# 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 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
• 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.