Course:Harris, Fall 07: Diary Week 14

Mon:

• Handed back Exam II.
  • Most difficulty was with explaining why there are only three regular tessellations of the plane.
    • Biggest hint: You have to know what the formula is for the angle in a regular n-gon (or derive the formula from what you know for the angle-sum of an n-gon).
  • For final exam: Explain why there are only five regular tessellations of the sphere.
    • Biggest hint: Same as above--except it's an inequality you get.
• examined concept of "how much infinity is out there" in the three geoemtries, using triangular tessellations
  • Sphere, with {3,5} regular tessellation (or any other!), eventually has no neighbors at large enough distances from a given point.
  • Plane, with {3,6} regular tessellation, has 4n neighbors at distance n from a given point.
  • Hyperbolic plane, with {3,7} regular tessellation, has 7, then 21 neighbors at distances 1, then 2, from a given point.
• Groups started in on Three Geometries Exploration, didn't finish.
• Discussion for next time: What is the shape of the universe?
  • possibilities:
    • spherical?
    • flat?
    • hyperbolic?
    • other? (what other?)
  • How might we tell?
  • Is this a sensible question?
    • How could you tell?

Wed:

• Groups finished up Three Geometries Exploration (20-25 minutes).
• Class discussed, "What is the shape of the universe?"
  • issue of dimension: What does 3-dimensional spherical or hyperbolic geometry mean?
    • This can be answered in terms of how much is "out there" at a given distance from a given point:
      • For spherical, far enough away, the universe shrinks back to a point
        • The universe plus (divine) heavens is given a three-dimensional spherical geometry in Dante's Paradiso)
      • For hyperbolic, the universe expands more quickly than Euclidean space does.
  • Could experiments help us to tell?
    • If the universe is finite in extent, then it can't be flat (i.e., Euclidean) or hyperbolic.
      • But what experiment would reveal finiteness of extent?
    • Experiments with triangles could distinguish among spherical, flat, and hyperbolic by measuring angle-sum.
      • Bigger triangles are better (more sensitive measurement).
      • What can be used for geodesics?
        • Maybe lasers.
    • Maybe experiments with parallel geodesics could be done:
      • Spherical geometry has no parallel geodesics.
      • Flat geometry has parallel geodesics that stay a constant distance apart.
      • Hyperbolic geometry has parallel geodesics, but none of them stay a constant distance apart.
  • Exercise (due Monday):
    • 1-2 page paper exploring these issues:
      • What are possible answers to "What is the shape of the universe?"
      • What are possible experiments that could help us know?
      • Is this necessarily a sensible question, and what might that mean?
    • If any student is interested in doing so, this could be expanded to something more substantial.
• Bring Visions of Symmetry for Dilation Exploration Friday.

Fri:

• Mentioned similarity transformations:
  • preserves angles
  • multiplies distances between points by a constant factor
  • has a central point of expansion or shrinkage
  • can incorporate isometries (rotations, reflections, etc.) in addition to shrinking/expanding
• Fractals are "self-similar" diagrams: Some similarity transformation leaves the diagram unchanged.
  • approximate examples in nature:
    • wrinkliness in sea shore (expanding the picture by multiple factors keeps it still looking just as wrinkled)
    • structure of a cloud (at all scales, it has same degree of fuzziness)
    • structure of a tree or leaf (repeated division and branching at finer and finer levels)
    • alveoli in the lungs (repeated division and branching at finer and finer levels)
• Fractal images are extremely easy to create with computer graphics, hence, an easy way to make invented landscapes or trees look natural.
• Groups did Similarity Exploration and nearly finished Iteration Exploration