Course:Harris, Fall 08: Diary Week 10

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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 - <math>\pi</math> (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 <math>\pi</math> radians) at the vertex V.
        • Thus, angle-sum = 2A + 180 (degrees) or 2A + <math>\pi</math> (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,
          • <math>\frac{area(triangle)}{area(sphere)} = \frac{A}{360}</math> (degrees) or
          • <math>\frac{area[triangle]}{area[sphere]} = \frac{A}{2\pi}</math> (radians)
      • Continuing in radian measure: Use <math>A = \frac{1}{2}defect</math>:
        • <math>\frac{area[triangle]}{area[sphere]} = \frac{defect}{4\pi}</math>
        • <math>defect = 4\pi\frac{area[triangle]}{area[sphere]}</math>
      • Now use the fact that for a sphere of radius R, <math>area[sphere] = 4\pi {R^2}</math>:
        • <math>defect = 4\pi\frac{area[triangle]}{4\pi {R^2}}</math> or
        • <math>defect = \frac{1}{R^2}area[triangle]</math>
      • 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.