Math 331 - Transformation Geometry: Syllabus and Homework

Syllabus

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

All problems are worth 10 points unless otherwise specified.

• Due on 2/5 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 2/12
  • 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.
  • Show that a linear isometry f preserves the Euclidean inner product, that is, (f(x),f(y))=(x,y). Start by rewriting the isometry property in terms of the scalar product. Then use the bilinearity of the latter, as well as the property ||f(x)||=||x||, that we proved.
• Due on 2/19
  • 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 2/26
  • 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 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 3/5 page 41/5.9
• Due on 3/12 page 50-51/6.3, 6.16
• Due on 3/31 page 61/7.10, page 68/8.2
• Due on 4/2 page 76/9.8
• Due on 4/9 page 146/13.27
• Proofs for the second test: the conjugate of a rotation by a reflection sigma_l rho_{C,theta} sigma_l is rho_{sigma_l(C),-theta} (p. 55, the 8 lines after fig. 7.2); the composition of three reflections sigma_r sigma_q sigma_p is a glide reflection when p and q intersect, but r does not pass through their intersection point (p. 63, the 6 lines before Theorem 8.2, and p. 64, the paragraph after Fig. 8.3); Leonardo's theorem: all the even isometries in a finite group of isometries are of the form 1, rho, ..., rho^n (p. 66); dilatations are either translations or dilations (p. 137-139, from Theorem 13.2 to 13.5, exclusive). The problems 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.
• Due on 4/23 page 164/14.4; and: 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 4/30 page 165/14.26 (see hint at back of book).
• Due on 5/5 page 197/16.7, 16.14