Unimodular transformations

We start by examining the transformations of a square lattice onto itself.

1 On some square lattice paper, choose four lattice points A, B, C and D so that ABCD forms a parallelogram, ᴨ, without lattice points inside the parallelogram or on its perimeter except at its vertices. Sketch the image of ᴨ under the translation of the lattice which maps A to B. Call this translation. Sketch the image of ᴨ under the translation of the lattice which maps A to D. Call this translation σ. Do and σ (ᴨ) have lattice points inside them or on their perimeters? Do or have lattice points inside them or on their perimeters for any integers m or nl Is the same true for the parallelogram?

2 With the notation of q 1, does every point of the plane lie within or on the perimeter of one or more of the parallelograms? Must each lattice point be a vertex of four such parallelograms?

3 Using the conventional coordinate system with rectangular cartesian axes for the plane, the set of square lattice points may be labelled with the set of all ordered pairs of integers. If ABCD is a parallelogram of lattice points as in q 1, we are free to choose A as the origin of our coordinate system. If B = (a, b) and D = (c, d), what are the coordinates of C if C is the vertex of the parallelogram opposite Al What is the image of the unit square ﹛(0, 0), (1, 0), (1, 1), (0,1)﹜ under the linear transformation

What are the images of the lattice points of the form (x, 0) under α? What are the images of the lattice points of the form (0, y) under α ? The lines x = k, y = l, parallel to the axes, with k and l integers, form a grid of unit squares. What is the image of this grid under the linear transformation α?

4 Does the linear transformation α of q 3, map the set of lattice points onto itself?