# Rotational Symmetry

**K-12:**
Materials at high school level.

A shape with rotational symmetry is a shape that looks the same even if you turn the shape around a little bit. Another way to think about rotational symmetry is to notice in the following examples how we see several copies arranged around a central point.

Below the Spiderwort flower for instance has three leaves arranged around the center of the flower. The Spiderwort has 3-fold rotational symmetry (120 degrees). The baby starfish has 4 legs arranged around the center of the body. This baby starfish has 4-fold rotational symmetry (90 degrees) The red knobbed starfish shown here has five equally spaced legs. It has 5-fold rotational symmetry (72 degrees). The Clematis shown has 8-fold rotational symmetry (45 degrees). It has 8 flower petals arranged around the center of the flower. The top left flower in the bunch is the one where the number of petals is most easily seen.

Spiderwort | Baby starfish | Red knobbed starfish |

Clematis | Chemistry | Yin-Yang symbol |

The benzene molecule is interesting. It almost has 6-fold rotational symmetry, but if you look closely you will notice that the two models on the left have some single lines in there that tusn it into 3-fold symmetry. The picture with the circle in the center really does have 6 fold symmetry. These symmetries play an important role in the advanced study of chemistry.

The Yin-yang symbol is included here to show you that colors used in an image may play a role. If we ignore the fact that one side is black and the other is white, then this image has 2-fold rotational symmetry. If we do look at the colors, the symbol does not have rotational symmetry. Some people would say that the yin-yang symbol has 2-fold rotational symmetry, but has no color rotational symmetry. In our discussions of patterns and their symmetry we tend to ignore the colors used.

To be a bit more precise we will use the following terminology.
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 $ 1/5 $ of a complete 360° turn. Since $ 1/5 \times 360^\circ = 72^\circ $, 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 *n* rotational symmetry if *1/n* of a complete turn leaves the figure unchanged. Another way to say this is that the figure has n-fold rotational symmetry.

You can compute the angle of rotation from the order of rotation:

**An order n rotation corresponds to a $ \frac{360^\circ}{n} $ angle of rotation**.

You can check for instance that with the Spiderwort flower we have an order 3 rotation (also called 3-fold rotation), and the angle of rotation would be computed by taking 360 and dividing it by 3:

$ \frac{360^\circ}{3}=120^\circ $