Symmetry and Moving Shapes

A shape can match itself in two quite different ways, and this track separates them before putting them to work. Folding comes first: a line of symmetry is a crease that lands one half exactly on the other. Turning comes second, and the order of rotational symmetry counts how many times a shape falls back onto itself in one full turn — a count that is never zero, because every shape returns to itself after a complete turn. The two are genuinely independent, and the track leans on the pair that proves it: a parallelogram turns onto itself but has no fold line at all, while an isosceles trapezium folds but will not turn. Only the regular polygons give the same answer both ways. The second half stops describing symmetry a shape already has and starts moving shapes deliberately. Reflection sends every corner straight across a mirror line and the same distance beyond it. Translation slides everything by the same two numbers and needs no centre or mirror at all, which makes it the simplest of the three. Rotation needs a centre, an angle and a direction, and the centre is the part people leave out — the same half turn about two different points sends a shape to two quite different places. All three leave every length and every angle untouched, so the image is always congruent to the object; what separates them is what else they preserve. Only reflection reverses the handedness, which is the reliable test when a picture could be a flip or a turn.