Mathematical_Recreations-Kraitchik-2e

Magic Squares 153
in one-, two-, or three-dimensional space whose co-ordinates
are integers, positive, negative, or zero. Any point of such
a lattice may be denoted by its co-ordinates, (x), (x, y) or
(x, y, z), depending on the dimension. We shall confine ourselves
almost wholly to lattices lying in a plane. We shall
suppose then that our basic lattice is the set of ali points
(x, y) having x and y = 0, ± 1, ±2, ...
It is useful to attach a second meaning to the symbol
(x, y), in that (x, y) may be thought of as representing the motion
of translation from the origin (0, 0) to the lattice point
(x, y), or the parallel motion in the same direction from any
lattice point (a, b) to the lattice point (a + x, b + y). In this
sense we shall call (x, y) a motion, and x and y its direction
numbers. The motion opposite to a motion (x, y) is (-x,
-y), and will be denoted by -(x, y). The result of two successive
motions, (x, y) and (t, u), is the motion (x + t, y + u),
which will be expressed by (x, y) + (t, u).
The motion
(nx, ny) resulting from n repetitions of the motion (x, y) will be
denoted by n(x, y). Thus we may have a single motion expressed
as a linear combination of other motions, as (x, y) =
aCt, u) + b(v, w) + c(p, q), where a, b, c are signed integers.
Keeping in mind the double significance of the symbol
(x, y), we may add a motion to a fixed point and interpret the
result as the point obtained by moving in the given way from
the given point. Thus (a, b) + (t, u) may be interpreted to
mean the result of moving t units horizontally and u units
vertically from the point (a, b) to the point (a + t, b + u).
This is particularly important when we add to a fixed point
the indefinite repetitions of one or more motions. Thus
(x, y) = (a, b) + ret, u), r = 0, ± 1, ±2, .. " may be used to denote
the points (x, y) obtained from (a, b) by indefinite
repetitions of the motion (t, u) in either direction. For
example, (x, y) = (1, 2) + r(3, -4) represents the points
(1, 2), (4, -2), (-2,6), (7, -6), (-5, 10), and so on.