Combining symmetries

A single shape can carry several symmetries at once: many lines of symmetry, a high order of rotational symmetry, or both. Understanding how these combine reveals the deep structure of a shape.

Concept

Take an equilateral triangle. It has:

$3$ lines of symmetry.
Rotational symmetry of order $3$ .

Take a square:

$4$ lines of symmetry.
Rotational symmetry of order $4$ .

Take a regular pentagon:

$5$ lines of symmetry.
Rotational symmetry of order $5$ .

Pattern: for a regular $n$ -gon, the number of lines of symmetry equals the order of rotational symmetry, and both equal $n$ .

Mismatched cases

For irregular shapes, the two counts can differ:

A rectangle (not a square): $2$ lines of symmetry, rotational order $2$ .
A rhombus (not a square): $2$ lines, rotational order $2$ .
A parallelogram: $0$ lines, rotational order $2$ .
A kite (not rhombus): $1$ line, rotational order $1$ .
The letter "S": $0$ lines, rotational order $2$ .
The letter "T": $1$ line, rotational order $1$ .

A useful rule

If a shape has $k$ lines of symmetry (with $k \geq 2$ ), then it must also have rotational symmetry of order at least $k$ . (You can show this by composing reflections , two reflections in a row equal a rotation.) So shapes can have:

Many lines and many rotations (regular polygons).
Some lines and some rotations (rectangles).
Some rotations but no lines (S, N, Z , like pinwheels).
One line but no rotations (kites, T, isosceles triangle).
Neither (most shapes).

But it is impossible to have many lines of symmetry without correspondingly high rotational symmetry. This is one of mathematics' deep facts about how reflections and rotations interact.

Symmetry of all kinds in real life

The Indian flag (saffron, white, green stripes with a wheel in the middle): has $1$ line of symmetry (vertical down the middle when held still), the wheel itself has rotational symmetry of order $24$ .

The Sri Yantra of Hindu tradition has $1$ vertical line of symmetry; the design is built around it.

Rangoli designs commonly have $4$ , $6$ , or $8$ lines and rotational symmetry of the same order , a deliberate choice that makes them so visually pleasing.

Worked examples

Example 1. State the number of lines of symmetry and order of rotational symmetry for a regular octagon.

$8$ lines, order $8$ .

Example 2. A figure has $3$ lines of symmetry. What can you say about its rotational symmetry?

It must have rotational symmetry of order at least $3$ . (If it has exactly $3$ lines arranged symmetrically, the order is exactly $3$ .)

Example 3. A shape has rotational symmetry of order $4$ . Does it necessarily have line symmetry?

Not necessarily. A pinwheel-shaped fan with $4$ curved blades has order $4$ but no line symmetry.

Example 4. The letter "I" , count its lines of symmetry and order of rotational symmetry.

$2$ lines (horizontal and vertical).
Rotational order $2$ ( $180°$ ).

Example 5. A regular polygon has rotational symmetry of order $7$ . How many lines of symmetry does it have?

For a regular polygon, lines $=$ order. So $7$ lines.

Try it yourself

State both line symmetries and rotational order for a regular hexagon.
State both for a rectangle (not square).
State both for a rhombus (not square).
State both for a kite (not rhombus).
The Olympic rings , line symmetries? Rotational order?
The Ashoka Chakra (24-spoke wheel on the Indian flag) , rotational order?
The letter "O" , line symmetries and rotational order.
Investigate: design a shape with $0$ lines of symmetry but rotational symmetry of order $5$ . (Hint: think of a flower with each petal curving the same way.)

Activity

Symmetry classifier. Find $10$ different objects or images at home (logos on cereal boxes, letters in a book, jewellery, rangoli designs). For each, write down: (a) number of lines of symmetry, (b) order of rotational symmetry. Sort them into the five categories: many of both / some of both / only rotations / only lines / neither.