Math 331 - Transformation Geometry: Syllabus and Homework

Syllabus

Annuncements:

• Due to the missed classes, the two tests are pushed back by one week, so they will be on March 2 and April 13.
• For reflections in R^n, I refer to the following source .

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

All problems are worth 10 points unless otherwise specified.

• Due on 2/9
  • 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).
  • Show that, given a linear map f, we have: (1) f is an isometry if and only if (2) f preserves the Euclidean inner product, that is, (f(x),f(y))=(x,y). You need to prove both implications. For (1) implies (2), start by rewriting the isometry property in terms of the scalar product, and then use the bilinearity of the latter. For (2) implies (1), replace both x and y with x-y in (2).
  • 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-->R^2, and write down the formulas describing them. Justify your choices.
• Due on 2/23
  • Prove that translations and halfturns map any line {x+tv : t in R} to a parallel line (same direction v).
  • page 21/3.10
• Due on 3/2
  • 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 (a,b,c,d) and the fact that a given diagonal bisects an angle (say the one formed by the sides a and b).
• Proofs for the test (from the Steinberger lecture notes on the web): 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 page 20-21/3.2 -- in class, 3.10 -- homework), formulas for various transformations (examples in class), geometric constructions based on reflections (examples in class and homework), compositions of reflections (examples in class and homework).
• Due on 3/23 page 41/5.9
• Due on 3/30 page 50-51/6.3, 6.16
• Due on 4/6 page 61/7.10, page 68/8.2, page 76/9.8.
• Due on 4/13 page 146/13.27
• 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); Theorem 14.1 -- the first two sentences (proof: the paragraph after the statement). 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 4/27 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).
• Due on 5/4 page 165/14.26 (see hint at back of book). You can assume the following two facts: (1) the sides of the tangential triangle are parallel to those of the orthic triangle of the original triangle (formed by the feet of the altitudes); (2) the mentioned altitudes are angle bisectors in the orthic triangle, so they intersect at the incenter. You get extra credit for proving these two facts (use quadrilaterals inscribed in a circle, and the fact that inscribed angles corresponding to the same arc are congruent).
• Optional practice problems from chapter 14 pages 164-165/ 14.3, 4, 7, 9, 12, 15, 17, 18, 19, 20, 22, 23*, 24*, 25* (the problems marked with * are harder).
• Due on 5/9 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