Course:Harris, Fall 07: Diary Week 16

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  • 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


  • 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