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 2/8 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 2/15
  • 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, for one of the 4 possible solutions).
• Test on 2/27 There will be 5 problems, and you need to choose 4.
  • (1) Reproducing one proof, out of the following (in the Steinberger lecture notes): Lemma 2.4.3 and Corollary 2.4.4, Proposition 2.5.1, Theorem 4.1.5.
  • (2) composition of translations and halfurns (practice problems on p.20-21 in the textbook).
  • (3) affine transformations (very similar to the problem solved in class today).
  • (4) mapping a triangle to a congruent one via a composition of at most 3 reflections (see homework problem 5.1).
  • (5) composing reflections, rotations, and translations by decomposing all of these into reflections (practice problems at the end of chapters 5 and 6).
• Due on 3/1 page 40/5.1 (read the text at the bottom of page 35 and top of 36); page 50/6.3.
• Due on 3/15 page 61/7.10, page 68/8.2.
• Due on 4/3 page 76/9.8.
• Due on 4/5 page 146/13.27, second and third questions (follow the general outline of the solution in class for the first question; 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 4/12
  • 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).
  • optional problem: 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.
• Practice problems for the second test (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; 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).
• Solutions to some practice problems: click here
• Proofs for the second test: 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); Theorem 14.1 except the last statement. The problems on the test will be concerned with: composition of 3 or more isometries (part of Chapter 6 and Chapter 7), glide reflections (Chapter 8), equations for isometries (Chapter 9), Menelaus + Ceva theorems and related properties of the triangle (Chapter 14). See the practice problems above.
• 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