Symmetry

A butterfly rests with both wings identical; a snowflake repeats itself six ways; the letters A, H, O look perfectly balanced. Mathematics makes "balanced" precise: a shape is symmetric when some move — a fold or a turn — makes it land exactly on itself.

Fold symmetry: reflection

Fold a shape along a line and the two halves match — that line is a line of symmetry, and the shape has reflection symmetry.

An isosceles triangle has 1 line of symmetry, an equilateral triangle has 3;
A square has 4, a rectangle has 2;
A general parallelogram folds onto itself along no line at all — its symmetry is rotational, see the next section;
Letter hunt: A, M, T, W fold along a vertical line; B, C, D, K fold along a horizontal one.

Turn symmetry: rotational

Rotate a shape about its center, and it lands on itself again — that is rotational symmetry. The number of positions (less than a full turn of $360^\circ$ ) that land on themselves is called the order.

A square lands on itself every $90^\circ$ , four times per full turn — order 4, because $360^\circ \div 4 = 90^\circ$ ;
An equilateral triangle lands on itself every $120^\circ$ — order 3, since $360^\circ \div 3 = 120^\circ$ ;
A regular pentagon has order 5, a regular hexagon order 6, and a circle lands on itself at any angle.

InteractiveRotational Symmetry

Angle60

Type：Acute= 1.047 rad

180° = π rad

Try it: the regular hexagon

A regular hexagon has 6 identical spokes. $360^\circ \div 6 = 60^\circ$ , so it lands on itself at $60^\circ$ , $120^\circ$ , $180^\circ$ , $240^\circ$ and $300^\circ$ — six positions per turn, order 6.

Point symmetry: a half-turn

A shape that lands on itself after a $180^\circ$ turn (half a circle) has point symmetry. A parallelogram does exactly this: turn it half a turn about the crossing point of its diagonals and every vertex returns home. The letters S, N and Z work the same way — turn them upside down and they are still themselves.

Find symmetry nearby

Nature is a symmetry expert: butterflies, leaves, faces and starfish are mostly reflection-symmetric;
Paper craft: fold a sheet a few times, make one cut, unfold — an instant reflection pattern;
Playing cards, car logos, packages: decide which are reflection-symmetric, which rotational, and count the order.

The tracing-paper test

Never guess a line of symmetry: lay tracing paper over the shape and fold along the candidate line — it only counts if the halves match. Try the diagonal of a rectangle: it does not match, so a rectangle has 2 lines of symmetry, not 4.

Check yourself

Quick quiz

1. What is the order of rotational symmetry of a square?
2. Which letter folds onto itself along a vertical line?
3. What is the smallest angle of rotational symmetry of an equilateral triangle?