MSM 705
Notes and exercises for section 4.3: Symmetry
A symmetry of a set T is an isometry that takes T onto itself.
A figure has reflection symmetry if there is a reflection isometry that maps the figure onto itself. More
informally, a figure has reflection symmetry is there is a line (sometimes called the mirror line or the line of
symmetry) that the figure can be “folded over” so that the halves exactly match.
Example 1: For each figure below, find all lines of reflection symmetry, if any. Identify the lines by sketching
them on the figure.
A figure has rotation symmetry if there is a rotation isometry that maps the figure onto itself. More
informally, a figure has rotation symmetry is there is a center point about which the figure can be rotated by a
number of degrees (less than a full turn) so that the rotated image matches the original figure exactly. A figure
1
has an order n rotational symmetry if of a complete turn leaves the figure unchanged. An order n rotation
n
360 o
corresponds to an angle of rotation of
.
n
Example 2: The equilateral triangle has an order 3 rotational symmetry, which corresponds to an angle of
rotation of 120o. Another angle of rotation is 240 o.
Example 3:
angles.
Find rotational symmetry for the rest of the figures in Example 1. Identify the order and the
An image has translational symmetry if there is a translation isometry that maps the image onto itself. This type of
symmetry only exists for an image pattern that is assumed to continue infinitely.
Example 4: Assuming the pattern continues infinitely, this image displays translational symmetry in a vertical
direction and in a horizontal direction. There is also translational symmetry along the diagonal. Sketch vectors
to represent the translation symmetry.
Example 5: Sketch a vector to represent the translational symmetry of this pattern.
A pattern displays glide reflection symmetry if there is a glide reflection isometry that maps the pattern onto
itself.
Example 6: The patterns below can be mapped onto themselves by combining a translation and a reflection.
These patterns also have translation symmetry. The vector representing the translation symmetry is longer than the vector
representing the glide rotation.
Exercises
1)
2)
Answer the following questions for the figure below:
✬
a.
Does the figure have any lines of reflection symmetry? If so, sketch them on the figure and state how
many there are. If not, explain.
b.
Does the figure have any rotational symmetry? If so, give the angles of rotation. If not, explain.
Identify which letters of the alphabet in the font below have reflection symmetry, and identify the lines of
symmetry. Then identify which letters have rotation symmetry, and identify the angles of rotation.
A
H
O
V
3)
B
I
P
W
C
J
Q
X
D
K
R
Y
E F G
L M N
S T U
Z
For each pattern below, assume it continues infinitely in the plane. Does the pattern display
translational symmetry? Glide reflectional symmetry? If so, represent the translation with a vector,
and show the reflection line if applicable.
a.
b.
c.