Class Log

Note: Numbers 1.1 etc. correspond to section numbers in the course text Geometry and Topology by Reid and Szendroi.

Monday 5/3/10. Review session for final exam.

Friday 4/30/10. We finished the description of rotations in 3 dimensions in terms of the quaternions. We used this to understand the geometry of the group SO(3) of rotations in 3 dimensions.

Wednesday 4/28/10. We introduced the quaternions and showed how they can be used to compute compositions of rotations in 3 dimensions.

Monday 4/26/10. We finished the proof of the classification of finite groups of rotations in 3 dimensions and realized each type as the group of rotational symmetries of a polyhedron. We filled out class evaluations.

Friday 4/23/10. We deduced that the finite rotation groups in 3 dimensions are either cyclic, dihedral, or the group of rotational symmetries of a regular polyhedron (tetrahedron, cube/octahedron, or dodecahedron/icosahedron).

Wednesday 4/21/10. We derived the counting formula for finite rotation groups in 3 dimensions, relating the order of the group and the order of the stabilizer for each orbit of poles.

Monday 4/19/10. No class (Patriots' day).

Friday 4/16/10. We identified the full group of symmetries of the dodecahedron as A5xZ2 using the 5 inscribed cubes together with the determinant. We began to study finite groups of rotations in 3 dimensions.

Wednesday 4/14/10. We identified the group of rotational symmetries of the dodecahedron with the alternating group A5 using the 5 inscribed cubes.

Monday 4/12/10. We finished the description of the symmetries of the cube.

Friday 4/9/10. We introduced the alternating group An (the subgroup of Sn consisting of even permutations) and began to describe the symmetries of the cube (or equivalently, its dual the octahedron).

Wednesday 4/7/10. We described the cycle notation for permutations and the sign of a permutation.

Monday 4/5/10. We recalled the definition of the symmetric group Sn and showed that it is generated by transpositions. We identified the group of symmetries of a tetrahedron with S4 by associating a symmetry with the induced permutation of the vertices.

Friday 4/2/10. We gave another proof of Euler's formula using spherical geometry. We described the dihedral group Dn of symmetries of a regular n-sided polygon.

Wednesday 3/31/10. We proved Euler's formula V-E+F=2 for polyhedra by an inductive argument.

Monday 3/29/10. We finished the description of the icosahedron in coordinates and obtained the dodecahedron as its dual. We discussed spherical projections of polyhedra.

Friday 3/26/10. We began the construction of the icosahedron in coordinates and discussed the golden ratio.

Wednesday 3/24/10. We described the cube and octahedron in coordinates. We discussed duality of polyhedra.

Monday 3/22/10. We proved that the list of regular polyhedra below is complete.

Friday 3/12/10. We discussed the regular polyhedra (Platonic solids): tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

Wednesday 3/10/10. We finished discussing 3.5-3.6.

Monday 3/8/10. We compared spherical geometry and Euclidean plane geometry (3.5,3.6).

Friday 3/5/10. We classified motions of the sphere using radial extension and the classification of motions of R3.

Wednesday 3/3/10. We worked out examples of motions of the sphere (3.4).

Monday 3/1/10. We proved the spherical cosine rule (3.2) and deduced the triangle inequality for spherical triangles (3.3).

Friday 2/26/10. We discussed basic notions of spherical geometry (3.1).

Wednesday 2/24/10. No class due to snow day.

Monday 2/22/10. We proved Theorem 2.6 (every reflection in Rn is a composite of at most n+1 reflections).

Friday 2/19/10. We discussed compositions of reflections in Euclidean geometry.

Wednesday 2/17/10. Prof Urzua described the circumcenter, incenter, and orthocenter of a triangle in Euclidean geometry (1.16.4).

Tuesday 2/16/10. Prof. Kazanova discussed some aspects of plane Euclidean geometry: congruence of triangles, the angle sum of a triangle, and the centroid (1.13, 1.16.2, 1.16.4).

Friday 2/12/10. I finished explaining how to describe motions of R3 geometrically given an algebraic description.

Wednesday 2/10/10. No class due to snow day.

Monday 2/8/10. I began to show how to describe motions of R3 geometrically given an algebraic description.

Friday 2/5/10. I showed how to describe a motion T of R2 geometrically given an algebraic description of the form T(x)=Ax+b.

Wednesday 2/3/10. I finished the proof of the classification of orthogonal matrices (1.11.2) and deduced the classification of motions of R3 (1.15).

Monday 2/1/10. I discussed examples of compositions of rotations in R3 and began the proof of the classification of orthogonal matrices (1.11.2).

Friday 1/27/10. I described the normal form for orthogonal matrices (1.11.2).

Wednesday 1/25/10. Prof. Tevelev covered 1.14, the classification of motions of R2.

Monday 1/23/10. Prof. Kazanova described the 2x2 orthogonal matrices (1.11.1).

Friday 1/21/10. I covered 1.5--1.10: Angles, and description of motions in terms of orthogonal matrices and translations. Also I gave a reminder of basic properties of orthogonal matrices (from Math 235 Linear Algebra, see also Appendix B).

Wednesday 1/19/10. I covered 1.1--1.4 : the metric on Rn, the triangle inequality, and shortest paths.