# Course:Harris, Fall 07: Diary Week 16

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- We talked about "multiplication" of symmetries.
- For instance, if R is a rotation and F is a flip, then RF means applying first the rotation and then the flip.
- In a rosette, we labeled the corners of the polygon, then applied each transformation.
- In a frieze, we labeled the top of the figures, then applied each transformation.

- For instance, if R is a rotation and F is a flip, then RF means applying first the rotation and then the flip.
- We talked about contrasting the three geometries we studied:
- similarities:
- vocabulary of
- objects: points, lines (geodesics), angles, polygons
- measurements: distance, angle-measure, area

- vocabulary of
- differences:
- relationships of measurements (such as angle-sums)
- relationships with parallel lines/geodesics

- similarities:
- We looked briefly at more general kinds of geometry.
- Our usual three geometries are uniform:
- spherical = uniform positive curvature
- flat = zero curvature
- hyperbolic = uniform negative curvature

- Many surfaces have varying curvature:
- An egg has one end (the pointy one) which has higher positive curvature than anywhere else on the egg.
- A golf ball has high positive curvature in each dimple and negative curvature around the edge of a dimple.

- Our usual three geometries are uniform:
- We considered ways in which math is used in art and art is used for math:
- Escher and Coxeter:
- Escher got help from Coxeter for hyperbolic regular tessellations.
- Coxeter got illustrations from Escher on depicting hyperbolic tessellations.

- Rennaisance workers with mathematically refined perspective, such as Michaelangelo and Bruneschelli

- Escher and Coxeter:
- In general, mathematics abstracts simplified notions that can be rigorously studied, notions derived from
- nature
- art
- science
- anything of human interest

Fri:

- gave exam; no students took more than 75 minutes, some finishing in 45 minutes
- found out that this Wiki site had gone down the evening before, right when students were perhaps doing their last-minute studying
- seems a "network switch" at the university failed
- to compensate for lack of studying ability, will modify grading scheme on final exam:
- initial grade levels: 84, 71, 58, 45 (i.e., 84 = A/B borderline, 71 = B/C borderline, etc.)
- final grade levels: 82, 67, 52, 37
- net effect, on average, was to add 1/4 grade level to students' final exam grades