Rotation symmetry of a plane lattice

When the various symmetry elements present in a shape are applied, it is found that one point is left unchanged by the transformations and when they are drawn on a figure, they all 

As described in Chapter 1, a crystal is defined by the fact that the whole structure can be built by the regular stacking of a unit cell that is translated but neither rotated nor reflected. The same is true for two-dimensional crystals or patterns. This imposes a limitation upon the combinations of symmetry elements that are compatible with the use of unit cells to build up a two-dimensional pattern or a three-dimensional crystal. To understand this, consider the rotational symmetry of the five unique plane lattices, described in the previous chapter. 

Suppose that a rotational axis of value n is normal to a plane lattice. It is convenient, (but not 

The solutions to this equation, for the lowest values of n, are given in Table 3.1. This shows that the only rotation axes that can occur in a lattice are 1, 2, 3, 4, and 6. The fivefold rotation 

This derivation provides a formal demonstration of the fact mentioned in Chapters 1 and 2, that a unit cell in a lattice or a pattern cannot possess five-fold symmetry. The same will be found to be true for three-dimensional lattices and crystals, described in Chapter 4. (Here it is pertinent to note that five-fold symmetry can occur in two- and three-dimensional patterns, if the severe constraints imposed by the mathematical definition of a lattice are relaxed slightly, as described in below in Section 3.7 and Chapter 8, Section 8.9. 

---