Course:Harris, Fall 07: Diary Week 8

Mon:

• groups finished up Explorations from last week
• discussed the nature of spherical geometry:
  • need spherical analogues of planar geometry:
    • fundamental objects:
      • points
      • line segments
      • lines
    • combinations of fundamental objects:
      • angles
      • polygons, etc.
    • measurements:
      • distances between points
      • lengths of line segments (or other curves)
      • measures of angles
      • areas of polygons, etc.
    • symmetries
  • some analogues:
    • points in plane --> points on sphere
    • lengths of curves in plane --> lengths of curves on sphere
    • line segment between two points in plane --> shortest curve between two points on sphere
    • distance between two points in plane --> length of shortest curve between two points on sphere
    • line in plane --> extending "spherical line segment" to a curve on sphere always shortest between any two of its points
    • angles formed by lines in plane --> angles formed by "spherical lines" on sphere
    • measure of angle in plane --> measure of "spherical angle" on sphere by looking "up close" to vertex of angle, where sphere is like plane
• groups began working on Spherical Easel Exploration
  • measurements of angles there is given in radians:
    • 1 radian is about $57^{\circ }$
    • a right angle is $\pi /2$ radians

Wed:

• showed how Spherical Easel can mark a selected triangle as existing on the "outside" of the three segments selected, producing angle measurements that may all be greater than 180 degrees (showing up as more than Pi in the Spherical Easel measurement window)
• mentioned that "spherical lines" are great circles, i.e., the circles of greatest possible size on the sphere
  • smaller circles don't correspond to lines in the plane, they just correspond to circles in the plane
• in any surface, the geodesics are the curves (in the surface) corresponding to lines in the sense of being the shortest curve between any pair of its points (technically: any pair of close-by points)
  • so the geodesics in a sphere are the great circles--any two of which intersect in an antipodal pair of points
• groups finished or nearly finished Spherical Easel exploration (interpreting "rhombus" as "quadrilateral with four equal sides"--doesn't have to be a parallelogram)
• when finished with that, groups went on to next exploration, continuing for Friday

Fri:

• groups worked on finishing up the first exploration (Spherical Easel), doing the second (Spherical Geometry), and getting onto the third (Spherical Geometry: Polygons)
  • most groups (maybe all) didn't have time to complete all the explorations
• still some uncertainty about what "lines" mean in spherical geometry: it means "great circles"
• didn't mention due-date for Exercizes; they will be due end of next week (which is short because of Fall Break on Monday)