3.2.4  Enumeration

The 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 = k-1, k, ¼, 2ab-a-b). 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 k-1 of two-sided internal mirrors be introduced incident to the cell-edges. If t = (ab-LCM(a,b)):(k(k-1)) = 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 k-1 only edge-incident mirrors, and a ¹ b, 

1. for k odd: (4k)^k-2t^k-1 + 2(4k)^{[(k-3)/ 2]}t^{[(k-1)/ 2]}; 

2. for k even: (4k)^k-2t^k-1 + (4k)^{[(k-2)/ 2]}zt^{[(k-2)/ 2]}, with z = x for aº0 mod 2k, bº k mod 2k, and z = x+y, for a ºbºk mod 2k,

(II) with k-1 edge-incident or edge-perpendicular mirrors, and a non equal to b, 

1. for k odd: 2(8k)^k-2t^k-1 + 4(8k)^{[(k-3)/ 2]}t^[(k-1)/2]; 

2. for k even: 2(8k)^k-2t^k-1 + 2(8k)^{[(k-2)/ 2]}zt^{[(k-2)/ 2]}, with z = x for aº0 mod 2k, bº k mod 2k, and z = x+y for a ºbºk mod 2k.

For a = b we have to put t = 1, z = 1, divide the numbers by 2, and get

(i) for k-1 only edge-incident mirrors 

1. for k odd: 2^2k-5k^k-2 + 2k-3k^{[(k-3)/ 2]}; 

2. for k even: 8k^2k-5k^k-2 + 2k-3k^{[(k-2)/ 2]},

(ii) with k-1 edge-incident or edge-perpendicular mirrors: 

1. for k odd: (8k)^k-2 + 2(8k)^{[(k-3)/ 2]}; 

2. for k even: (8k)^k-2 + (8k)^{[(k-2)/ 2]}.

Even for some smaller rectangles (e.g., a = 6, b = 3), and minimal number of mirrors (k-1 = 2), the number of the different curves obtained is very large. For example, there are 52 different arrangements of two edge-incident mirrors in a rectangle 6×3 producing perfect curves. Among them, only 8 are symmetrical- 4 mirror-symmetrical, and 4 point-symmetrical. 

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 mirror-schemes. There is only one for k = 2 and 28 for k = 3.