What Is the Order of Symmetry?


The order of symmetry can be found almost everywhere in our world. While it may not be immediately noticeable, once you start looking for it, you will begin to notice symmetry everywhere. The order of symmetry is, simply put, the number of times a shape can fit perfectly into itself during a complete rotational turn. This basic mathematical concept provides an important basis to understanding nature and the world around us.

The Order of Symmetry

  • Another name for the order of symmetry is rotational symmetry, since this concept examines how symmetrical an object is as it rotates. Put differently, rotational symmetry shows how many times a shape can, without overlapping, make an exact copy of itself. The number of times it can do this is called its order.

Examples of Rotational Symmetry

  • A square has a rotational order of four, because if one corner of a square is pinned down and the square is turned 360 degrees, it will match its initial shape and position four times during a full turn. A rectangle, on the other hand, will have an order of two. This is because, in a full rotation, a rectangle can only make two exact copies of itself without overlapping itself.

Prove It to Yourself

  • Symmetry can perhaps best be explained by attempting to recreate it yourself. To prove to yourself that a square has the rotational symmetry of order four, cut a perfect square out of paper. Place this square on top of another piece of paper and pin down one of its corners. Now outline your square with pencil. Turn your square until it is no longer overlapping your pencil outline. Outline your square again. Continue doing this until you have made one complete rotation around the pin. Unpin your square and you should see a two-by-two square grid that looks much like a window with four panes.

Reflection Symmetry and Point Symmetry

  • Rotational symmetry is only one form of symmetry. The form of symmetry people are often most familiar with is known as reflection symmetry. This is the kind of symmetry that can be found when looking at a butterfly. If you were to draw a line down the body of a butterfly, the wings on each side would seem identical; therefore a butterfly's wings are symmetrical. Point symmetry is another form of symmetry, where each part of a shape has a matching point, but in the opposite direction. You can think of this as the kind of symmetry you would find when looking at letters reflected in a mirror.


  • Photo Credit Ryan McVay/Digital Vision/Getty Images Jupiterimages/BananaStock/Getty Images
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