Euclid tried (unsuccessfully!) to formulate a series of postulates for the geometry of a (flat, infinite) piece of paper. His abstract model was supposed to accurately reflect the world around us in that his postulates were to be "self-evident''. However, his most famous postulate, the parallel postulate, turns out to be independent of the others. This leaves room for non-Euclidean geometries in which the other postulates are satisfied but which have too many or too few parallel lines.
This course investigates several types of geometry which differ fundamentally from the geometry of a piece of paper. We will study the classic alternatives to the parallel postulate, namely geometries with too many parallel lines (hyperbolic geometry) or too few parallel lines (elliptic geometry). Specific models for these geometries will be discussed, in addition to their general properties. Some of these models will be investigated using computer graphics, but no prior experience with computers will be assumed, and every effort will be made to make the course self-contained.
We will also study the beautiful model of geometry called taxicab geometry, which is based on the notion that the "taxicab" distance between two points is not in a straight line "as a crow flies" but rather along streets "as a taxicab drives". This model has many surprising and "non-Euclidean" features, but actually does satisfy Euclid's postulates, including the parallel postulate. (It does not, however, satisfy modern versions of Euclid's postulates, and is therefore classified as non-Euclidean.)
This is a writing-intensive class (WIC), and considerable time will be spent learning how to write mathematical text, culminating in a final paper which will be assessed on both content and presentation.