Course:Harris, Fall 08: Diary Week 10

Mon:

• Discussion: How are planar and spherical geometry different?
  • One interpretation: What are the different subject matters of the two geometries?
    • Example:
      • Planar geometry examines lines.
      • Spherical geometry examines great circles.
  • Second interpretation: When considering analogous terms, how are the terms related to one another differently in planar geometry than in spherical?
    • Example of analogous terms:
      • lines in the plane <--> great circles in the sphere
        • extent:
          • Lines are infinite.
          • Great cirlces are finite.
        • parallel:
          • Parallel lines exist in the plane.
          • Parallel great circles do not exist in the sphere.
  • The second interpretation is the deeper one, and that is the one we will most be concerned with.
  • But to get to that, you have to first know which terms in each geometry are to be considered analogous.
    • point <--> point
    • line <--> great circle
      • Both are called geodesics.
    • between <--> ???
      • We still need to consider what "between" ought to mean on a sphere.
    • line segment <--> great circle arc
      • Both could be called a geodesic segment.
    • polygon <--> polygon
      • Just be sure to express it in terms of geodesic segments joined at their endpoints, etc.
      • In particular, "rectangle" can be used for any quadrilateral with four right angles, in any geometry.
        • In the plane it is also a parallelogram, but that need not be part of the definition of the word.
• Can there be rectangles in a sphere?
  • Consider first a triangle on a sphere:
    • The spherical segments connecting the three points bow outwards (on the sphere) from the three line segments that connect the points going through the interior of the sphere.
    • Therefore, each of the angles on the spherical triangle is greater than the corresponding angle of the corresponding line-segment-triangle of those same three points.
    • Therefore, the angle-sum of the spherical triangle is greater than the angle-sum of the line-segment-triangle.
      • In other words, the angle-sum(spherical triangle) > 180 degrees.
      • And the further apart the points on the sphere are, the more bowed-out the spherical segments, and the greater the deviation from 180 degrees.
  • Since a quadrilateral can be divided into two triangles, this means
    • angle-sum(spherical quadrilateral) > 360 degrees.
  • Therefor, a spherical rectangle is impossible, as four right angles would mean angle-sum = 360 degrees.
• We had about 20 minutes left to do the Spherical Polygon Exploration.
• Preliminary sketches for the Escher Art Project were handed in, and the first Spherical Exercises were collected.

Wed:

• Preliminary sketches for the Escher Art Project were returned.
• We discussed defects of spherical triangles:
  • For a spherical triangle,
    • defect = angle-sum - 180 (for degrees)
    • defect = angle-sum - $\pi$ (for radians)
  • On a given sphere, defect is larger for larger triangles:
    • For smaller triangles, there is less difference between the spherical segments and the line segments between the vertices, so less bowing out of the spherical sides, so a closer correspondence between spherical and linear angles, so a closer correspondence between spherical and linear angle-sum.
  • More specifically, defect is proportional to area of the triangle.
  • We can see what the proportionality constant is by concentrating on biangular triangles:
    • A biangle can be considered a triangle by using the two vertices of the biangle (call them North and South Pole) and a third vertex V in one of the two sides.
    • This yields the three angles as the same angle A at North Pole and South Pole and 180 degrees (or $\pi$ radians) at the vertex V.
      • Thus, angle-sum = 2A + 180 (degrees) or 2A + $\pi$ (radians).
      • So defect = 2A (in either case).
    • The ratio of the biangle to the entire sphere is the same as the ratio of the angle A to one revolution (best seen by looking from the vantage point of North Pole).
      • In other words,
        • ${\frac {area(triangle)}{area(sphere)}}={\frac {A}{360}}$ (degrees) or
        • ${\frac {area[triangle]}{area[sphere]}}={\frac {A}{2\pi }}$ (radians)
    • Continuing in radian measure: Use $A={\frac {1}{2}}defect$:
      • ${\frac {area[triangle]}{area[sphere]}}={\frac {defect}{4\pi }}$
      • $defect=4\pi {\frac {area[triangle]}{area[sphere]}}$
    • Now use the fact that for a sphere of radius R, $area[sphere]=4\pi {R^{2}}$:
      • $defect=4\pi {\frac {area[triangle]}{4\pi {R^{2}}}}$ or
      • $defect={\frac {1}{R^{2}}}area[triangle]$
    • We've shown this only for biangular triangles, but it's true generally.
• We considered isometries of spheres:
  • Isometry means a rigid motion, preserving
    • distances and
    • angles.
  • Isometries of the plane:
    • translations (by some distance along a direction)
    • rotations (around a point)
    • reflections (across a line)
    • glide-reflections (by some distance along an axis, a line)
  • Isometries of the sphere:
    • translations = rotations (by some angular amount around a point = along a great circle)
    • reflections (across a great circle)
    • gilde-reflections (by some angular amount along an axis, a great circle)
• Groups started the Spherical Isometries Explorations, not quite finishing it in 20 minutes.

Fri:

• Art Project due in one week (Friday, Nov. 7)
  • Finished art work:
    • no grid lines
    • excellence of execution counts
  • Report:
    • What group of symmetries was used?
      • Include a listing of types of symmetries in the work.
  • Comparisons with specific Escher tessellations are good.
  • How did your choice of group of symmetries effect the finished artwork?
  • How did your artistic vision effect your choice of group of symmetries?
• Why are great circles the geodesics of spheres?
• Finishing up the Spherical Isometries Exploration
• Betweenness on spheres.