Wheel Symmetry

What you need to know to understand this type of symmetry

Basis of these symmetry groups

• Circles– How many degrees in a circle?– Importance of the radii and the diameters?– The role of the center of the circle

Basic Properties of Circles

• All circles comprise 360 degrees

• A radius is a line segment with one endpoint on the circle and the other endpoint at the center of the circle

• A diameter is a line segment whose endpoints are on the circle and intersects the center of the circle

One type of symmetry that occurs

• Rotational Symmetry– There must be an angle that the shape is rotated

through and a point about which the angle is centered

• The angle of rotation is the angle between two radii of a circle

• The center of rotation is ALWAYS the center of the circle

A possible type of symmetry

• Reflective Symmetry– A line that acts as a mirror may be present– This line must be a diameter of the circle

Classifying the Symmetry Groups

• Only rotational symmetries are present– These groups are called CYCLIC– Each of the rotations are by the same number of

degrees

Classifying the Symmetry Groups

• Rotational and reflective symmetries are present– These groups are called DIHEDRAL– All rotations are by the same degree

measurement – There is a mirror along each rotational radius

and halfway between each radius– There are the same number of mirror lines as

rotations

Notation to represent the groups

• Cyclic groups– Named by the number of rotations

• Four 90 degree rotations: C4

• Ten 36 degree rotations: C10

• Dihedral groups– Named by the number of rotations (Note: there are the

same number of reflection mirrors)• Three 120 degree rotations and three lines of reflection: D3

• Six 60 degree rotations and six lines of reflection: D6

Examples of Wheel Symmetry

• The picture to the right is of an automobile hubcap. It represents a wheel symmetry called D5.

• There are five rotational symmetries and five lines of reflection.

Examples of Wheel Symmetry

• This hubcap is an example of a C7 symmetry

• There are seven rotations each measuring 360/7 degrees (or 51 3/7 degrees)

Examples of Wheel Symmetry

• This hubcap is an example of a D8 symmetry

• Do you see the eight 45 degree rotations and the eight lines of reflection?

Which symmetry groups are seen below?