## Patterns in the plane and beyond : symmetry in two and three by B.G. Hann, M.A. Thomas

By B.G. Hann, M.A. Thomas

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The contents of this booklet were used in classes given via the writer. the 1st was once a one-semester path for seniors on the collage of British Columbia; it used to be transparent that stable undergraduates have been completely in a position to dealing with common workforce thought and its software to easy quantum chemical difficulties.

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6 Patterns exhibiting six-fold rotation There are two remaining all-over pattern classes in which the highest order of rotational symmetry is 6 (60 degrees rotation). The patterns are constructed using an hexagonal lattice unit bounded by two equilateral triangles, as previously seen with patterns exhibiting three-fold rotation. All-over pattern class p6, illustrated in Figure 22, exhibits sixfold rotation points at each corner of the hexagonal lattice unit cell. Centres of three-fold rotation are located at the centres of the triangular cells, and centres of two-fold rotation are present at the midpoints of the triangular cell edges.

The fundamental region is one-quarter of the unit cell and two-fold rotational points are positioned at the centre of the unit, at each of the unit corners and the midpoints of the unit sides. Figure 13 illustrates the construction of pattern class p2gg on a square lattice. 22 Constructed on either a rectangular or a square lattice, pattern class p2mg presents two alternating and parallel reﬂection axes intersecting at right angles with parallel glide-reﬂection axes. Centres of two-fold rotation are found on the glide-reﬂection axes positioned at the centre of the unit cell, at each of the unit corners and the midpoints of the unit sides.

Centres of threefold rotation occur at the centres of the two triangular units and also at the corners of the rhomboid cell where the reﬂection axes intersect. The fundamental region is equal to one-sixth of the area of the unit cell. 5 Patterns exhibiting four-fold rotation There are three classes of all-over patterns which exhibit a highest order of rotation of 4 (90 degrees rotational symmetry). These are classes p4, p4mm, and p4gm, all of which are based on a square lattice structure. All-over pattern class p4, constructed using a square unit cell, exhibits no reﬂection or glide-reﬂection, presenting only two- and four-fold rotation.