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Motifs can also be repeated by non‐translational symmetry operations. Many patterns can be repeated by rotation (n). Rotation (n) is a symmetry operation that involves the rotation of a pattern about an imaginary line or axis, called an axis of rotation (A), in such a way that every component of the pattern is perfectly repeated one or more times during a complete 360° rotation. The symbol “n” denotes the number of repetitions that occur during a complete rotation. Figure 4.3 uses triangle motifs to depict the major types of rotational symmetry (n) that occur in minerals and other crystals. The axis of rotation for each motif is perpendicular to the page. Table 4.1 summarizes the major types of rotational symmetry.

Table 4.1 Five common axes of rotational symmetry in minerals.

Type	Symbolic notation	Description
Onefold axis of rotation	(1 or A1)	Any axis of rotation about which the motif is repeated only once during a 360° rotation (Figure 4.3 (1))
Twofold axis of rotation	(2 or A2)	Motifs repeated every 180° or twice during a 360° rotation (Figure 4.3 (2))
Threefold axis of rotation	(3 or A3)	Motifs repeated every 120° or three times during a complete rotation (Figure 4.3 (3))
Fourfold axis of rotation	(4 or A4)	Motifs repeated every 90° or four times during a complete rotation (Figure 4.3 (4))
Sixfold axis of rotation	(6 or A6)	Motifs repeated every 60° or six times during a complete rotation (Figure 4.3 (5))