## 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.

Class log

**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 S_{n} of permutations of n objects. Cycle notation for permutations.
Generation of S_{n} by transpositions, the sign of a permutation,
the alternating group A_{n} of even permutations. The symmetry groups of the regular polyhedra:
explicit geometric description and algebraic description via permutation groups.
Finite groups of rotations in R^{3} : classification and examples.

Quaternions and compositions of rotations in R^{3}. The geometry of the group of rotations in R^{3}.

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