# Course:Harris, Fall 07: Diary Week 16

Mon:

• 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.
• We talked about contrasting the three geometries we studied:
• similarities:
• vocabulary of
• objects: points, lines (geodesics), angles, polygons
• measurements: distance, angle-measure, area
• differences:
• relationships of measurements (such as angle-sums)
• relationships with parallel lines/geodesics
• 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.
• 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
• 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