4.3 Symmetry

Learning objectives

4.3A Line and rotational symmetryidentify lines of symmetry and the order of rotational symmetry of a given two-dimensional figure

A line of symmetry divides a 2D figure into two mirror-image halves. Folding along the line would make corresponding points coincide.

The order of rotational symmetry is the number of times a figure matches its starting position during one full $360^\circ$ turn, including the final return.

Task	Reliable check
test a symmetry line	compare equal perpendicular distances on both sides
find all lines	test vertical, horizontal and diagonal candidates
find rotational order	rotate by the smallest matching angle
connect angle and order	$\text{order}=360^\circ/\text{smallest angle}$

A regular hexagon has rotational order 6. A non-square rhombus has two diagonal lines of symmetry and rotational order 2.

A figure always matches after $360^\circ$ , so its rotational order is at least 1. Order 1 means no non-trivial rotational symmetry.

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