Mathematical_Recreations-Kraitchik-2e

154 Mathematical Recreations
In full generality the points (x, y) =
(a, b) + ret, u) are
those having co-ordinates x = a + rt, y = b + ru for all integral
values of r. This shows that these points are just those
points of the basic lattice which lie on the line with slope 7
through the point (a, b). These points obviously form a onedimensional
lattice in the plane. It is equally obvious that
every one-dimensional lattice composed of points in the basic
plane lattice can be represented by such a notation. For if
Ca, b) and Ce, d) are any two consecutive points on the onedimensional
lattice we can take (t, u) = (c - a, d - b) and
write (x, y) = (a, b) + r(e - a, d - b).
Hence we shall call the set of points with co-ordinates
(x, y) = (a, b) + ret, u) the (one-dimensional) lattice with
origin (a, b) generated by the motion (t, u), or having
direction numbers t, u. If the lattice goes through (0, 0)
it may just as well be expressed by ret, u), unless we have
particular occasion to think of some one of its points as a
starting point.
From what has been said about the one-dimensional case
it is easy to guess the interpretation of the expression (a, b) +
ret, u) + s(v, w). If (t, u) and (v, w) are motions in different
lines, which will be the case if and only if t: u ~ v: w, that is if
tw - uv ~ 0, the points whose co-ordinates are given by (x, y)
= (a, b) + ret, u) + s(v, w) form a two-dimensional lattice
of points drawn from the fundamental lattice, and every such
lattice can be so expressed. (If the directions of the motion
are not distinct it can be shown that the expression generates
a one-dimensional lattice, which may then be expressed by a
single motion.) Such a lattice will be said to have (a, b) as
origin and to be generated by the motions (t, u) and (v, w).
As before, if (0, 0) lies in the lattice we shall usually denote
the lattice by ret, u) + s(v, w). For example, the basic lattice
may be represented by (x, y) = rei, 0) + s(O, 1).