Rotation

Watch the second hand sweep round a clock. Watch a Ferris wheel carry its cars up one side and down the other. Ride the waltzers at a fair and feel yourself whirl. In every case something is turning about a fixed point — the middle of the clock, the hub of the wheel, the centre of the ride. Nothing gets bigger, smaller, squashed or flipped; it just spins round.

That turning motion is a rotation. In maths a rotation takes a whole shape and swings it around a fixed point by a chosen angle. The image is exactly the same size and shape as the original — it has simply been turned.

Three things describe a rotation

A wheel can turn a little or a lot, this way or that, around this pin or that one. So to describe a rotation completely you always need three ingredients:

a centre — the fixed point the shape turns around (the hub);
an angle — how far it turns, e.g. 90^\circ, 180^\circ or 270^\circ;
a direction — clockwise or anticlockwise.

The shape keeps its size, its angles and its orientation — a rotation is not a mirror image. Every point simply stays the same distance from the centre as it swings around, like a car on a fairground ride staying the same distance from the middle the whole way round.

Mathematicians measure angles the anticlockwise way as the "positive" direction — the opposite way to a clock's hands. So a turn of +90^\circ means a quarter-turn anticlockwise, and -90^\circ means a quarter-turn clockwise. It feels backwards at first, but once you know the rule it never trips you up again.

Rotating about the origin

When the centre is the origin (0, 0), each turn has a tidy coordinate rule — no protractor needed. You just shuffle and flip the signs of the two numbers:

For a point (x, y) rotated about the origin:

90^\circ anticlockwise: (x, y) \to (-y, x);
180^\circ: (x, y) \to (-x, -y);
90^\circ clockwise: (x, y) \to (y, -x).

A neat check: a 90^\circ clockwise turn is the same as a 270^\circ anticlockwise turn — the shape ends up in exactly the same place. Turning a shape four times by 90^\circ brings it all the way back home to where it began.

Worked example 1 — a quarter-turn clockwise

Rotate the point (4, 1) by 90^\circ clockwise about the origin.

The clockwise rule is (x, y) \to (y, -x).
Here x = 4 and y = 1.
So the image is (y, -x) = (1, -4).

The point started up on the right and swings down to the bottom — exactly where a clockwise quarter-turn should carry it. Its distance from the origin hasn't changed: it was \sqrt{4^2 + 1^2} away before, and it still is.

Seeing it on a grid

Take the triangle with corners (1,1), (3,1) and (1,2), and rotate it 90^\circ anticlockwise about the origin. Each corner follows (x, y) \to (-y, x), so it swings a quarter-turn round the marked centre while keeping its shape. Step through and watch it swing.

Worked example 2 — a half-turn (the special one)

Rotate the triangle with corners (1,1), (4,1) and (1,3) by 180^\circ about the origin. The rule is (x, y) \to (-x, -y), so:

(1, 1) \to (-1, -1);
(4, 1) \to (-4, -1);
(1, 3) \to (-1, -3).

Every corner jumps to the diagonally opposite spot. And here is the magic of a half-turn: it doesn't matter whether you turn clockwise or anticlockwise — you land in exactly the same place either way. A half-turn is the one rotation where the direction makes no difference at all.

Worked example 3 — describing a rotation

Sometimes you are shown a shape and its image and asked to describe the rotation fully. That means finding all three ingredients. Suppose a triangle at (1, 0), (3, 0), (1, 1) maps to an image at (0, 1), (0, 3), (-1, 1).

Centre: test the origin. Each point's distance from (0,0) is unchanged — the origin works, so it is the centre.
Angle and direction: the corner (1, 0) went to (0, 1). That matches the rule (x, y) \to (-y, x), which is a 90^\circ anticlockwise turn.

So the full description is: a rotation of 90^\circ anticlockwise about the origin. All three ingredients named — that is what "describe fully" means.

The classic slip is to give an incomplete answer. "A rotation of 90^\circ" is not enough — round which point, and which way? To describe a rotation completely you must state all three:

the centre (people forget this most often),
the angle, and
the direction — clockwise or anticlockwise.

The only exception is a 180^\circ half-turn, where clockwise and anticlockwise give the same image, so you may leave the direction out. For every other angle, drop any one ingredient and the answer is marked wrong.

Everywhere, once you look. A snowflake has six-fold rotational symmetry — turn it by 60^\circ and it looks identical, six times over in a full circle. Car wheels, clock faces, fans, windmills and the recycling symbol are all built on rotation, and many flowers arrange their petals in perfect rotational patterns.