Order of Rotation

Definition

The order of rotation refers to the number of times a shape can be rotated around a central point and still look the same within a full rotation of 360 degrees. This concept is closely linked to symmetry, as it helps identify the degree of rotational symmetry a shape possesses, whether in two-dimensional figures like polygons or three-dimensional objects like spheres and cubes.

5 Must Know Facts For Your Next Test

1. The order of rotation can be calculated by dividing 360 degrees by the angle of rotation that maps the shape onto itself.
2. For example, a square has an order of rotation of 4 because it can be rotated 90 degrees (360/90 = 4) and still appear unchanged.
3. In contrast, an equilateral triangle has an order of rotation of 3, as it matches its original position after rotating 120 degrees.
4. Three-dimensional shapes can also exhibit rotational symmetry, such as a sphere, which has infinite order since it looks the same from any angle.
5. Identifying the order of rotation helps in understanding more complex geometric properties and classifications based on symmetry.

Review Questions

• How do you calculate the order of rotation for a regular polygon, and what does this tell us about its symmetry?
  • To calculate the order of rotation for a regular polygon, you divide 360 degrees by the angle of rotation that brings it back to its original position. For instance, in a hexagon, which has an internal angle of 60 degrees, the order of rotation is 360/60 = 6. This calculation indicates that the hexagon exhibits rotational symmetry, reflecting how many symmetrical positions it can occupy when rotated around its center.
• Compare the order of rotation between two shapes: a rectangle and an equilateral triangle. What implications does this have for their respective symmetries?
  • A rectangle has an order of rotation of 2, as it can only match its original appearance after being rotated 180 degrees. In contrast, an equilateral triangle has an order of rotation of 3 because it returns to its original shape after rotating 120 degrees. This comparison shows that while both shapes possess rotational symmetry, the triangle's higher order indicates more symmetrical positions and thus richer symmetry properties than the rectangle.
• Evaluate how the concept of order of rotation applies to both two-dimensional and three-dimensional shapes. What are some examples that illustrate these differences?
  • The concept of order of rotation applies similarly in both two-dimensional and three-dimensional shapes but highlights different complexities. For instance, a two-dimensional shape like a regular pentagon has an order of rotation of 5 due to its five identical sides and angles. In contrast, a three-dimensional object like a cube possesses an order of rotation of 24 because it can be rotated in various ways across multiple axes while appearing unchanged. This difference illustrates how rotational symmetry is assessed in different contexts, emphasizing the broader implications in geometry.

Related terms

Rotational Symmetry: A characteristic of a shape that looks the same after a certain amount of rotation around a central point.

Central Point: The fixed point around which a shape is rotated to determine its order of rotation.

Angle of Rotation: The degree measure through which a figure is rotated around a central point to achieve the same appearance as its original position.