MATH 466 Groups and Geometry (Spring 2002)
2001- 02 Spring
MATH 466: Groups and Geometry
Final Exam: Date, time and place are announced. Contents: Only
reflection groups.
1.1 - 1.9, 1.12 - 1.14 and 2.1 - 2.7 (inclusive).
The following is what we have covered so far or what we will cover in the class.
- Week 1: (2 hours) Finite and infinite dihedral groups.
- Week 2: (2 hours) An orthogonal transformation of the Euclidean plane (R^2) is either a rotation at the origin or a reflection whose axis passes through the origin. An element of the Euclidean group (i.e. an isometry of the Euclidean plane) is a rotation or a reflection or a translation or a glide reflection.
- Week 3: (1 hour) The Euclidean group is generated by reflections. The combinations of various
isometries were examined.
- Week 4: (3 hours) The Euclidean group is a semidirect product of
the group of translations and the group of isometries that fix the origin. A finite subgroup of the Euclidean group is either cyclic or dihedral.
- Week 5: (3 hours + Exam 1) The symmetry groups of the Platonic solids were studied. A finite subgroup
of SO3 is cyclic or dihedral or the symmetry group
of one of the Platonic solids.
The exam will cover the contents of the first 4 weeks only. Solving the questions of the
first exercise sheet
will be an excellent preparation for the exam.
- Week 6: (3 hours) Frieze groups were
discussed and classified.
- Week 7: (2 hours) We proved that there are 5 types of lattices. A wallpaper group is defined to be a subgroup of the Euclidean group whose translation subgroup H is generated by two independent tranlsations and whose point group J is finite. It is possible to construct a lattice for a given wallpaper group by using the 'minimal generators' of H. Moreover J acts on this lattice. The order of a non-identity rotation in a wallpaper group is 2, 3, 4 or 6.
- Week 8: (3 hours) Some examples of wallpaper groups were studied. We started the classification of wallpaper groups.
- Week 9: (3 hours + Exam 2) The classification was completed.
- From Week 9 on, I follow the book by Humphreys. The sections that were covered from Chapter I are:
1.1 - 1.9, 1.12 - 1.14 (inclusive).
The proof of theorem 1.9 was postponed to the extra class.
- I shall cover 2.1 - 2.7 (and maybe 2.8 - 2.10 if time permits) from Chapter II.
- Exam 3. 16 May 2002, Thursday at 17:40 in M-05.
Contents: 1.1 - 1.9, 1.12 - 1.14 (inclusive) and 2.1.
LECTURES, GRADING ETC.
HINTS TO EXAM QUESTIONS
BOOKS
COURSE PLAN
SOME LINKS ON SYMMETRY
Visit this page frequently to see the future announcements and changes.