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Understanding Wheel Symmetry: Key Concepts and Properties Explained

Wheel symmetry, a vital facet of geometry, includes rotational and reflective symmetries primarily associated with circles. With 360 degrees in a circle, essential elements include radii, diameters, and the circle's center. Two main types of symmetry are explored: **Cyclic** groups, involving only rotation, and **Dihedral** groups, which incorporate both rotation and reflection. The notation for these groups, such as C4 for four 90-degree rotations, serves as a fundamental classification. Examples include automobile hubcaps displaying D5 and C7 symmetries, illustrating how these concepts manifest in everyday designs.

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Understanding Wheel Symmetry: Key Concepts and Properties Explained

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  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?

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