# Course:Harris, Fall 08: Diary Week 8

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- 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.

- First attempt at sketches is due this Friday.
- 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

- This is to be Escher-like:
- 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.

- Identify

- timeline:
- 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<math>\pi</math> radians, so
- <math>180^\circ = \pi</math> radians, <math>90^\circ = \frac{\pi}{2}</math> radians

- For "rectangle" read "quadrilateral with four right angles".

- The program uses radian measure:
- 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*.

- Thus, they are the largest possible circles on the sphere, called
- 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?

- What similarities do they bear to Euclidean lines?

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.

- When this fails to happen, either
- 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.

- 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.
- 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.

- Easy to spend too much time on the question of "between" without getting to the later questions.
- 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.

- As a consequence, the Spherical Geometry Exercises won't be due till next week.

- There was time to look only at Spherical Geometry Exploration.