1. Wheel Symmetry What you need to know to understand this type of symmetry

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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.

10. 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)

11. 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?

12. Which symmetry groups are seen below?