4) Symmetry

The word symmetry, like many mathematical terms, comes from a Greek word, "symmetria", meaning "measured together". When we measure something and find it is the same on one side as on the other, we say that it is symmetrical. The boat below appears to be symmetric: it is the same on the left as on the right.

Artists often use symmetry in their paintings and other artwork, as you can see in this stained glass window:

Many plants and animals have this kind of symmetry. If we drew a vertical line down the middle, the left side of each would be a "mirror image" of the right side. Actually, this is only one kind of symmetry: we call this "reflection symmetry". Reflection Symmetry: The word "reflection" comes from the idea of the mirror reflection, just like the reflection of the bridge in the water below:

Many animals and plants have reflection symmetry. This coconut tree frond, for example, is the same on one side as on the other, and so is the fern:

The butterflies painted on this fabric have reflection symmetry, as do many other living things:

All regular polygons have reflection symmetry. They have more than one line of reflection symmetry, however, unlike most animals. In the three regular polygons below, all lines of reflection symmetry have been drawn. Different that there are different types of lines, and they are drawn in different colors:


Rotation Symmetry

Besides reflection symmetry, there is another kind of symmetry called rotation symmetry. The design below does not have reflection symmetry, because there is no way that you can fold it over and have one half match the other. But it does have a kind of symmetry called rotation symmetry. Draw a diagram as much like this one as you can, or print this page. Then trace your diagram onto tracing paper and place the tracing on top of the original so that they line up perfectly Put a pin or compass point at the center of the figure, and then rotate the tracing until it matches the original. How far did you have to go before they matched?

You should have found that the tracing matched the original after being rotated one-sixth of the way around. If you were to rotate it all the way around to the beginning, this would be a rotation of 360 degrees. So a rotation of one-sixth of the way would be a rotation of 60 degrees (360 divided by 6 equals 60 degrees), and we would say "this figure has 60 degree rotation symmetry about the center point"

Many plants, particularly flowers, and some ocean animals, have rotation symmetry.

Some geometric figures, plants, and animals have both types of symmetry, like the white Spider Lily flower below, if we ignore the slight difference in the spacing of the middle petals on the left. The flower has six lines of reflection symmetry, one through the center of each petal. It also has 60 degrees of rotation symmetry

Go to Topic 5 Constructions