Math 461

Math 461 - Fall 2009
Geometry I
MWF 10:10-11:00, LGRT 119

Instructor:	Tom Braden	email:	braden@math.umass.edu
Office hours:	Tues 2-3, Thurs. 3-4, and by appointment
Office:	LGRT 1240	Phone:	545-1732

Link to homework page
Some web resources and programs

Here are solutions to the midterm.

In this class we will explore several flavors of plane geometry; besides the familiar Euclidean geometry, we will study projective, spherical, and hyperbolic geometry. All of these geometries have concepts like points, lines, triangles, circles, etc, but they can behave in very different and sometimes surprising ways. Linking all of these notions of geometry together will be the notion of transformations, the motions that preserve properties like distance and angles in each geometry. This will lead naturally to studying symmetries of a geometric figure, which are the transformations which leave that figure unchanged.

Text: Michael Henle, Modern Geometries: Non-Euclidean, Projective and Discrete, second edition, Prentice-Hall.

Topics: Plane geometry using complex numbers. Defining geometries using transformations -- the Kleinian approach. Möbius transformations of the complex plane. The Poincaré model of the hyperbolic plane -- lines, circles, horocycles and hypercycles, length, area. Elliptic geometry (a.k.a. spherical geometry). Projective geometry; representing Euclidean, hyperbolic, and elliptic geometry in the projective plane. Discrete symmetry groups -- groups generated by rotations and reflections.

Grading: your grade will be based on:

Homework and quizzes: 35%
An evening midterm, on Wednesday October 28 (tentative): 30%
A final exam, at the time scheduled by the University: 35%

Homework Rules and Guidelines:

Here are the rules for collaborating on homework problems:

I. You must list the names of all people with whom you discussed each specific problem.
II. You MUST write up your solutions completely independently.
Homework will be due on Fridays at the start of lecture, unless otherwise stated. Late homework will not be accepted without a valid reason (illness, etc.), but the lowest homework grade will be dropped.