges and compasses. In fact the presentation in most high school
geometry courses has changed very little  at least in principle  from that
in Euclid's "Elements", published around 300 B.C. 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 socalled "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.
In this course, we will 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.
News
Friday 14 September: Many things updated on the Schedule page.
Tuesday 28 August: I've posted the reading assignments for Week #1 and
the homework for Wednesday. You can follow the link from the Schedule page.
Monday 27 August: Welcome to 21101 The Four Pillars of Geometry.
