11.3: Unit Cells and Lattices in Two Dimension

Figure 11.16: Some possible shapes for floor tiles

What possible shapes can unit cells have? Mineralogists begin answering this by considering only two dimensions. This is much like imagining what shapes we can use to tile a floor without leaving gaps between tiles. Figure 11.16 shows some possible tile shapes. In the right-hand column of this figure, a single dot has replaced each tile to show the plane lattice that describes how the tiles repeat. This figure does not show all possible shapes for tiles or unit cells, but as we will see later the number of possibilities is quite small.

Figure 11.17: Tiles with gaps between them

What happens when gaps occur between tiles? Figure 11.17 shows the two possibilities: the gaps may be either regularly (11.17a) or randomly (11.17b) distributed. If regularly distributed, we can define a unit cell that includes the gap, as we have in drawing 11.17a. The two vectors, t₁ and t₂ and the lattice they create describe how the unit cell repeats to make the entire tile pattern.

If gaps between tiles are random, the entire structure is not composed of identical building blocks that fit together in a regular way. It is not repetitive and we cannot describe it with a unit cell and a lattice. It does not, therefore, represent the symmetry of a possible crystal structure and we need not consider it further.

Figure 11.18: Floor tiles and a complex pattern

Various complex patterns can appear on tiled floors, but the tiles are usually simple shapes such as squares or rectangles. For example, suppose we wish to tile a floor in the pattern shown in Figure 11.18a. We could use L-shaped tiles, shown in red in Figure 11.18b. However, parallelogram- or square-shaped tiles would get us the same pattern (Figures 11.18c and d). Other shapes, too, would work.

We can make any repetitive two-dimensional pattern, no matter how complicated, with tiles of one of four fundamental shapes: parallelogram, rhomb (a parallelogram with sides of equal length), rectangle, and square. These shapes are relatively simple compared to more complex ones we could choose, and they reveal symmetry that is present. So, they are used by mineralogists and preferred by tile makers.

Figure 11.19: The five possible two-dimensional shapes

For reasons we will see later, we usually distinguish two types of rhombs (parallelograms with sides of equal length): those with nonspecial angles between sides and those with 60^o and 120^o angles between sides. For the rest of this chapter, we will refer to the general type of rhomb as a diamond, and the term rhomb will be reserved for shapes with only 60^o and 120^o angles (and sides of equal length). So, Figure 11.19 shows the five basic shapes, the only ones needed to discuss two-dimensional unit cells, and their symmetries.

We can now explain why we only considered 1-, 2-, 3-, 4-, and 6-fold rotational symmetry in the previous chapter. Two-dimensional shapes with a 5-fold axes of symmetry, for example, cannot be unit cells because they cannot fit together without gaps. We can verify this by drawing equal-sided pentagons on a piece of paper. No matter how we fit them together, space will always be left over. The same holds true for equal-sided polyhedra with seven or more faces. They cannot fit together to tile a floor. And if the shapes cannot fit together in two dimensions, they cannot fit together in three dimensions.