Rotational Symmetry

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K-12: Materials at high school level.

If points on a figure are equally positioned about a central point, then we say the object has rotational symmetry. A figure with rotational symmetry appears the same after rotating by some amount around the center point.


The angle of rotation of a symmetric figure is the smallest angle of rotation that preserves the figure. For example, the figure on the left can be turned by 180° (the same way you would turn an hourglass) and will look the same. The center (recycle) figure can be turned by 120°, and the star can be turned by 72°. For the star, where did 72° come from? The star has five points. To rotate it until it looks the same, you need to make <math>1/5</math> of a complete 360° turn. Since <math>1/5 \times 360^\circ = 72^\circ</math>, this is a 72° angle rotation.

Using degrees to describe the rotation amount is inconvenient because the precise angle is not obvious from looking at the figure. Instead, we will almost always use the order of rotation to describe rotational symmetry:

Order of rotation 
A figure has order <math>n</math> rotational symmetry if <math>1/n</math> of a complete turn leaves the figure unchanged.

As with the star, you can compute the angle of rotation from the order of rotation:

An order <math>n</math> rotation corresponds to a <math>\frac{360^\circ}{n}</math> angle of rotation.

In the three examples above, the hourglass has order 2 rotation symmetry, the recycle logo has order 3 rotation symmetry, and the star has order 5 rotation symmetry.