MATH 430 (Euclidean and Non-Euclidean Geometries)
| DESCRIPTION |
Hilbert's axioms for Euclidean Geometry. Neutral
Geometry: The
consistency of the hyperbolic parallel postulate and the inconsistency
of the elliptic parallel postulate with neutral geometry. Models of
hyperbolic
geometry. Existence and properties of isometries. |
| PREREQUISITES |
MATH 140-141, or equivalent |
| TOPICS |
Logical deficiencies in Euclidean Geometry
Flawed proofs
Correction of some flawed proofs by clearly stating
certain postulates and giving rigorous deductions
Overview of the structure of Euclidean Geometry
Importance of Euclid's Parallel Postulate as opposed
to the other postulates
Equivalence of certain postulates (Play fair, etc)
to Euclid's Parallel Postulate
Absolute or Neutral geometry. Work of Legendre
and Saccheri
Negation of Euclid's Parallel Postulate;
non-Euclidean
geometry.
Discussion of models for non-Euclidean geometry
Axiom systems
Incidence axiom and ruler postulate
Betweenness
Segments, Rays and Convex sets
Angles and Triangles
Pasch's Postulate and Plane Separation Postulate
Perpendiculars and inequalities
SAS postulate
Parallel postulates
Models for non-Euclidean geometry
Proof of the consistency of non-Euclidean geometry
by means of models
|
| TEXT |
Text(s)
typically used in this course. |
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