COURSE NUMBER AND TITLE: MATH 4310 Modern Geometry

CREDIT HOURS: 3

CATALOG DESCRIPTION: A modern treatment of geometry primarily from the metric approach, but with some reference to the Euclidean Synthetic approach. Topics include parallelism, similarity, area, constructions, non-Euclidean, and finite geometries.

PREREQUISITE(S): MATH 2030 or permission of instructor

SUGGESTED TEXT(S):
Elementary Geometry from an Advanced Standpoint by Edwin E. Moise
Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg

COURSE OUTLINE:

1. Axiom systems and the real number line
   • Field Axioms for the real numbers
   • Order relations and ordered fields
   • Positive integers and mathematical induction
   • The Archimedean Postulate and Euclidean completeness
2. Incidence and metric geometries
   • Hilbert’s axioms of incidence
   • The Ruler Postulate
   • Segments, Rays, Angles and Triangles
   • Convex quadrilaterals
   • Hilbert’s axioms of betweenness; The Plane Separation Postulate
   • Hilbert’s axioms of segment and angle congruence.
   • Triangle congruence; The SAS Postulate and Triangle congruence theorems
   • Geometric inequality; The Exterior Angle Theorem; The Triangular inequality
3. Absolute Geometry
   • Sufficient conditions for parallelism
   • Saccheri and Lambert quadrilaterals
   • The Saccheri-Legendre Theorem
4. Non-Euclidean Geometries
   • The parallel postulates in hyperbolic and elliptic geometries.
   • Hyperbolic geometry and Poincare model.
   • Elliptic geometry and the Riemann model.
5. Euclidean Geometry and parallel projections
   • The Euclidean parallel postulate
   • Parallel projections
   • The comparison and similarity theorem.
6. Optional Additional Advanced Topics may include:
   • Similarities between triangles
   • Polygonal regions and area
   • Constructions by ruler and compass
   • The surd field and the surd plane.