Prerequisites:
MATH 221 and MATH 351 or permission of instructor
Frequency: Every Fall semester
Credit: 3 credits
Recent text: A Course in Modern Geometries,
Cedarberg (Springer-Verlag).
Recent instructors:
J. Faulkner; J. Howland; G. Keller; H. Ward
Student body:
An occasional 3
-year, but mostly 4
-year mathematics majors
and graduate students in mathematics and mathematics education
Topics and goals:
The two main goals of this course are to develop a knowledge and
appreciation of the wide scope of geometry beyond high school geometry
and to familiarize the student with axiomatics and mathematical
modeling.
Euclid's fifth axiom states that through any point P not on a line
l there is exactly one line m which does not meet 
, i.e., m
is parallel to l. Historically, this axiom was considered to be less
``self-evident'' than the other axioms for Euclidean geometry, and many
attempts were made to prove it from the other axioms. Specifically, one
can assume the axiom is false either by assuming that there are no
parallel lines or by assuming that there are several parallel lines, and
then trying to arrive at a contradiction. Indeed, in this way, one can
develop geometries with various nonintuitive properties---such as all
triangles have an angle sum of less than 180
or similar
triangles are congruent---but one does not arrive at a logical
contradiction. In fact, by modeling non-Euclidean geometry in Euclidean
geometry and vice versa, one can show that a contradiction in the axioms
of either geometry would lead to a contradiction in the axioms of
the other, so the two geometries are equally valid logically,
although they contradict each other.
A literal extension of the Euclidean plane is the real projective plane
which is obtained by adding points and a line at ``infinity.'' It is
useful in studying properties invariant under projection.
Topics in the course are selected from an introduction to mathematical
modeling, Euclidean geometry, affine and projective geometry, and
hyperbolic and other non-Euclidean geometries. Methods of linear
algebra are commonly used.
Relation to other courses: The only other undergraduate geometry
course, MATH 572 (Introduction to Differential Geometry), develops the
interrelation between geometry and calculus, while MATH 570 primarily
concerns basic geometric concepts such as incidence, parallelism,
conics, order, and angles.