Course:Harris, Fall 07: Diary Week 16

Mon:

• We talked about "multiplication" of symmetries.
  • For instance, if R is a rotation and F is a flip, then RF means applying first the rotation and then the flip.
    • In a rosette, we labeled the corners of the polygon, then applied each transformation.
    • In a frieze, we labeled the top of the figures, then applied each transformation.
• We talked about contrasting the three geometries we studied:
  • similarities:
    • vocabulary of
      • objects: points, lines (geodesics), angles, polygons
      • measurements: distance, angle-measure, area
  • differences:
    • relationships of measurements (such as angle-sums)
    • relationships with parallel lines/geodesics
• We looked briefly at more general kinds of geometry.
  • Our usual three geometries are uniform:
    • spherical = uniform positive curvature
    • flat = zero curvature
    • hyperbolic = uniform negative curvature
  • Many surfaces have varying curvature:
    • An egg has one end (the pointy one) which has higher positive curvature than anywhere else on the egg.
    • A golf ball has high positive curvature in each dimple and negative curvature around the edge of a dimple.
• We considered ways in which math is used in art and art is used for math:
  • Escher and Coxeter:
    • Escher got help from Coxeter for hyperbolic regular tessellations.
    • Coxeter got illustrations from Escher on depicting hyperbolic tessellations.
  • Rennaisance workers with mathematically refined perspective, such as Michaelangelo and Bruneschelli
• In general, mathematics abstracts simplified notions that can be rigorously studied, notions derived from
  • nature
  • art
  • science
  • anything of human interest

Fri:

• gave exam; no students took more than 75 minutes, some finishing in 45 minutes
• found out that this Wiki site had gone down the evening before, right when students were perhaps doing their last-minute studying
  • seems a "network switch" at the university failed
  • to compensate for lack of studying ability, will modify grading scheme on final exam:
    • initial grade levels: 84, 71, 58, 45 (i.e., 84 = A/B borderline, 71 = B/C borderline, etc.)
    • final grade levels: 82, 67, 52, 37
    • net effect, on average, was to add 1/4 grade level to students' final exam grades