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4 5 rotAtions And reflection point symmetries There are two further basic expressions of symmetry, namely rotation and reflection. Each of these forms of symmetry relies on the notion of congruence, that is to say, a general correspondence between each part of an element, however expressed below. In simple rotational symmetry the component parts are laid out, at regular intervals, around a central point 14. Because the elements in these symmetries are simple unreversed replicas of each other, they are described as being directly congruent. In reflection symmetry, by contrast, the reversed elements are arranged about a mirror line, and so are oppositely congruent 5,6 . Because the central point or line remains fixed in reflections and rotations, these are collectively known as point symmetries. In its most basic form rotational symmetry involves just two components arranged around a centre. Ordinary playing cards are of this kindany cut through the centre of a card results in two identical halves. The triskelion symbol consists of three rotated parts a swastika of four, and so onwith no upward limit to the number, other than the amount of repeats that can be arranged around a given centre. Rotational and reflection symmetries can also be combined, in which case the lines of reflection intersect at a central point of rotation. Figures and objects of this kind are described as having dihedral symmetry 7. 2. Playingcards are probably the most familiar example of 2rotation symmetry, demonstrating a selfcoincidence of 180o note that there is no reflection here 1. The simplest form of rotation around a centre, using just two elements 3. Rotational symmetry may involve any number of elements 5. Reflection about a line 7. Dihedral symmetry 4. Motifs using 3, 4 and 5 rotational symmetry , with selfcoincidences of 120o, 90o and 72o respectively 6. Motifs with only reflection symmetry are among the most common 8. Motifs demonstrating dihedral symmetry, combining reflection and rotation
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