  The dynamical system generated by the iterated calculation of the high order
gaps between neighboring terms of a sequence of natural numbers is remarkable
and only incidentally characterized at the boundary by the notable
Proth-Glibreath Conjecture for prime numbers.
  We introduce a natural extension of the original triangular arrangement,
obtaining a growing hexagonal covering of the plane. This is just the base
level of what further becomes an endless discrete helicoidal surface. %
Although the repeated calculation of higher-order gaps causes the numbers that
generate the helicoidal surface to decrease, there is no guarantee, and most
often it does not even happen, that the levels of the helicoid have any
regularity, at least at the bottom levels.
  However, we prove that there exists a large and nontrivial class of sequences
with the property that their helicoids have all levels coinciding with their
base levels. This class includes in particular many ultimately binary sequences
with a special header. % For almost all of these sequences, we additionally
show that although the patterns generated by them seem to fall somewhere
between ordered and disordered, exhibiting fractal-like and random qualities at
the same time, the distribution of zero and non-zero numbers at the base level
has uniformity characteristics. Thus, we prove that a multitude of straight
lines that traverse the patterns encounter zero and non-zero numbers in almost
equal proportions.
