3.2.4 EnumerationThe problem is to finding the number of all different monolinear curves (i.e. the corresponding mirror arrangements) which can be derived from a rectangular grid RG[a,b] with the sides a, b, covered by k curves, for a given number of mirrors m (m = k1, k, ¼, 2abab). Unfortunately, we are very far from the general solution of this problem. The reason is that: every introduction of an internal mirror changes the whole structure, so it behaves like some kind of Game of Life or cellular automata, where a local change results in the global change. Till now we have only a few combinatorial results (Harary and Palmer, 1973), obtained for some particular cases by S. Jablan, and generalized by G. Baron. Let a rectangular grid RG[a,b], k = GCD(a,b), be given, and let the minimal number k1 of twosided internal mirrors be introduced incident to the celledges. If t = (abLCM(a,b)):(k(k1)) = 4xy (LCM least common multiple), x = [a/ 2k], y = [b/ 2k], we have the following results, where conditions for a, b, and the number of curves are given for different values of k: (I) with k1 only edgeincident mirrors, and a ¹ b,
(II) with k1 edgeincident or edgeperpendicular mirrors, and a non equal to b,
For a = b we have to put t = 1, z = 1, divide the numbers by 2, and get (i) for k1 only edgeincident mirrors
(ii) with k1 edgeincident or edgeperpendicular mirrors:
Even for some smaller rectangles (e.g., a = 6, b = 3), and minimal number of mirrors (k1 = 2), the number of the different curves obtained is very large. For example, there are 52 different arrangements of two edgeincident mirrors in a rectangle 6×3 producing perfect curves. Among them, only 8 are symmetrical 4 mirrorsymmetrical, and 4 pointsymmetrical. G. Baron also derived formulas for the case a = b with the larger groups of symmetries constructed for k = a = b equal 2 or 3 and the maximum number of mirrors all different mirrorschemes. There is only one for k = 2 and 28 for k = 3.
