MA 625 A, Fall 2005 (Stevens Institute of Technology)

MA 625, Fall 2005
FUNDAMENTALS OF GEOMETRY

Communications to the class

Dec 9: Grades will be posted on or around Mon Dec 12.

Dec 9: Check the list of internet resources on the Poincaré disc, below.

Course information

Brief description: This is a course on the foundations of Euclidean geometry and its extensions, the non-Euclidean geometries. The course will be elementary in the sense that it will only deal with basic, non-advanced topics, with which most students are familiar from high school. On the other hand, as the name suggests, the focus will be on the rigorous foundation and derivation of the elementary concepts and results of geometry, something that might require quite a change of perspective for some. (In fact, one might use this class as a trainer on mathematical thinking and logic.) To this purpose, the subject will be presented with an attention to the historical context. Another peculiarity of the course is that no calculus is needed.

Time & Place: Monday, Wednesday 4:30--5:45, Morton 205.

Lecturer: Marco Lenci. Office: Kidde 219B. Tel.: x-5453. E-mail: mlenci@math.stevens.edu.

Office hours: The official office hours are posted here, but students are welcome to meet me at other times as well, by appointment.

(Tentative) syllabus:

Geometry before Euclid
Euclid's first four postulates
The fifth postulate
Problems with Euclid's postulates
Models
Incidence geometry
Hilbert's axioms
Axioms of betweenness
Axioms of congruence
Axioms of continuity
Axioms of parallelism
Neutral geometry
Saccheri-Legendre Theorem
Angle sum of a triangle
Equivalence of parallel postulates
Attempts to prove the fifth postulate
The birth of non-Euclidean geometry
Independence of the parallel postulate
Elliptic geometry on the sphere
Hyperbolic geometry
The Poincaré disc
The Lobachevsky upper half-plane
Consistency of hyperbolic geometry

Textbook: M. J. Greenberg, Euclidean and Non-Euclidean Geometry, 3rd ed.,W. H. Freeman & Co. (A good further reading is H. S. M. Coxeter, Introduction to geometry, Wiley.)

Homework, exams and grading policy: There will be two types of homework in this class: occasional homework given while lecturing, and officially assigned homework. The first type can be worked out by the student at his/her discretion and possibly given to me for comments. The second type will consist of two homework sheets, each containing a short list of problems, given at two different times during the semester. These will have a due date and will be graded, the grade counting in (small) part towards the class grade.
Also, there will a midterm exam and a final exam, whose dates we will agree on during the semester.
Both the official homework and the exams will receive letter grades, to emphasize that the final grade for the class will depend more on the progress shown by the student than on his or her average (not that this should give anybody license to flunk an exam!).

Internet resources on the Poincaré disc

NonEuclid, applet to play with the Poincaré disc, by Joel Castellanos.
Wilson Stothers' Hyperbolic Geometry Page. Explanations, theorems and applets.
An applet to easily draw and move segments and polygons in the Poincaré disc, by Paul Garrett.
Hyperbolic Tessellation Applet, by Don Hatch.
Hyperbolic Tessellation with applet, by David E. Joyce.
A list of tutorial articles on the Poincaré disc and related subjects.
Much more stuff from Google.

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