Symmetries in two dimensions
We would now like to discuss some of the properties of crystals from the point of view of their internal symmetries. The main feature of a crystal is that if you start at one atom and move to a corresponding atom one lattice unit away, you are again in the same kind of an environment. That’s the fundamental proposition. But if you were an atom, there would be another kind of change that could take you again to the same environment—that is, another possible “symmetry.” Figure 30–7(a) shows another possible “wallpaper-type” design (though one you have probably never seen). Suppose we compare the environments for points $A$ and $B$. You might, at first, think that they are the same—but not quite. Points $C$ and $D$ are equivalent to $A$, but the environment of $B$ is like that of $A$ only if the surroundings are reversed, as in a mirror reflection.  There are other kinds of “equivalent” points in the pattern. For instance, the points $E$ and $F$ have the “same” environments except that one is rotated $90^\circ$ with respect to the other. The pattern is quite special. A rotation of $90^\circ$—or any multiple of it—about a vertex such as $A$ gives the same pattern all over again. A crystal with such a structure would have square corners on the outside, but inside it is more complicated than a simple cube.  Now that we have described some special examples, let’s try to figure out all the possible symmetries a crystal can have. First, we consider what happens in a plane. A plane lattice can be defined by the two so-called primitive vectors that go from one point of the lattice to the two nearest equivalent points. The two vectors $\FLPone$ and $\FLPtwo$ are the primitive vectors of the lattice of Fig. 30–1. The two vectors $\FLPa$ and $\FLPb$ of Fig. 30–7(a) are the primitive vectors of the pattern there. We could, of course, equally well replace $\FLPa$ by $-\FLPa$, or $\FLPb$ by $-\FLPb$. Since $\FLPa$ and $\FLPb$ are equal in magnitude and at right angles, a rotation of $90^\circ$ turns $\FLPa$ into $\FLPb$, and $\FLPb$ into $-\FLPa$, giving the same lattice once again.  We see that there are lattices which have a “four-sided” symmetry. And we have described earlier a close-packed array based on a hexagon which could have a six-sided symmetry. A rotation of the array of circles in Fig. 30–5(a) by an angle of $60^\circ$ about the center of any circle brings the pattern back to itself.  What other kinds of rotational symmetry are there? Can we have, for example, a fivefold or an eightfold rotational symmetry? It is easy to see that they are impossible. The only symmetry with more sides than four is a six-sided symmetry. First, let’s show that more than sixfold symmetry is impossible. Suppose we try to imagine a lattice with two equal primitive vectors with an enclosed angle less than $60^\circ$, as in Fig. 30–8(a). We are to suppose that points $B$ and $C$ are equivalent to $A$, and that $\FLPa$ and $\FLPb$ are the two shortest vectors from $A$ to its equivalent neighbors. But that is clearly wrong, because the distance between $B$ and $C$ is shorter than from either one to $A$. There must be a neighbor at $D$ equivalent to $A$ which is closer than $B$ or $C$. We should have chosen $\FLPb'$ as one of our primitive vectors. So the angle between the two primitive vectors must be $60^\circ$ or larger. Octagonal symmetry is not possible.  What about fivefold symmetry? If we assume that the primitive vectors $\FLPa$ and $\FLPb$ have equal lengths and make an angle of $2\pi/5=72^\circ$, as in Fig. 30–8(b), then there should also be an equivalent lattice point at $D$, at $72^\circ$ from $C$. But the vector $\FLPb'$ from $E$ to $D$ is then less than $\FLPb$, so $\FLPb$ is not a primitive vector. There can be no fivefold symmetry. The only possibilities that do not get us into this kind of difficulty are $\theta=60^\circ$, $90^\circ$, or $120^\circ$. Zero or $180^\circ$ are also clearly possible. One way of stating our result is that the pattern can be left unchanged by a rotation of one full turn (no change at all), one-half of a turn, one-third, one-fourth, or one-sixth of a turn. And those are all the possible rotational symmetries in a plane—a total of five. If $\theta=2\pi/n$, we speak of an “$n$-fold” symmetry. We say that a pattern with $n$ equal to $4$ or to $6$ has a “higher symmetry” than one with $n$ equal to $1$ or to $2$.  Returning to Fig. 30–7(a), we see that the pattern has a fourfold rotational symmetry. We have drawn in Fig. 30–7(b) another design which has the same symmetry properties as part (a). The little comma-like figures are asymmetric objects which serve to define the symmetry of the design inside of each square. Notice that the commas are reversed in alternate squares, so that the unit cell is larger than one of the small squares. If there were no commas, the pattern would still have fourfold symmetry, but the unit cell would be smaller. The patterns of Fig. 30–7 also have other symmetry properties. For instance, a reflection about any of the broken lines $R$–$R$ reproduces the same pattern.  The patterns of Fig. 30–7 have still another kind of symmetry. If the pattern is reflected about the line $Y$–$Y$ and shifted one square to the right (or left), we get back the original pattern. The line $Y$–$Y$ is called a “glide” line.  These are all the possible symmetries in two dimensions. There is one more spatial symmetry operation which is equivalent in two dimensions to a $180^\circ$ rotation, but which is a quite distinct operation in three dimensions. It is inversion. By an inversion we mean that any point at the vector displacement $\FLPR$ from some origin [for instance, the point $A$ in Fig. 30–9(b)] is moved to the point at $-\FLPR$.  An inversion of pattern (a) of Fig. 30–9 produces a new pattern, but an inversion of pattern (b) reproduces the same pattern. For a two-dimensional pattern (as you can see from the figure), an inversion of the pattern (b) through the point $A$ is equivalent to a rotation of $180^\circ$ about the same point. Suppose, however, we make the pattern in Fig. 30–9(b) three dimensional by imagining that the little $6$’s and $9$’s each have an “arrow” pointing out of the page. After an inversion in three dimensions all the arrows will be reversed, so the pattern is not reproduced. If we indicate the heads and tails of the arrows by dots and crosses, respectively, we can make a three-dimensional pattern, as in Fig. 30–9(c), which is not symmetric under an inversion, or we can make a pattern like the one shown in (d), which does have such a symmetry. Notice that it is not possible to imitate a three-dimensional inversion by any combination of rotations.  If we characterize the “symmetry” of a pattern—or lattice—by the kinds of symmetry operations we have been describing, it turns out that for two dimensions $17$ distinct patterns are possible. We have drawn one pattern of the lowest possible symmetry in Fig. 30–1, and one of high symmetry in Fig. 30–7. We will leave you with the game of trying to figure out all of the $17$ possible patterns.  It is peculiar how few of the $17$ possible patterns are used in making wallpaper and fabrics. One always sees the same three or four basic patterns. Is this because of a lack of imagination of designers, or because many of the possible patterns are not pleasing to the eye? 