Assignment 8 (Due Monday, 11/9)
This weeks begins a new focus for the course. We will spend about three weeks beginning a study of transformations of the plane. This will bring in some new ideas beyond similar figures. The objects we study will sometimes be the motions themselves. On the other hand, transformations are a special kind of functions on the plane. You have studied such functions in multivariable calculus and linear algebra, so they are familiar in other contexts, at least.
Reading: Read Bix, Section 7 very thoroughly.
Other problems: In addition to the assigned problems, you should understand 7.4, 7.12, and 7.13.
Email extra credit. (Yet another experiment trying to promote online dialog about geometry. ) Several problems are listed at the end which will provide extra credit if share the answers over the email list. Instructions are given with the problems.
Problems to Turn In (5 Problems -- 65 Points)
General Instructions for 8.1 and 8.2
The answers should be in the form of a drawing or sketch on plain or graph paper with the reflection lines (solid lines) and glide reflection lines (dashed lines) clearly marked. Also, indicate centers of rotation and draw arrows for translations.
8.1 List the symmetries (10 points).
Bix, Problem 7.1. Do problems: a), b), c), d), e). Follow the general instructions above.
Also, at the end of each problem, write down and Put in a BOX: (1) the total number of reflections, (2) the total number of other symmetries.
Note: Look for a pattern or patterns relating the number of isometries which are reflections to the number which are not. What is it?
8.2 List the symmetries (20 points)
Bix, Problem 7.2. Do problems a), b), c), d), e).
Follow the general instructions above.
8.3 Inverses of certain isometries (5 points)
Bix 7.5. Tell the answer and illustrate with a sketch..
8.4 Three Reflections Theorem and Related Facts (20 points)
The problems 7.8-7.11 develop a very important theorem called the Three Reflections Theorem. Two are given here. Two are given later for extra credit. Figures for this will be produced in lab.
8.5 Sketchpad symmetric patterns (10 points)
You will produce some symmetric "frieze" patterns in lab on Friday, 11/6.
Print out 3 examples which have the same group of symmetries of a), c), d), of 8.2. Labe each one clearly a, c, or d, and turn them in.
Extra Credit Questions
You get the credit if you are the first person who sends in a correct answer. To give other people a chance, you can only send in one email problem answer per 24 hour period. However, you can always comment on or correct answers (including your own) which are already sent to the list.
The first answer for a figure gets the credit. However, you may continue the discussion if you feel an answer is incorrect or incomplete; you will get credit if you are right.
8.1 Four Extra Credit Problems (5 points each message)
Answer the same question as 8.1 for one of the figures l), n), o), r) and send to the email list.
8.2 Five Extra Credit Problems (10 points each message)
Answer the same question as above for one of the figures f), g), h), j), l) and send to the email list. You will have to find words to describe the nature of the symmetries.
8.4 Two Extra Credit Problems