Symmetry

The word symmetry stems from Greek, meaning (very roughly) ‘having the same measure’ or ‘being in agreement’. Put simply, symmetry is concerned with whether a shape still looks the same if you move it in some way, and if so, how. There several types of symmetry, but the two that come up most often are reflectional and rotatational.

Reflectional symmetry

This is the type of symmetry we are most used to: there is some line through the shape, such that one side is a reflection of the other. Butterflies have reflectional symmetry, for example.

Any such line is called a ‘line of symmetry’. In the case of the butterfly, there is only one line of symmetry (shown above in purple). If we were to draw a different line through the butterfly, one side would not be a mirror image of the other. Our faces (approximately) have reflectional symmetry, but it can sometime be surprising to see how not-symmetric they are, for example by placing a mirror on a photo, or by using a computer to create a face of both left halves or both right.

Some shapes have more than one line of symmetry, for example this ice crystal.

We could place a mirror along any of the purple lines and the reflection would look the same as the other half of the shape.

Rotational symmetry

For a shape to have rotational symmetry, it needs to be the same after being rotated (spun round) part way. The ice crystal above has rotational symmetry: if we spin it round, there are six positions in which it looks the same. Therefore, we say it has rotational symmetry of order six.

The butterfly does not have rotational symmetry, because if we spin it round all the way, it doesn’t get back to looking the same until it’s been rotated back to its original position.

There are shapes that have rotational symmetry but not reflectional symmetry, such as the icon on the Isle of Man flag, which has rotational symmetry of order three.

These sorts of shapes are not easy to find in nature.