The Crystallographic Restriction 
In order for this lattice to possess rotational symmetry of order k it is necessary for each lattice point to map to another under rotation through an angle q = 2p/k. In other words, for every pair of integers m,n, there must be another pair of integers M,N such that 
Letting subscripts r and i denote the real and imaginary parts of u and v, we can expand this expression and separate the real and imaginary parts to give the two conditions 

Solving these equations for M and N in terms of m and n, we get 



Furthermore, if we let f denote the angle between the basis vectors (i.e., the phase difference between the complex numbers) u and v, then we have the expressions for the scalar and vector products 

where 
The determinant k_{1}k_{4}  k_{2}k_{3} is identically equal to 1. We also have the relations 
Eliminating f from the first and third, we also have 
From this we also get 
To illustrate, we can set q equal to p/3 = 60 degrees, which is one of the four possible (nontrivial) values. In this case we have 
and so k_{1} + k_{4} = 1. Now it appears we have some freedom of choice, because we can select integers k_{2}, k_{3} such that k_{2}k_{3}  sin(q)^{2} is a square integer m^{2}, and then compute the values of f, u/v, and k_{4}  k_{1}, with which we can infer the individual (integer) values of k_{1} and k_{4}. For any odd integer m we can put 
and then we have 
Thus k_{1} = (1m)/2 and k_{4} = (1+m)/2. The individual coefficients k_{2} and k_{3} can be any factorization of (m^{2}+3)/4, one positive and one negative. The angle f between the two basis vectors is given (in this example) by 
The factorization of k_{2}k_{3} is needed only to determine the ratio of the lengths of the basis vectors using the relation 
The simplest possibility is to set k_{2}k_{3} = 1 by putting k_{2} = 1 and k_{3} = 1. This corresponds to m = 1, which gives basis vectors of equal length, and the angle between them is f = 60 degrees. We also get k_{4} = 1 and k_{1} = 0, so the lattice transformation is M = n, N = m + n. This gives the expected triangular lattice with 6fold rotational symmetry shown below. 
On the other hand, if we take m = 5 we get k_{2}k_{3} = 7, and we can select the factorization k_{2} = 1, k_{3} = 7. In this case we get k_{1} = 2, k_{4} = 3, and 
It may seem as if this gives a different lattice from the previous case, but if we plot the lattice points given by these expressions we get the figure shown below. 
It is the same lattice as before, merely expressed in terms of different basis vectors. It follows that the "fundamental region" of a lattice is not unique, because the same lattice can be described in terms of different sets of basis vectors, corresponding to different fundamental regions. In order to arrive at a less arbitrary definition, we would stipulate that the basis vectors must be the smallest possible. 