Groups and Symmetry pp 145-154 | Cite as

# Lattices and Point Groups

## Abstract

Figure 25.1 shows a repeating pattern of hexagons which, if continued indefinitely,*fills out the whole plane*. The pattern has a certain amount of symmetry. For example, if we apply either of the translations τ_{1}, τ_{2} or reflect in the *x*-axis, or rotate anticlockwise through π/3 about the origin, then hexagons go to hexagons and the pattern is preserved. By shading in part of each hexagon, as in Figure 25.2, we produce a new design which is “less symmetrical” because the rotational symmetry has been destroyed. As usual, the symmetry is measured by a group, in this case the appropriate subgroup of *E*2 whose elements are the isometries of the plane which send a given pattern to itself. We shall classify the groups which can arise in this way as symmetry groups of two dimensional repeating patterns or, as we shall call them, * wallpaper patterns*. If you find hexagons rather dull for a wallpaper, then try the designs shown in Figure 25.3. Both exhibit precisely the same symmetry as the pattern of (unshaded) hexagons.

## Keywords

Symmetry Group Point Group Finite Order Infinite Order Crystallographic Group## Preview

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