Rotational Symmetry and Turns in Degrees

The order of rotational symmetry is how many times a shape looks exactly the same during one full turn. Turn a square a quarter turn and it looks unchanged, and the same at a half turn and three quarters, and of course at the full turn — order four. A rectangle that is not a square only matches at a half turn and a full one, so its order is two. Count the positions in one full turn, including the finish, and do not count the start twice.

The smallest turn that brings a shape back to looking the same is three hundred and sixty divided by its order. Order four gives ninety degrees, order three gives a hundred and twenty, order six gives sixty. A regular shape's order is simply its number of sides, so a regular pentagon has order five and first matches after seventy-two degrees. The rule works backwards as well: a shape that first matches after forty-five degrees has order three hundred and sixty divided by forty-five, eight.

To find how far a shape has turned after several steps, multiply the step angle by the number of steps. A shape of order five steps seventy-two degrees at a time, so two steps is a hundred and forty-four and three steps is two hundred and sixteen. A square turned three steps has gone two hundred and seventy degrees. And if the steps take you past a full turn, take off three hundred and sixty: five steps of a square is four hundred and fifty, which looks the same as ninety.

The two kinds of symmetry are independent. A pinwheel turns onto itself every ninety degrees, so its order is four, yet it cannot be folded onto itself at all: no lines. A kite folds down its long diagonal but only a full turn brings it back, so its order is one. A square has both, four lines and order four. And a leaning parallelogram is the neat case worth remembering: no lines of symmetry whatever, but order two, because a half turn puts it right back.

Everything here comes from one relationship: the order multiplied by the step angle is three hundred and sixty. Given the order, divide to get the angle; given the angle, divide to get the order; given a number of steps, multiply. A regular hexagon has order six and a step of sixty degrees; a square turned three steps has gone two hundred and seventy. Quiz and exercises now.