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.