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
Elementary Geometry from an Advanced Standpoint by Edwin E. Moise
Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg
- 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
- Incidence and metric geometries
- Hilbertís axioms of incidence
- The Ruler Postulate
- Segments, Rays, Angles and Triangles
- Convex quadrilaterals
- Hilbertís axioms of betweenness; The Plane Separation Postulate
- Hilbertís axioms of segment and angle congruence.
- Triangle congruence; The SAS Postulate and Triangle congruence theorems
- Geometric inequality; The Exterior Angle Theorem; The Triangular inequality
- Absolute Geometry
- Sufficient conditions for parallelism
- Saccheri and Lambert quadrilaterals
- The Saccheri-Legendre Theorem
- Non-Euclidean Geometries
- The parallel postulates in hyperbolic and elliptic geometries.
- Hyperbolic geometry and Poincare model.
- Elliptic geometry and the Riemann model.
- Euclidean Geometry and parallel projections
- The Euclidean parallel postulate
- Parallel projections
- The comparison and similarity theorem.
- 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.