The `Glimpses of Algebra and Geometry' (Springer, New York, 1998, ISBN 0-387-
98213-2) has been taught at Rutgers in four courses, three undergraduate and 
one `bridge course' offered both for undergraduate and graduate students. The 
following are suggested syllabi for these courses.


Undergraduate Mathematics Seminar (50:640:491 at Rutgers):

Course Material: Sections 1-8 from the Glimpses. 

Course Objectives: To get the students acquainted with some of the most 
important results in number theory as gently as possible without building
extensive machinery and with generous historical background. Prerequisite:
Calculus. 

Topics: Introduction to number theory and modular arithmetic. Early Pythagoreans
on irrational numbers; Hermite's proof of irrationality of e and pi. Special
Topic: Approximations of pi from Archimedes to Gauss. Pythagorean triples and 
the congruent number problem. Rationality: rational points in cubic curves; the 
group of rational points on elliptic curves; Mordell's theorem. 
Fermat's last theorem; history. Special topic: Proofs of Fermat's last theorem
for low exponents. Transcendence and the Liouville numbers. Complex arithmetic.
The proof of the fundamental theorem of algebra. Special topics: Four 
additional proofs. Geometric constructions. History: Ferro-Tartaglia-Cardano. 

In addition to homework, the students are required to choose a topic from a 
list of subjects given by the instructor, and, in the second half of the course,
make a presentation. In this, the students should demonstrate their abilities 
to work on a mathematical subject independently with some guidance of the
instructor. Library and the Internet can be used (cf. the list of web sites 
in the Glimpses). 


Undergraduate Geometry (50:640:435 at Rutgers):

Course Material: Sections 5-6, 9-13, 17 from the Glimpses.

Course Objectives: To introduce the students to modern geometry without
extensive buildup in axiomatism and with emphasis in 3-dimensional 
geometry. Complex language in geometry is emphasized. Hyperbolic geometry
uses Moebius transformations. 

Topics: Complex arithmetic. Euclidean isometries on the plane. Discrete
subgroups of isometries on the plane; 2-dimensional crystallography. Moebius 
transformations. Complex linear transformations. Hyperbolic geometry. The 
Poincare models. Special topics: Hyperbolic polygons and tessellations.
The five Platonic solids and their symmetries. Classification of finite 
groups of isometries in space. The golden ratio. Compound polyhedra. Special 
topics: Fibonacci numbers. Polyhedral Numbers. The Fullerene molecule. 
Klein's Icosahedron Book. 

This course can be taught in a traditional way; the use of 3-dimensional 
computer generated images/transparencies is recommended. 3-dimensional 
animation modules at some web sites can be used (cf. the list of web sites
at the end of Section 17).


Glimpses of Mathematics (56:645:580 at Rutgers):

Course Material: Sections 3, 7, 8, 17-23 from the Glimpses. 

This is a bridge course between undergraduate and graduate mathematics 
studies. The course discusses some spectacular discoveries in number theory, 
classical geometry, topology, graph theory, and modern algebra in an easily 
understandable manner. The focus is on the interconnections between these 
disciplines. The course uses the latest software technology and computer 
graphics to illustrate fundamental concepts and ideas that shaped the 
development of algebra and geometry from the Pythagoreans to the twentieth 
century. The students are assumed to be familiar with complex arithmetic, and 
some basic concepts in geometry. The textbook `Glimpses of Algebra and Geometry'
was designed to be suitable for this course; for more information, see the
Preface of the book.  

Topics: Rationality, elliptic curves, Fermat's last theorem. The fundamental
theorem of algebra, the five Platonic solids and their symmetries. Topology
of surfaces. The four color theorem. Quaternions and their use in 3-dimensional
geometry.

This course can be taught in a traditional way; the use of 3-dimensional 
computer generated images/transparencies is recommended. Discussion on the
historical background of the topics is also recommended. 


Geometry and Numbers (Freshmen Undergraduate Honors Seminar) 
(50:525:121:05 at Rutgers):

Course Material: Sections 1-6 from the Glimpses.

This is a slower paced course for freshmen who have no substantial background
in mathematics but who are interested in mathematics. Most of the material
taken from the Glimpses is from the clubsuit sections. The course is currently
running; for more information see the honors_news and honors_syllabus_01 in
the Honors_Courses directory.   


For more general information see the web sites:

1. http://mathsgi01.rutgers.edu/~gtoth/
  
   Current Directories: 
   
   1. Glimpses: Addenda, Errata, Solutions_Manual (in LaTeX, Postscript, HTML), 
   70_Colored_Pictures, Enrichment, Web sites, two Maple animation files.
    
   2. Visualizing_Mathematics (7 Maple worksheets containing graphics of the 
   Glimpses in Maple V, releases 3-4)

   3. Honors_Courses: honors_news, honors_syllabus_01  

2. http://carp.rutgers.edu/math-undergrad/
   Directories: science-vision.html 
                (3 downloadable Maple V worksheets, in release 3) 
                visual_math.html 
                (7 downloadable Maple worksheets, in release 3)
                geomview.html 
                (Sample Geomview pictures)
                icos.html 
                (Graphics for the icosahedron)