Order of rotational symmetry

The order of rotational symmetry of a geometric figure is the number of times you can rotate the geometric figure so that it looks exactly the same as the original figure. You only need to rotate the figure up to 360 degrees. Once you have rotated the figure 360 degrees, you are back to the original figure.

Order of rotational symmetry of 1

Let us start with a shape that has an order of rotational symmetry of 1. A rotational symmetry of order 1 means that the shape will look like its original only once after you rotated the shape 360 degrees. The arrow you see above has a rotational symmetry of order 1.

Notice that you could not get the original arrow until you rotated the arrow 360 degrees.

You do not need to do 90 degrees rotation each time. You can rotate a figure any amount of degrees you like to see if you will get the original figure.

When the order of rotational symmetry is 2

Order of rotational symmetry of 2

In our example above, we rotated a rectangle 90 degrees each time. Notice that we were able to get the original shape twice. The first time we got the original image, we got it with a rotation of 180 degrees and the second time, we got it with a rotation of 360 degrees. Since we were able to return the original shape 2 times, the rectangle has rotational symmetry of order 2.

In fact, all rectangles have an order of rotational symmetry of 2.

Example of rotational symmetry of order 6.

Order of rotational symmetry of 6

In this last example above, we rotated a hexagon 60 degrees each time. Each 60 degrees rotation returns the original shape as you can see above. Since we were able to return the original shape 6 times, the hexagon has rotational symmetry of  order 6.

Other examples of order of rotational symmetry.

  • Each 90 degrees rotation of a square will return the original square, so a square has an order of rotational symmetry of 4. Notice that 4 times 90 degrees = 360 degrees.
  • Each 120 degrees rotation of an equilateral triangle will return the original equilateral triangle, so an equilateral triangle has an order of rotational symmetry of 3. Notice that 3 times 120 degrees = 360 degrees.
  • Each 45 degrees rotation of an octagon will return the original octagon, so an octagon has an order of rotational symmetry of 8. Notice that 8 times 45 degrees = 360 degrees.

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