Geometry
Chapter 10 Answers
Transformations and Patterns
The motion of a sailboat can be tracked by using a series of isometries.
Below, you will describe the isometries found in diagrams of sailboat movement.
In these diagrams, you will see roundabout paths to get from one place to
another. These are necessary in sailing to optimize the power of the wind.
Part A, Isometries
1. Go to the Match Racing Web site and scroll down to the first diagram after the title "Pre-start circling manoeuvres." Have you ever thought about how sailboats start a race? The sailors cannot just sail their boats to the starting line and stop until the race begins. One of the tactics for gaining optimal starting position is called "circling." One type of circling is described in this diagram.
a. Use transformations (reflections, rotations, and translations) to describe the series of isometries which occur from position #1 to position #5 for sailboat A. Be careful! Each of the shifts in position requires at least two different transformations. If the transformation is a reflection, describe the line of reflection. If it is a rotation, describe the center, angle and direction of rotation. If it is a translation, give the distance of the translation in terms of boat length, and describe the direction of the translation.
[From position #1 to #2 - translation one boat length forward followed by a rotation of 90° clockwise centered at the stern (back) of the boat; from position #2 to #3 - translation one boat length forward followed by a rotation of 90° clockwise centered at the stern; from position #3 to #4 - translation about 1 1/2 boat lengths forward followed by a rotation of about 3° clockwise centered at the stern; from position #4 to #5 - translation about 1 1/2 boat lengths forward followed by a rotation of about 3° clockwise centered at the stern.]b. Which of the three basic transformations which are isometries will probably not be used to describe the motion of a sailboat. Why?
[Reflections won't be used because sailboats can only go forward or turn. They cannot reflect themselves over an axis.]
Part B, Compositions of Transformations
2. Now, you will look at some instructions about steering a sailboat into the wind. Go to the Practical Aspects of Sailing Web site for basic instructions on sailing.
a. Scroll down to figure 23 titled "Beating Upwind." This diagram shows how to make forward motion when you are sailing into the wind. Describe the sailboat's path using the language of transformations as you did in #1a.
[A translation about 1 1/2 boat lengths forward followed by a rotation of 90° clockwise centered at the stern; another translation about 1 1/2 boat lengths forward followed by a rotation of 90° counterclockwise centered at the stern; another translation forward of about half a boat length.]b. Describe one transformation which is mathematically equivalent to the composition of transformations described above. Is this a viable path for the sailboat?
[A transformation of about three boat lengths directly into the wind. This is not possible for the sailboat because it would involve sailing at an angle to the boat's line of symmetry, which is extremely difficult. It would also involve the sailboat sailing directly into the wind, which is also extremely difficult.]
Islamic art is famous for its use of geometric patterns. The Alhambra, a
palace in Granada, Spain, has some of the most beautiful examples of symmetry
in Islamic art. It is decorated with carvings and tile patterns on most
of the ceilings, walls, and floors.
Part A, Frieze Patterns
1. Go to the Spain Web site to see a frieze pattern which occurs on the wall of the Alhambra.
a. Sketch a minimal piece that can be translated to create the pattern.
Answer:
b. What types of symmetry does this pattern have?
[horizontal translation, vertical translation (with a glide), reflection symmetry through a line 135° to the horizontal]
Part B, Wallpaper Patterns
2. Go to the Spain II to see a frieze pattern which could also cover the whole plane to become a wallpaper pattern.
a. What symmetries does the frieze pattern have?
[possibly translation symmetry (unable to determine this without more of the pattern - be careful with the overlapping sections in the center knots!); rotational symmetry through angles of 90°, 180°, and 270°; possibly vertical line symmetry (again unable to determine this without more of the pattern)]b. What symmetries would the wallpaper pattern have?
[possibly translation symmetry; rotational symmetry through angles of 90°, 180°, and 270°; possibly vertical and horizontal line symmetry]
3. Go to the Spain
III to see a wallpaper pattern in the Alhambra. What symmetries
does it have?
[translation symmetry; rotational symmetry through
angles of 90°, 180°, and 270°; line symmetry through vertical
lines, horizontal lines and lines at 45° and 135° to the horizontal
axis; glide reflection symmetry]
