## Euclidean and Non-Euclidean Geometry

NOTE: The "Main Topics" entries are not exhaustive, and all entries referring to future dates are speculative.
Date Main topics and activities Written assignment for next day Reading assignment for next day
Thu 1/2 Hello. Course introduction, syllabus, and questionnaire. Transformations, isometries, congruences, symmetries (with practice at the board), compositions, inverses. Introduction to Euclid and I.1. Classwork: Beginning Euclid. #1-5 on symmetry handout -- Symmetry handout
-- Euclid: definitions, postulates, common notions, and I.1-2 (w/proofs)
Fri 1/3 Sticks and circles. Illustrating a geometric statement. Proposition I.2. Name That Isometry. Classwork: compositions of reflections. The multiplication table of the symmetry group of a square. Euclid I.4-10, with classwork. HW 2 statements of Euclid I.3-10 (except for I.7,9,10, reading the proofs is optional)
Sat 1/4 Building isometries in various ways as compositions. Orientation. Classwork. Straightness. Basic terminology for spheres. Geodesics. Classwork: Paths on spheres. -- HW 3
-- (from now on) corrections stapled to the back of graded HW
-- [HT]: Ch. 1-2
-- Euclid I.9-21 (statements)
Mon 1/6 Discussion of Euclid I.11-21, and classwork. Debating the validity of Euclidean postulates in spherical geometry. Some aspects of spherical geometry, and classwork. Euclid I.22-35 (statements)
Tue 1/7 Midterm 1. Archimedes. Some bonus solutions. Spherical isometries and circles, with classwork. Girard's theorem and its proof. HW 5 -- Statements up to I.35, and the proof of I.35
-- Java-enabled proof of Girard's theorem
-- skim pp. 143-157 and Problem 7.1 of [HT], ignoring hyperbolic geometry
-- nearing completion of Flatland
Wed 1/8 Review of classwork/homework problems involving circumferences of spherical circles (like lines of latitude). Euclid I.27-34 (the basic theory of parallels) and I.35 (the first theorem on area). Archimedes' theorem for the area of a spherical circle. Classwork. The failure of some Euclidean triangle theorems for spherical triangles. Some theorems of Archimedes about spheres and cylinders. Consequences and extensions of Girard's theorem. Classwork. Ch. 6 and 9 in [HT]
Thu 1/9 Midterm 2. Movie and music. Platonic solids. Classwork. Discussion of various points from classwork and lecture. Holonomy. Regular tilings. Classwork. -- HW 6, including the one-page essay on Flatland
-- a 5-7-5 haiku about your favorite Platonic solid (on a separate sheet)
-- Statements of I.36-48, and the proof of I.47
-- finish Flatland
Fri 1/10 Review of tilings. Euclid I.36-48, the proof of the Pythagorean theorem, and some concluding comments on Euclid. Curvature on curves, with classwork practice. Holonomy, uniform curvature, and generalizations of Girard's theorem. Gaussian curvature. Gauss's Theorema Egregium, with applications to giftwrapping, maps, and pizza. The development of noneuclidean geometry, including some differences between euclidean, spherical, and hyperbolic geometry. HW 8 [the triangle pattern is here] -- Handout on Janos Bolyai
-- Handout on non-euclidean geometry
Sat 1/11 Final exam. Poetry slam and prize. Flatland: movie clips and discussion. Other proofs of the Pythagorean theorem, its spherical version, Pythagorean triples, and Fermat's last theorem. The Shape of Space and the unknown geometry of the universe. Redos may be submitted to my mailbox by noon Monday 1/13

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