Rotational symmetry

Some shapes look the same after you turn them. A square turned by $90°$ looks identical. A regular hexagon turned by $60°$ looks unchanged. This is rotational symmetry , a shape's stability under turning.

Concept

A shape has rotational symmetry if you can rotate it by some angle (less than $360°$) about a fixed point , called the centre of rotation , and the shape looks exactly the same as before.

The smallest angle of rotation that brings the shape back to its original position is called the angle of rotational symmetry. The number of times the shape returns to itself in one full $360°$ turn is called the order of rotational symmetry.

A clean relationship:

$\text{order} = \frac{360°}{\text{angle of rotation}}$

So if the smallest angle is $60°$, the order is $360° / 60° = 6$.

Examples

• Square. Rotate by $90°$, $180°$, $270°$, $360°$ , all give the same shape. Order $= 4$.
• Equilateral triangle. Rotate by $120°$, $240°$, $360°$. Order $= 3$.
• Regular hexagon. Rotate by $60°$, $120°$, $180°$, $240°$, $300°$, $360°$. Order $= 6$.
• Regular $n$-gon. Order $= n$. Smallest angle $= \frac{360°}{n}$.
• Letter "S" or "Z". Rotate by $180°$. Order $= 2$.
• Letter "N". Rotate by $180°$. Order $= 2$.
• Circle. Any angle works. Order is "infinite".

What about order $1$?

A shape has order $1$ if the only rotation that brings it back is the full $360°$ turn , i.e., it has no rotational symmetry. To keep language clean, we say such a shape has order $1$ (or "no rotational symmetry").

Line symmetry and rotational symmetry , separate ideas

• The letter "T" has line symmetry but no rotational symmetry (order $1$).
• The letter "S" has rotational symmetry (order $2$) but no line symmetry.
• The letter "H" has both (line symmetry with $2$ axes, rotational symmetry of order $2$).
• The letter "F" has neither.

A regular polygon with $n$ sides has both: $n$ lines of symmetry and rotational symmetry of order $n$.

Worked examples

Example 1. Find the order of rotational symmetry of a square.

• Smallest rotation that gives the same square: $90°$.
• Order $= 360° / 90° = 4$.

Example 2. Find the order of rotational symmetry of the letter "Z".

• Rotate by $180°$ , looks the same.
• $90°$ does not work. So smallest is $180°$.
• Order $= 360° / 180° = 2$.

Example 3. What is the order of rotational symmetry of a rectangle (not a square)?

• $180°$ rotation gives the same rectangle.
• $90°$ does not.
• Order $= 2$.

Example 4. A regular hexagon has rotational symmetry of order $6$. What is the smallest angle of rotation?

• $360° / 6 = 60°$.

Example 5. Does a parallelogram have rotational symmetry?

• A general parallelogram (not rectangle/rhombus/square) has order $2$ , $180°$ rotation brings it back.

Try it yourself

1. Find the order of rotational symmetry of an equilateral triangle.
2. Find the order of rotational symmetry of a regular pentagon. What is the smallest rotation?
3. Does the letter "S" have rotational symmetry? Order?
4. Does the letter "T" have rotational symmetry?
5. List four English letters with rotational symmetry of order $2$.
6. The Olympic rings logo , does it have rotational symmetry? Why or why not?
7. A shape has rotational symmetry of order $5$. What is the smallest angle of rotation?
8. Investigate: find a shape with rotational symmetry of order $3$ but no line symmetry. (Hint: spinning fan blades.)

Activity

Pinwheel. Cut out a square of paper. Draw lines from each corner toward the centre, but stop just before reaching it. Now fold every other corner toward the centre and pin them down. Stick a pin through the centre. You've made a pinwheel , and the design has rotational symmetry of order $4$ but no line symmetry (the curving of the blades breaks any potential mirror).