Course:Harris, Fall 08: Diary Week 13

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Mon:

  • Exam II


Wed:

  • Class generally did not do well on the "explain" problems, 3 and 6, of the exam.
    • Students may gain extra credit (likely applied to the Exercises section of grades) by submitting corrections on those problems. Come see me before you submit.
  • The Poincare disk has "ideal points": the points on the boundary (not really part of the hyperbolic model).
    • Any two ideal points determine a unique geodesic that goes from one to the other.
    • If P, Q, and R are ideal points (i.e., on the boundary), then the geodesics PQ and PR form a {0}^{\circ } angle at P.
      • This can be seen by remembering that both PQ and PR must intersect the boundary at 90^{\circ }.
    • An ideal polygon is formed from geodesics (not just geodesic segments, but entire geodesics) which come together at ideal points.
      • So all angles in ideal polygons are {0}^{\circ }.
      • So all ideal triangles have defect = 180^{\circ }.
      • That implies all ideal triangles have the same area, \pi .
  • Groups took 20 minutes to do the Hyperbolic Ideal Tessellations Exploration.


Fri:

  • I mentioned that differences among the three geometries would be a primary theme of the end of the course.
    • We used KaleidoTile to examine what happens when trying for regular tesselations of various Schlaefli types, {n,k}:
      • In KaleidoTile, one can pick 3 numbers for "Choose Symmetries"
        • The first is best left at 2 (I don't know what it does).
        • If the point in the control triangle is very close to the left-hand corner, then
          • the second number is n = # of sides in the regular polygon chosen for regular tessellation, and
          • the third number is k = # of polygons around a vertex.
      • We found that with n = 3 (triangles), we got
        • for k = 3, it tesselates a sphere (tetrahedral)
        • for k = 4, it tesselates a sphere (octohedral)
        • for k = 5, it tesselates a sphere (icosohedral)
        • for k = 6, it tesseallates the plane
        • for k = 7 or higher, it tessellates hyperbolic space.
          • We could tell that was hyperbolic because
            • The triangles had inward-bending sides (i.e., angle-sums were less than in the plane).
            • For a given line L and a point P not on L, there were multiple lines through P, all of them parallel to L.
    • Conclusion: Hyperbolic geometry can tolerate high numbers of regular polygons around a vertex, but not the plane or the sphere; the sphere takes the lowest number of regular polygons around a vertex.
  • Groups took 25 minutes to do the Three Geometries Exploration (needing some assistance with isometries).
    • We largely omitted semi-regular tessellations.
  • Next week, Monday will be devoted to trying to create with paper and tape that tessellation of 7 regular triangles around a vertex, thereby building ourselves a hyperbolic plane.