## Math 462 : Geometry II

### Spring 2010

Instructor: Paul Hacking
Email: hacking@math.umass.edu
Office: LGRT 1235H
Office Hours: Wednesdays and Thursdays, 4:00PM--5:00PM, in my office LGRT 1235H.
Classes: MWF 11:15AM-12:05PM in LGRT 121.
LGRT = Lederle Graduate Research Tower map, photo

Course description

This course is a continuation of Math 461: Geometry I. We study euclidean and spherical geometry, with particular emphasis on the group of symmetries in each case.

Textbook

The course text is Geometry and Topology by Miles Reid and Balazs Szendroi. googlebooks.

We will also use the text from Math 461, Modern Geometries, 2nd ed., by Michael Henle. googlebooks

It is important to read the textbook as well as attending the lectures -- there is not enough time in class to cover all the material.

Homework

There will be weekly homework, due at the beginning of Friday's class (first homework due 2/5/10).

The homework and quizzes make up 35% of your course grade.

Homework sets

Exams

There will be one midterm exam and one final exam.

The midterm exam will be held on Wednesday 3/10/10 at 7PM in LGRT 0113. There will be a review session on Tuesday 3/9/10 at 7PM in LGRT 121. Midterm review questions are here.

The final exam will be held on Tuesday 5/11/10 at 10:30AM in LGRT 321. There will be a review session on Monday 5/10/10 at 7PM in LGRT 121. Final review questions are here.

Calculators and the textbook are not allowed on exams and quizzes. You are allowed one sheet of notes (8.5x11, both sides).

The exams make up 65% of your course grade -- 30% for the midterm and 35% for the final.

Syllabus

Numbers refer to sections of the course text Geometry and Topology by Reid and Szendroi.

1 Euclidean geometry: Metric (distance function), classification of motions (symmetries). Sample theorems of plane geometry.

2 Composition of maps: Examples. Reflections generate all Euclidean motions.

3 Spherical geometry: Metric, triangles, and classification of motions. Comparison of spherical and Euclidean geometry.

Polyhedra: Examples. The regular polyhedra: classification and description in coordinates. Duality of polyhedra. Euler's formula V-E+F=2.

Symmetry groups: The dihedral group of symmetries of a regular n-sided polygon. The symmetric group Sn of permutations of n objects. Cycle notation for permutations. Generation of Sn by transpositions, the sign of a permutation, the alternating group An of even permutations. The symmetry groups of the regular polyhedra: explicit geometric description and algebraic description via permutation groups. Finite groups of rotations in R3 : classification and examples.

Quaternions and compositions of rotations in R3. The geometry of the group of rotations in R3.

This page is maintained by Paul Hacking hacking@math.umass.edu