Rotational Symmetry

Rotational symmetry: summary and recommendation

What reputable sources say

• The NCETM glossary defines rotational symmetry as: a 2-D shape has rotational symmetry about a point if it matches itself after a rotation through some angle greater than 0° and less than 360°. It defines the order as the number of distinct matches in a full turn. Saying "order 1" is technically true because every shape matches after 360°, although this is not very informative.
• GCSE revision materials commonly avoid "order 0". They either write "no rotational symmetry" or accept "order 1" for the trivial full-turn only case. Many teachers prefer to say "no rotational symmetry" to avoid confusion.
• General references agree that order means a positive integer count of matching orientations in a full turn.

Which description is best

• "Order 0" is not correct in school mathematics because order is defined as a positive integer.
• "Order 1" is technically defensible, since the shape matches after 360°, but it often misleads pupils into thinking there is a non-trivial rotation.
• "No rotational symmetry" is the clearest classroom language when there is no match for any angle strictly between 0° and 360°.

How to describe very irregular shapes

Use plain wording: asymmetric with no rotational symmetry. This is precise and avoids implying any special order.

My recommendation for secondary teaching

1. Teach the NCETM definition that requires some angle strictly between 0° and 360°.
2. When a shape only matches after a full turn, write no rotational symmetry.
3. If an exam board allows "order 1", you may mention it as a technical aside, yet reserve "order n" for n ≥ 2 in everyday teaching.
4. For especially uneven shapes, describe them as asymmetric with no rotational symmetry.

In short

Avoid "order 0". Prefer "no rotational symmetry". Acknowledge "order 1" only as a technical note where mark schemes permit it.