Course:Harris, Fall 08: Diary Week 8

Mon:

• I handed back Exam 1. Common problems:
  • Spotting reflections in a frieze pattern (particularly if no other symmetries, other than translations).
  • Spotting glide-reflections in a wallpaper pattern (particularly if no other symmetries, other than translations or rotations).
• For anything (like corrected exercises, etc.) to count on the mid-term grades, it must be in by Friday.
• Class should have done the Polygonal Tessellation Exercises for today, but no one did; get them in by Wednesday.
• We looked in some detail at Tessellations by Recognizable Figures, as these (plus the associated Exploration) are the means by which to create the Art Project, coming up very quickly.
• We devoted some 35 minutes to Escher-Like Tessellations Explorations, left over from last week.
  • It appears that Geometer's Sketchpad is rather laborious for using these techniques, though it gives the most accurate result (assuming one can print it--been having difficulty there).

Wed:

• Tessellation Art Project
  • timeline:
    • First attempt at sketches is due this Friday.
      • I'd like several from each person; we'll talk about them in class.
    • The "official" preliminary sketches are due next Friday.
      • I want at least two ideas, preferably three or four, from each person.
      • I'll comment upon them; the goal is that at least one will be acceptable for you to proceed upon for your project.
    • The final project is due two weeks thereafter.
  • artistic content:
    • This is to be Escher-like:
      • It's a tessellation by recognizable figures (i.e., not just polygons).
      • The spaces between figures are themselves figures.
    • Pay some attention to artistic interest in the pattern (p4m is probably not a good idea, too bland).
    • Artistic vision counts.
    • Artistic execution counts:
      • good materials
      • careful execution
  • mathematical content (written report!):
    • Identify
      • the symmetries present.
      • the symmetry group.
    • Discuss how your artistic vision influenced your choice of pattern.
    • Discuss how your choice of pattern influenced the art.
• Spherical Geometry
  • This marks the beginning of our exploration of geometries other than the Euclidean plane.
  • We spent about 35 minutes on Spherical Easel Exploration.
    • The program uses radian measure:
      • A full circle has 2$\pi$ radians, so
      • $180^{\circ }=\pi$ radians, $90^{\circ }={\frac {\pi }{2}}$ radians
    • For "rectangle" read "quadrilateral with four right angles".
  • What the program labels as "spherical lines":
    • They are potential equators (if we just rotate the sphere appropriately).
    • They each divide the sphere equally in two.
    • They are circles.
    • They each have diameter = sphere diameter.
      • Thus, they are the largest possible circles on the sphere, called great circles.
    • So why are they called spherical lines?
      • What similarities do they bear to Euclidean lines?
        • This will be a continuing theme to our discussions over the next few weeks.
      • What differences are there from Euclidean lines?
        • Not just "they're not lines", but how do they act differently than Euclidean lines, what specific differences?

Fri:

• On collecting the groups' spherical geometry explorations, I noticed that everyone seemed to think that it's impossible to have an angle of 90 degrees on a sphere. I don't know where this idea came from; it's pretty easy to see that two "spherical lines" can be put at any angle you wish.
• We spent a good deal of time looking at suggestions for the Escher art project.
  • The hardest thing to do is to make each line for a figure do double duty: It shapes a figure equally importantly on each side of the line.
    • When this fails to happen, either
      • One figure is seen as overlapping the one next to it, or
      • There is unused space ("background") between the figures.
    • So avoid both of those: no overlapping, no background space.
  • Some of the sketches used two different figures; this makes for a very striking appearance.
    • This is one possible approach to avoiding background space: Make that a second figure.
• We had 30 minutes left to do spherical explorations.
  • There was time to look only at Spherical Geometry Exploration.
    • Easy to spend too much time on the question of "between" without getting to the later questions.
      • We'll spend time on "between" in class.
  • Spherical Geometry: Polygons will be looked at next week (leaving us still one Exploration behind schedule, as we've been for a couple weeks now).
    • As a consequence, the Spherical Geometry Exercises won't be due till next week.
      • Start looking at them for Wednesday, anyway, so that we can talk about these items in class.