Math 444/487 Geometry

Week 9

Monday 11/26 - Friday 11/30

    444 Home Page

  1. Week 10/01-05

  2. Week 10/08-12

  3. Week 10/15-19

  4. Week 10/22-26

  5. Week 10/29-11/2

  6. Week 11/05-09

  7. Week 11/12-16

  8. Week 11/19-23

  9. Week 11/26-30

  10. Week 12/03-07

  11. Week 12/10-14

    12.

Monday 11/26 444

  • Proof that 3 noncollinear points determine an isometry.
  • Every isometry is the composition of 1, 2, or 3 line reflections.
  • Definition of glide reflections.
  • Proof that line reflection followed by half-turn = A_180R_m is a glide reflection.
  • Assignment Due Wed, 11/28

Wednesday 11/28 444

Reading: You should aim to finish reading Brown, Chapter 2 by the end of the week.

  • Proof that a composition of 3 line reflections (if the lines are not concurrent or parallel) is a glide reflection. Case proved in class when 3 lines form a triangle. Case when 2 lines are parallel left as exercise.Proof follows from "mirror adjustment" so that triple product is seen to be equal to the composition of a line reflection and a half turn.
  • Given 2 rotations A_u and B_v, showed construction using "mirror ajustment" of the center C of the rotation C_(u+v) = B_vA_u. This means that a transformation given as the compostion of 4 line reflections can also be expressed as the composition of 2 line reflections.
  • Started in-class analysis of a tesselation of the plane by squares and non-regular octagons.

 

Wednesday 11/28 487 Lab

Lab has two parts.

  • Part A is about Glide Reflections.
  • Part B generates symmetric patterns with 90-degree rotations.

Friday 11/30 444

Quiz 2. This will be a Quiz on transformations. Important points will include

  • definitions from Brown
  • conceptual analysis of ST, knowing isometries S and T
  • construction of geometric data for ST
  • construction of ST(P) for some P.

Study Guide. Follow this link to Math 444 exams from last year. Look at the transformation questions on Major Quiz 2 and the Final Exam. Follow these links for hints and comments.