Course:Harris, Fall 08: Diary Week 9

Mon:

• No class, Fall break

Wed:

• 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!

Fri:

• 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.