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 n rotational symmetry if <math>1/n</math> of a complete turn leaves the figure unchanged.

There are quite a few shapes you know that have rotational symmetry. Below the Spiderwort flower has 3-fold rotational symmetry (120 degrees). The baby starfish has 4-fold rotational symmetry (90 degrees) and the red knobbed starfish shown here has 5-fold rotational symmetry (72 degrees. The clematis shown has 6-fold rotational symmetry (60 degrees).

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.

Spiderwort-Blue-Flower.jpg Baby-starfish.jpg Red-knobbed-starfish.jpg
Spiderwort Baby starfish Red knobbed starfish
Clematis2.jpg Benzene structure.png Yin-yang.svg
Clematis Chemistry Yin-Yang symbol

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.