# Syllabus: Geometry

## Math 306, Spring 2006, UMBC

Let none enter here who are ignorant of geometry - inscribed above the entrance to Plato's school of philosophy

UMBC Course Description: Topics of this course are to be selected from foundations of geometry, modern Euclidean geometry, non-Euclidean geometry, projective geometry and its subgeometries.
Pre/co-requisite: MATH 301.

During this offering of the class I plan to start with a short discussion of Euclidean geometry, as seen through a translation of the Elements. We will then follow the general flow of the first seven chapters of Greenberg's book:

• Euclid's Geometry
• Logic and Incidence Geometry
• Hilbert's Axioms
• Neutral Geometry
• History of the Parallel Postulate
• The Discovery of Non-Euclidean Geometry
• Independence of the Parallel Postulate

This course assumes an initial familiarity with Euclidean geometry at a high school level. While a greater skill with this topic will come out of this class, that is not the central topic. The basic topics of the class are:

• A selection of axiomatic geometries
• Axiomatic systems in general
• Formal logic

### Course Outline

1. Logic & An Example - (1 week)
[Mese83, Chap 1] [Gree93, Chap 2] [Cede00, Chap 1] [DeLo04]
2. Euclid & History - (2 weeks)
[Gree93, Chap 1] [Eucl-H56] [Eucl-H02] [Eucl-J] [Eucl-T] [Coxe89, Chap 1]
3. post-Euclid & History - (3 weeks)
[Hart00, Chap 2] [Gree93, Chaps 3 & 5] [Cede00, Chap 2] [Hilb71]
4. Neutral, Hyperbolic & Spherical Geometries - (3 weeks)
[Gree93, Chaps 4, 6 & 10] [Cede00, Chap 2] [Coxe89, Chaps 15 & 16]
5. Projective Geometry - (2 weeks)
[Mese83, Chaps 2-4] [Cede00, Chap 4] [Coxe89, Chap 14]
6. Transformations, Symmetries & Tesselations - (2 weeks)
[Gree93, Chap 9] [Cede00, Chap 3] [Coxe89, Chaps 4, 7 & 15] [John01] [Mathworld: Tesselation] [Wiki: Tesselation]
7. Special Topics in Geometry - (1 week)
8. Further Topics - (as possible)
• Differential/Riemannian Geometry [Lanc70]
• Formal Logic [DeLo04]
• Hilbert to Incompleteness & Gödel
• Topology: Geometric & Algebraic
• Algebraic Geometry

Robert Campbell, campbell@math.umbc.edu
January 29, 2006