Line and Rotational Symmetry

Draw a rectangle that is not a square. How many fold lines map it onto itself? Now try the diagonals, carefully. Do they work?

Line symmetry is a fold: a reflection in some line maps the shape exactly onto itself. Rotational symmetry is a turn: a rotation about some centre maps it exactly onto itself. The order counts how many positions in one full turn work.

A shape has line symmetry if there is a line you could fold it along so that the two halves land exactly on each other. That line is an axis of symmetry.

Equivalently, and more usefully later, reflecting the shape in that line produces the same shape occupying exactly the same position. Every point moves, but the set of points is unchanged.

Some shapes have none, some have one, some have several, and a circle has infinitely many, since every line through its centre works.

Test a candidate line rather than trusting the picture. The diagonals of a rectangle look promising and are not axes: folding along one sends a long side onto a short side, and they do not match.

These are worth knowing on sight rather than rediscovering under pressure.

Triangles: equilateral $3$, isosceles $1$, scalene $0$.

Quadrilaterals: square $4$, rectangle $2$ through the midpoints of opposite sides, rhombus $2$ along the diagonals, parallelogram $0$, kite $1$ along one diagonal, isosceles trapezium $1$.

Regular polygons: a regular polygon with $n$ sides has exactly $n$ axes.

Notice that the rectangle and the rhombus both have two axes, but in different places. The rectangle's run through the midpoints of opposite sides; the rhombus's run along its diagonals. That difference is often what a question is actually testing.

A shape has rotational symmetry if turning it about some centre, by less than a full revolution, leaves it looking exactly as it was.

The order is the number of positions in a complete turn where this happens, and the starting position is counted. So a shape with no rotational symmetry has order $1$, because the full $360°$ turn always works.

A parallelogram has order $2$: turn it $180°$ about the point where its diagonals cross and every vertex lands where another vertex was. A square has order $4$, matching $90°$, $180°$, $270°$ and $360°$.

Order $1$ is the correct way to say a shape has no rotational symmetry. Saying order $0$ would mean not even a full turn worked, which is impossible.

The positions are evenly spaced around the full turn, so the smallest angle that works is

$\text{angle} = \dfrac{360°}{\text{order}}$

An order of $6$ gives $60°$, an order of $4$ gives $90°$, an order of $3$ gives $120°$, and an order of $2$ gives $180°$.

The relationship runs both ways, so a shape that first repeats after a $72°$ turn has order $\dfrac{360}{72} = 5$.

The reason the positions are evenly spaced is worth a moment. If a turn of angle $\theta$ works, then doing it twice works too, and three times, and so on. Those repeated turns generate evenly spaced positions, and they must close up exactly at $360°$, so $\theta$ divides $360°$ exactly.

It is tempting to think one kind of symmetry brings the other with it. Two shapes show that it does not.

A parallelogram that is neither a rectangle nor a rhombus has $0$ axes and order $2$. Rotational symmetry without any line symmetry.

An isosceles trapezium has $1$ axis, through the midpoints of its two parallel sides, and order $1$. Line symmetry without any rotational symmetry.

So the two counts are separate pieces of information, and a description giving only one of them is incomplete.

There is one direction that does hold: a shape with two or more axes always has rotational symmetry, because reflecting twice in two different lines is the same as a rotation. But a single axis brings nothing with it.

Five items. Give the number of axes and the order of rotational symmetry for each.

1. 1 An equilateral triangle.

   Show answer
   Three axes, each from a vertex to the midpoint of the opposite side.$3$ axes, order $3$

2. 2 A rectangle that is not a square.

   Show answer
   The axes go through the midpoints of opposite sides, not along the diagonals.$2$ axes, order $2$

3. 3 A scalene triangle.

   Show answer
   No fold works and no turn short of a full revolution works.$0$ axes, order $1$

4. 4 A regular octagon.

   Show answer
   A regular $n$-gon has $n$ axes and order $n$.$8$ axes, order $8$

5. 5 A shape first maps onto itself after a rotation of $45°$. What is its order?

   Show answer
   $\dfrac{360°}{45°}$Order $8$