Symmetry

A shape has symmetry when some — other than just leaving it alone — carries it exactly onto itself. There are two kinds to hunt for:

: spinning the shape about its center by some angle less than $360^\circ$ lands it back on itself.
: flipping the shape across some line through its center lands it back on itself.

Hunt: the square

Make a prediction: Before you start hunting: how many symmetries does a square have — counting every rotation and every reflection, but not the do-nothing move?

478Infinitely many

A square is a symmetry jackpot. Try every proposal below — the explorer checks each one exactly and keeps count. Can you find all 7?

Tap a move to test whether it carries the shape exactly onto itself.

Found 0 of 7 symmetries

Rotations

Reflections

Hunt: the isosceles triangle

Not every shape is so generous. This isosceles triangle has exactly one symmetry — and it isn't a rotation. (And a scalene triangle would have none at all: "irregular" shapes are part of the standard too.)

Tap a move to test whether it carries the shape exactly onto itself.

Found 0 of 1 symmetries

Rotations

Reflections

Counting symmetries

For a regular polygon with $n$ sides, the pattern is beautiful: $n - 1$ rotations (by multiples of $360^\circ / n$ ) and $n$ lines of symmetry. The square fits perfectly: $3$ rotations $+$ $4$ lines $= 7$ symmetries.

Worked example: Count the symmetries of a regular pentagon

Start with the lines: how many does a regular n-gon have?
Before the next step: How many lines of symmetry does a regular n-gon have?
nn − 12n

Irregular shapes break the pattern in interesting ways — a rectangle keeps only the half-turn and two of the four lines, and a generic parallelogram keeps the half-turn but no lines at all.

Symmetry

Hunt: the square

Hunt: the isosceles triangle

Counting symmetries

Practice