ENUMERATION 

The problem is: find the number of all different monolinear curves (i.e. the corresponding mirror arrangements) which could 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, ..., 2abab). Unfortunately, we are very far from the general solution of this problem. Reason for this is: 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. For example, if the possible positions for an additional internal mirror are orange (the left figure), after introducing that new mirror (turquoise mirror), the possible positions for the next (orange) mirror depend from its position. Till now we have only few combinatorial results, obtained for some particular cases by the author, and generalized by G.Baron. Let be given a rectangular grid RG[a,b], k = gcd(a,b) (gcd  greatest common divisor), and let be introduced the minimal number k1 of twosided internal mirrors incident to the celledges. If t = (ablcm(a,b)):(k(k1)) = 4xy (lcm  lowest common measure), x = a:(2k), y = b:(2k), we have the following results, where for different k are given the conditions for a, b, and the number of curves: with k1 only edgeincident mirrors, and a non equal to b, for k odd (4k)^{k2}t^{k}^{1} + (4k)^{(k2)/2}}zt^{(k2)/2}, for k even, with z=x for a = 0 mod 2k, b = k mod 2k, and z = x+y for a = b = k mod 2k; with k1 edgeincident or edgeperpendicular mirrors, and a non equal to b, for k odd 2(8k)^{k2}t^{k}^{1} + 2(8k)^{(k2)/2}zt^{(k2)/2}, for k even, with z = x for a = 0 mod 2k, b = k mod 2k, and z = x+y for a = b = k mod 2k; For a = b we have to put t = 1, z = 1, divide the numbers by 2, and get for k1 only edgeincident mirrors: for k odd 8k^{2k5}k^{k}^{2} + 2^{k3}k^{(k2)/2},
for k even;
for k odd (8k)^{k2} + (8k)^{(k2)/2}, for k even. 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 large enough. G.Baron also derived formulas for the case a = b with the larger groups of symmetries and, finally, 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. 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. 