# H6X9H

Relaxation

The assumption of D being smaller than N can be relaxed. Suppose for the moment that N < D” < 2 * N  (optionally after decreasing D” modulo 2*N). The figure with this value of D” is similar to the figure with D = 2*N – D” (apart from a figure’s translation). The proof is based on operations on the full array of directions. The recipe for pair (N, D) can be written as [ R f(R,L) ] where f(R,L) denotes the partial recipe from the second direction to the Nth direction. Note that the first direction of the recipe (R) is an arbitrary choice; we could have chosen L instead of R as well. The partial recipe f has an anti-symmetry property. If the first direction of f is R, then the last direction is L, and vice versa. And this property holds also for the other pairs of directions in f. Consider the full array of directions for the figure with D” = 2*N – D. This array equals [ R f(L,R) ]6, where f(R,L) is the partial recipe based on D. When we mirror (with respect to the initial orientation) drawing this array, then we get the array [ L f(R,L) ]6 . Note that the drawing shows an anti-clockwise rotation. Now, start the drawing from the second direction so that the (cyclic) array becomes [ f(R,L) L ]6 . Finally, traverse the drawing in opposite direction starting at the last direction. This means that all directions are reversed from R to L, and from L to R . Furthermore, note that the clockwise rotation of the drawing is restored. Due to the anti-symmetry of f(R,L), the array becomes [ R f(R,L) ]6, being the array of the figure with D.