Mathematical_Recreations-Kraitchik-2e

156 Mathematical Recreations
then lie under the points (x, y) for which the values of x differ
from the numbers 1 to m by some fixed multiple of m, and
the values of y differ from the numbers from 1 to n by some
fixed multiple of n. Since, after all, the repetitions of the
figure are essentially the figure itself, we may then consider
a cell of the figure to be covered by all the points (a + rm,
b + sn), where (a, b) is the point covering the cell in its initial
position, and rand s are any two integers.
From one point of view, then, this procedure associates
with each cell of the fundamental configuration a lattice
(a, b) + rem, 0) + s(O, n) of points from the basic lattice.
From another point of view, and this is the one that
we shall usually adopt, the different points of the lattice
(a, b) + rem, 0) + s(O, n) are not considered as being distinct.
That is, two points whose x-co-ordinates differ by a multiple
of m and whose y-co-ordinates differ by a multiple of n are
considered to be identical. This can be expressed by saying
that the x-co-ordinates are taken with respect to the modulus
m, and the y-co-ordinates with respect to the modulus n.
Such a lattice will be called a basic lattice with respect to the
modulus em, nJ. When, as is most often the case, m = n,
we shall speak of the basic lattice with respect to the modulusm.
Very few modifications of the theory are necessary in order
to take care of modular lattices. One point is worth noting,
however. Suppose we have in the basic lattice with respect
to the modulus em, nJ a lattice (a, b) + ret, u) + s(v, w) and
a lattice (a, b) + r'(t, u) + s'(v, w), where r == r', modulo m,
and s == s', modulo n. If these two lattices are drawn in the
nonmodular plane they may look different. But if all the
points covered by each lattice are brought back to the corresponding
points of the fundamental rectangle it will be
found that each of the two lattices covers precisely the same
set of points. Such lattices will not, then, be considered dis-