Course Assistant: Matt Gibbs
Office: Weber 21; Phone: 491-3958; email: mgibbs@holly.colostate.edu
Matt Gibbs' Office Hours: Monday 2-3; Tuesday 11-12; Thursday 11-12.

Text: Euclidean And Non-Euclidean Geometry, an analytic approach, by Patrick J. Ryan
Cambridge University Press (1986); ISBN 0-521-27635-7

This course will discuss the basic notions of euclidean and non-euclidean geometry, in two dimensions. We will focus on the three main examples: euclidean, spherical/projective, and hyperbolic geometry. Each will take up approximately one-third of the course material. In the discussion of each of these three geometries, we will briefly touch on the synthetic approach,
and quickly turn to a more analytic treatment, with coordinates.  Finally we will discuss changes of coordinates in each geometry, leading to the groups of rigid motions of each geometry.  Tesselations or tilings in each geometry will cap each unit.

Syllabus:
1. Introduction to Geometry.
    The ingredients of geometry.  Axioms and Synthetic Geometry.  Models for a Geometry.
    Finite Geometries.  Coordinate Geometry.
2. Euclidean Geometry
    Definition of the Euclidean Plane.  Points, lines, circles, distance, angle.
    Analytic Geometry in the Euclidean Plane.  Cartesian Coordinates.  Trigonometry.
    Pythagorean Theorem and Descartes' Formula.
    Complex numbers as a model for the Euclidean plane.  Basic complex arithmetic.
    Changes of coordinates: the general linear group, orthogonal transformations, affine transformations.
    Structure of the group of rigid motions.  Translations, rotations, reflections, glides.
    Tilings of the Euclidean plane.  Ribbon Patterns Wallpaper patterns.
3. Projective and Spherical Geometry.
    Basic notions of spherical geometry: distance, angle, trigonometry.
    Spherical Pythagorean Theorem and Spherical Descartes' Formula.
    Group of Rigid Motions of the sphere: SO(3) and O(3).
    The Projective plane, homogeneous coordinates, distance and angle.
    Tilings of the sphere and projective plane.  Regular solids.
4. Hyperbolic Geometry.
    Poincare Disc Model for Hyperbolic Geometry.
    Lines, circles, distance, angle, and trigonometry in the hyperbolic plane.
    Hyperbolic Pythagorean Theorem and Hyperbolic Descartes' Formula.
    Upper-half plane model for hyperbolic geometry.
    Linear fractional transformations and rigid motions of the hyperbolic plane.
    Tilings of the hyperbolic plane.

Grading:  There will be regular homework assigned from the text.  There will be three examinations. There will be a computer project, which will be Web-based. Relative Importance:
Homework: 20%
Exams: 20% each
Project: 20%
See description of Project HERE.
 
Homework:
Due Monday, 1/24:
1. Write and Submit a proof of the Pythagorean Theorem.
2. Write and Submit a proof of the theorem that in euclidean geometry, the sum of the interior angles in a triangle is 180 degrees.
Due Monday, 1/31:
Definition: Let A and B be sets, and let f:A -> B be a function.

• We say that f is 1-1, or injective, or is an injection, if whenever f(a1)=f(a2) then a1 = a2.
• We say that f is onto, or surjective, or is a surjection, if for every b in B there is an a in A such that f(a)=b.
• We say that f is bijective, or is a bijection, if it is both 1-1 and onto.

Let P be a projective plane, that is, a set of points and lines satisfying the four axioms

• For every pair of distinct points p, q there is a unique line containing both.
• Every pair of distinct lines L, M meet in a unique point.
• Each line contains at least 3 points.
• There are at least 3 non-collinear points.