Instructor: Danny Calegari 432 Science Center

• Half-course (fall term 2000). MWF at 12 in Room 309 Science Center
• Office hours M 1:30 - 2:30
• Prerequisite: Mathematics 21 a,b
• Assessment: 50% homework, 50% take-home final exam
• Email dannyc

Course Assistant: Kathy Paur 321b Science Center

• Office hours Tu 4 - 5
• Section Th 6 - 7 in Room 321b Science Center
• Email paur

An introduction to spherical, Euclidean and hyperbolic geometry in two and three dimensions, with an emphasis on the similarities and differences between these flavors of geometry. The most important tool in analyzing these geometries will be a study of their symmetries; we will see how this leads naturally to basic notions in group theory and topology. Topics to be covered might include classical tessellations, the Gauss-Bonnet theorem, scissors congruence, orbifolds, and fibered geometries.

• Course notes for class; available as a .tex file, or as a .ps file. (Under revision. Updated May 4th, 2005)
• Syllabus for class; available as a .tex file, or as a .pdf file.
• Homework 1 - groups; available as a .tex file, or as a .pdf file.
• Homework 2 - the Euclidean plane; available as a .tex file, or as a .pdf file.
• Homework 3 - spherical geometry; available as a .tex file, or as a .pdf file.
• Homework 4 - hyperbolic geometry; available as a .tex file, or as a .pdf file.
• Homework 5 - surfaces and fundamental groups; available as a .tex file, or as a .pdf file.
• Homework 6 - properly discontinuous groups; available as a .tex file, or as a .pdf file.
• Final exam; available as a .tex file, or as a .pdf file.

• Three dimensional geometry and topology W. Thurston Princeton Mathematical Series, 35. Princeton University Press, 1997
• Classical tessellations and three-manifolds J. Montesinos Universitext, Springer-Verlag, 1987 out of print

• Invitation to geometry H. Coxeter John Wiley and sons, 2nd ed. 1989
• Transformation geometry M. Jeger Allen and Unwin, 2nd ed. 1969 out of print
• Foundations of hyperbolic manifolds J. Ratcliffe Graduate texts in mathematics 149, Springer-Verlag, 1994
• Algebraic topology M. Greenberg and J. Harper Perseus Books, 1982
• Braids, links, and mapping class groups J. Birman Annals of Math. Studies no. 82, Princeton University Press
• Unpublished chapters 13a, 13b and 13c of Thurston's notes on orbifolds.

last updated: 22nd December 2000 If you have any comments on the material in this page, or if you wish to comment on the material in the course, contact Danny Calegari via email The image to the right is Heaven and Hell (Circle limit IV) by Maurits C. Escher, the well-known 20th century Dutch artist. It depicts a tessellation of the hyperbolic plane by two kinds of tiles, angels and devils. The model of the hyperbolic plane used is the Poincaré (or conformal) model. Danny Calegari's home page