Math 331
Exam 3
Spring 2013
The following are wallpaper patterns. On each one, indicate the following
with colored ink:
Shortest translations, v and w , in two dierent directions, that preserve the pattern and form the boundary of a fundamental region R
Math 331
1. Let
Exam 1 Solutions
Spring 11
be the line y = 3 x + 2. Compute 0, explicitly.
3
Solution: Write 0, = m n with m . Thus,
3
slope(m) = slope( ) = 3.
Since m n = 0, m is the line y = 3 x. Since 0, = m n ,
3
the directed angle from n to m is , so
Math 331
Exam 1 Solutions
Fall 2012
1
2
1. Let be the line y = 3 x 3 . Write = (0, ) in standard
3
form (i.e., as a translation, rotation, reection, or glide reection in
standard form).
Solution: Write (0, ) = n m with m . Then the slope of m
3
1
equals t
Math 331 Exam 1 Solutions Spring 2013
1. Let be the line y = 3x 2. Let = (0, ) . Write in standard
3
form (i.e., as a translation, rotation, reection, or glide reection in
standard form).
Solution: Write
(1.1)
(0, ) = m n
3
with m .
Then m n = 0, so m is
Math 331
Exam 2
Spring 10
1. The following are rosette patterns. On each one, indicate the
following with colored ink:
all lines of symmetry;
the shortest rotation preserving the pattern. Write the angle
of rotation beside the arrow.
Below the gure, wri
Math 331
Exam 2
Spring 11
1. The following are rosette patterns. On each one, indicate the
following with colored ink:
all lines of symmetry;
the shortest rotation preserving the pattern. Write the angle
of rotation beside the arrow.
Below the gure, wri
Math 331
Exam 2
Spring 2012
1. The following are rosette patterns. On each one, indicate the following
with colored ink:
all lines of symmetry;
the shortest rotation preserving the pattern. Write the angle of
rotation beside the arrow.
Below the gure, w
Math 331
Exam 2
Spring 2013
1. The following are rosette patterns. On each one, indicate the following
with colored ink:
all lines of symmetry;
the shortest rotation preserving the pattern. Write the angle of
rotation beside the arrow.
Below the gure, w
Math 331
Exam 3
Spring 10
The following are wallpaper patterns. On each one, indicate the
following with colored ink:
Shortest translations, x and y , in two dierent directions, that
preserve the pattern and form the boundary of a fundamental
region R fo
Math 331
Exam 3
Spring 11
The following are wallpaper patterns. On each one, indicate the
following with colored ink:
Shortest translations, v and w , in two dierent directions, that
preserve the pattern and form the boundary of a fundamental
region T fo
Math 331
Exam 3
Spring 2012
The following are wallpaper patterns. On each one, indicate the following
with colored ink:
Shortest translations, v and w , in two dierent directions, that preserve the pattern and form the boundary of a fundamental region R
Math 331
Exam 1 Solutions
1
1. Let be the line y = 3 x +
Solution: We write
2
.
3
Spring 10
Compute 0, explicitly.
3
0 , = m n ,
3
with n| , m n = 0, and the directed angle from n m equal
1
to . Thus, n is the line y = 3 x and m is the x-axis.
6
Since n|