Lattices and Point Groups
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 τ1, τ2 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 E2 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.
KeywordsSymmetry Group Point Group Finite Order Infinite Order Crystallographic Group
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