Geometry is a wide-ranging subject, finding its way into pretty much every branch of math and many fields outside math. This course is an attempt to both appreciate what characterises a geometrical approach and see some of these dispersed, surprising incarnations.

In putting together a program of study that achieves this it has been somewhat tricky to pare things down. Clearly, we must start with Euclid. We have to see how the axiomatic method exemplified by Euclid (and, for Euclidean geometry, refined by Hilbert) relates to topics in the foundations and philosophy of mathematics. We have to see the powerful applications that bringing algebra in, via Cartesian coordinates, has. We need to trace the story of perspective from Renaissance artists through the powerful projective geometry and into the strange worlds of complex and finite geometries. Non-Euclidean geometry, and its use in relativity, is a must---what shape is the universe anyway? The topological approach to surfaces; the Mandelbrot set and fractal geometry; other cultures' approaches to geometry; the art of M.~C.~Escher; Platonic and Archimedean solids; symmetry groups, wallpaper patterns and crystallography; Klein's Erlanger Programm; aperiodic tilings...

Meeting twice a week for 80 minutes we should finish sometime in 2014.

More practically, I'm turning to you. We {\em will} start with Euclid. After that I'll introduce the various topics in sufficient detail to give you glimpse of their nature and we'll choose as a group which paths to follow. The role of algebra, perspective and projective planes, and the existence of non-Euclidean geometries are three topics with which we must make at least some acquaintance; the main text for the course---John Stillwell's {\em The Four Pillars of Geometry}---treats each of these very well. Other than that, start lobbying for what you want to investigate further.

As decisions are made I'll update the "syllabus" link on the left.

Graded homework will be assigned approximately once every two weeks (there will be ungraded class preparation in addition to this). There will be two in-class quizzes during the semester and an optional third quiz during reading days---your best two will be worth 5\% each. The final project will consist of a write-up and a class presentation. The final will be a 24-hour open-book take-home exam. Attendance, class participation and prompt submission of homework are expected. Your performance in these areas will influence your final result by up to one letter grade.

You are expected to be aware of the college's policy on academic integrity and to abide by it. It can be found here. Please come and talk to me if anything is unclear.

http://cs.marlboro.edu/ courses/ spring2010/geometry/ home
last modified Thursday January 21 2010 12:41 pm EST