Math 331 - Transformation Geometry: Syllabus and Homework

Syllabus

Textbook for most assignments: George Martin, Transformation Geometry. An Introduction to Symmetry

All problems are worth 10 points unless otherwise specified.

• Due on 2/1 Let f:R^3->R^2 and g:R^2->R^3 be given by f(e1)=e1+e2, f(e2)=2e1-e2, f(e3)=e2; g(e1)=e1-e2+e3, g(e2)=e1+3e2-2e3. Compute (fog)(3e1+2e2).
• Due on 2/8 Give an example of: 1) an isometry which is not a linear map; 2) a linear map which is not an isometry; 3) an affine map which is neither an isometry nor a linear map (you might combine the previous two examples). It is easiest to think of maps on the plane R^2 to R^2, and write down the formulas describing them (in terms of 2 by 2 matrices). Justify your choices.
• Due on 2/20
  • Prove that translations and halfturns map any line {x+tv : t in R} to a parallel line (same direction v).
  • page 21/3.10 (consider halfturns in R^n, and follow the hints in class).
• First test: 2/22. Proofs to know for the test (from the Steinberger lecture notes on the web, also covered in class): Lemma 2.4.3 and Corollary 2.4.4, Proposition 2.5.1, Theorem 4.1.5. Other problems on test based on: compositions of translations and halfturns (see textbook, pages 20-22/3.1 to 3.18), formulas for various transformations (examples and facts discussed in class).
• Due on 3/8
  • page 40/5.1 (read the text at the bottom of page 35 and top of 36).
  • Use a reflection to construct (with the ruler and compass) a quadrilateral given all the side lengths (arbitrary a,b,c,d) and the fact that a given diagonal bisects an angle (say the one formed by the sides a and b).
• Due on 3/22 page 50-51/6.3, 6.16; practice problem (not be handed in): page 41/5.9
• Due on 3/29 page 61/7.10, page 68/8.2.
• Due on 4/5 page 76/9.8.
• Optional practice problems (some were solved in class): page 60-61/7.1--7.14, 7.17; page 68-70/8.1--8.4, 8.7c--h, 8.9--8.11, 8.13, 8.14; page 76-77/9.1--9.12.
• Solutions to practice problems: click here
• Proofs for the second test: Theorem 7.5 part 1 (proof: see the paragraph before the statement, on p. 54); Theorems 6.6 + 6.11 (proof: the paragraphs before Theorems 6.4 and 6.9); Theorem 8.4 (proof: p. 63, the 6 lines before Theorem 8.2, and p. 64, the 2 paragraphs after Fig. 8.3); Theorem 13.3 + 13.5 (proof: p. 137-139, from Theorem 13.2 to 13.5, exclusive). The problems on the test will be concerned with: composition of 2 rotations or a translation and a rotation, composition of 3 or more isometries/glide reflections, equations for isometries. See the practice problems above.
• Due on 4/19 page 146/13.27 (use proof by contradiction to show that the commutation cannot happen: assume it does, and plug in a particular point for which the computation is easy).
• Due on 4/26
  • pick arbitrary D on the side BC of the triangle ABC, consider the bisectors of the two angles at D and their intersections E,F with sides AC and AB, then show that AD, BE, CF are concurrent (via Ceva's theorem).
  • 14.22 Hint: Assume that the tangent from A intersects side BC at D; I proved in class that the triangles ADB and ADC are similar; write down the three equal ratios, and use them to compute the corresponding ratio in Menelaus' theorem; then plug in the other ratios (just use similar formulas, without further computations), and show that the product is 1.
• Optional practice problems from chapter 14 pages 164-165/ 14.3, 4, 7, 9, 12, 15, 17, 18, 19, 20, 22, 23*, 24*, 25*, 26 (the problems marked with * are harder).
• Due on 5/3 page 197/16.7, 16.14
• Optional practice problems from chapter 16 pages 196-197/ 16.2, 3, 4, 6, 11, 13, 15, 20, 21, 22, 24.
• Solutions to practice problems from Chapters 14 and 16: click here