10.2: Connecting Transformations and Symmetry
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Humans have long associated symmetry with beauty and art. In this section, we define symmetry and connect it to rigid motions.
A symmetry of an object is a rigid motion that moves an object back onto itself.
There are two categories of symmetry in two dimensions, reflection symmetries and rotation symmetries.
A reflection symmetry occurs when an object has a line of symmetry going through the center of the object, and you can fold the object on this line and the two halves will “match.” An object may have no reflection symmetry or may have one or more reflection symmetries.
A rotation symmetry occurs when an object has a rotocenter in the center of the object, and the object can be rotated about the rotocenter some degree less than or equal to 360° and is a “match” to the original object. Every object has one or more rotation symmetries.
D-Type Symmetry: Objects that have both reflection symmetries and rotation symmetries are Type \(D_n\) where \(n\) is either the number of reflection symmetries or the number of rotation symmetries. If an object has both reflection and rotation symmetries, then it is always the same number, \(n\), of each kind of symmetry.
Z-Type Symmetry: Objects that have no reflection symmetries and only rotation symmetries are Type \(Z_n\) where \(n\) is the number of the rotation symmetries.
Identify the reflection and the rotation symmetries of the pentagon. The five dashed lines shown on the figure below are lines of reflection. The pentagon can be folded along these lines back onto itself and the two halves will “match” which means that the pentagon has a reflection symmetry along each line.
Figure \(\PageIndex{1}\): Reflection Symmetries of a Pentagon
Also, there are five vertices of the pentagon and there are five rotation symmetries. The angle of rotation for each rotation symmetry can be calculated by dividing 360° by the number of vertices of the object: . So, if you rotate the upper vertex of the pentagon to any other vertex, the resulting object will be a match to the original object, and thus a symmetry.
Figure \(\PageIndex{2}\): Rotation Symmetries of a Pentagon
72° 144° 216°
288° 360°
When an object has the same number of reflection symmetries as rotation symmetries, we say it has symmetry type Therefore, the pentagon has symmetry type because it has five reflection symmetries and five rotation symmetries.
Identify the rotation and the reflection symmetries of the smiley face. There is one line of reflection that will produce a reflection symmetry as shown below, and the only rotation symmetry is 360°, also shown below.
Figure \(\PageIndex{3}\): The Smiley Face has Symmetry Type
360°
Figure \(\PageIndex{4}\): Some Letters with Symmetry Type
The following letters are all examples of symmetry type since they each have only one axis of reflection that will produce a symmetry as shown below, and they each have only one rotation symmetry, 360°.
B C A E T
Identify the rotation and reflection symmetries of a pinwheel.
Figure \(\PageIndex{6}\): There are Five Rotation Symmetries of the Pinwheel
We find the angle by dividing 360° by five pinwheel points;. The rotation symmetries of the pinwheel are 72°, 144°, 216°, 288°, and 360°.
72° 144° 216°
288° 360°
When an object has no reflection symmetries and only rotation symmetries, we say it has symmetry type . The pinwheel has symmetry type .
Identify the rotation and the reflection symmetries of the letter S.
Solution
Figure \(\PageIndex{7}\): The Letter S
There are no reflection symmetries and two rotation symmetries; 180° and 360°, therefore the letter S has symmetry type
S S 180° and 360°
Identify the rotation and reflection symmetries of card the eight of hearts.
Solution
The card shown below has no reflection symmetries since any reflection would change the orientation of the card. At first, it may appear that the card has symmetry type . However, when rotated 180°, the top five hearts will turn upside-down and it will not be the same. Therefore, this card has only the 360° rotation symmetry, and so it has symmetry type .
Figure \(\PageIndex{9}\): A Design and the Letter K
a. b. K
a. The design has symmetry type , no reflection symmetries and six rotation symmetries. To find the degrees for the rotation symmetries, divide 360° by the number of points of the design: . Thus, the six rotation symmetries are 60°, 120°, 180°, 240°, 300°, and 360°.
b. The letter K has symmetry type , no reflection symmetries and one rotation symmetry (360°).