In the last unit every transformation moved a shape somewhere new. Symmetry asks a different question: is there a transformation that leaves the shape looking exactly as it was?
Symmetry is a shortcut. If a shape has a line of symmetry, knowing one half tells you the other half for free, so you only have to measure, design or calculate half of it. A builder can draw half a house plan and mirror it.
Line symmetry
A line of symmetry is a line you could fold the shape along so that the two halves land exactly on top of each other. A shape can have none, one, two, several, or in the case of a circle, infinitely many.
Do not work out lines of symmetry by staring. Trace the shape, cut it out and fold it. The parallelogram is the classic trap: its diagonals look as though they should work, and they do not.
Rotational symmetry
A shape has rotational symmetry if you can turn it about its centre by less than a full turn and it looks exactly the same. The order is the number of times it looks the same during one complete turn. A square has order 4.
Every shape looks the same after a full 360 degree turn, so that tells you nothing. A shape with order 1 is said to have no rotational symmetry. Rotational symmetry means order 2 or more.
The two kinds are independent
| Shape | Lines of symmetry | Order of rotational symmetry |
|---|---|---|
| Square | 4 | 4 |
| Rectangle | 2 | 2 |
| Parallelogram | 0 | 2 |
| Kite | 1 | 1 (none) |
| Equilateral triangle | 3 | 3 |
| Isosceles triangle | 1 | 1 (none) |
| Regular hexagon | 6 | 6 |
| Circle | Infinitely many | Infinite |
The parallelogram has no lines of symmetry, yet it has rotational symmetry of order 2. The kite is the opposite: one line of symmetry but no rotational symmetry. The two kinds of symmetry are genuinely separate ideas.
Regular polygons follow one rule
For a regular polygon with n sides there are exactly n lines of symmetry, and the order of rotational symmetry is also n. A regular pentagon has 5 of each, a regular octagon has 8 of each.
For an even number of sides, the lines either join opposite corners or the midpoints of opposite sides. For an odd number, every line runs from a corner to the midpoint of the opposite side, because there is never a corner directly opposite.
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