Course:Harris, Fall 08: Diary Week 9

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  • No class, Fall break


  • Preliminary sketches for the Tessellation Art Project are due on Friday
  • Comparison of Euclidean planar geometry and spherical geometry:
    • point:
      • Same in both.
    • line:
      • "spherical line"
        • Definition: a circle on the sphere whose plane goes through the center of the sphere.
          • Alternative definition: a circle on the sphere of largest possible radius (hence, "great circle").
        • Different from planar line:
          • It has a finite length.
          • Anything else?
        • Similar to planar line:
          • The shortest path between any two points lies along a spherical line.
          • Anything else?
    • line segment:
      • Arc of a great circle: much the same as for planar geometry (except worrying about what "between" means).
    • length of a line segment:
      • Same as for planar geometry.
    • angle (and its angle measure) between two segments sharing an endpoint:
      • Use the tangent plane at that common endpoint to project the spherical segments onto that plane.
      • Then it's exactly the same as planar geometry.
    • distance:
      • Length of the line segment between two points.
        • Does the same definition work in both?
    • between:
      • In the plane, P is between Q and R on the line L means P, Q, and R are points on L and
        • When traversing L from one infinite "end" to the other we come across first Q and then P and then R, or first R and then P and then Q.
          • That doesn't apply on the sphere, since a line doesn't have infinite ends.
        • dist(Q,P) + dist(P,R) = dist(Q,R).
          • We can at least state that on the sphere--but is it what we want "between" to mean?
          • Topic for discussion next time!


  • Dr. Anneke Bart took over the class, as I was away at Notre Dame for a relativity conference.
    • Discussion of the axiomatic approach to geometry, with emphasis on the differences between
      • the plane and
      • the sphere.