*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

**Logic & An Example**- (1 week)

[Mese83, Chap 1] [Gree93, Chap 2] [Cede00, Chap 1] [DeLo04]**Euclid & History**- (2 weeks)

[Gree93, Chap 1] [Eucl-H56] [Eucl-H02] [Eucl-J] [Eucl-T] [Coxe89, Chap 1]**post-Euclid & History**- (3 weeks)

[Hart00, Chap 2] [Gree93, Chaps 3 & 5] [Cede00, Chap 2] [Hilb71]**Neutral, Hyperbolic & Spherical Geometries**- (3 weeks)

[Gree93, Chaps 4, 6 & 10] [Cede00, Chap 2] [Coxe89, Chaps 15 & 16]**Projective Geometry**- (2 weeks)

[Mese83, Chaps 2-4] [Cede00, Chap 4] [Coxe89, Chap 14]**Transformations, Symmetries & Tesselations**- (2 weeks)

[Gree93, Chap 9] [Cede00, Chap 3] [Coxe89, Chaps 4, 7 & 15] [John01] [Mathworld: Tesselation] [Wiki: Tesselation]**Special Topics in Geometry**- (1 week)*Riemannian Geometry*[Lanc70]*Finite Geometries*(aka Combinatorics or Finite Fields) [Hart00, Sect 14]*Fractal Geometry*

[Cede00, Chap 5]*Minkowski Geometry*(geometry of Special Relativity)

[Jenn97]*Platonic & Archimedian Solids*

[Hart00, Chap 8] [Coxe89, Chap 10] [John01] [Mathworld: Platonic] [Mathworld: Archimedean] [Wiki: Platonic] [Wiki: Archimedean]*Klein's Erlangen Program*(Geometries in terms of their groups of transformations)

[Wikipedia] [Klein04]

**Further Topics**- (as possible)

Robert Campbell, campbell@math.umbc.edu

January 29, 2006