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 N^{th}
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.