  Let $V$ be society whose members express preferences about two alternatives,
indifference included. Identifying anonymous binary social choice functions
with binary functions $f=f(k,m)$ defined over the integer triangular grid
$G=\{(k,m)\in \mathbb{N}_0\times\mathbb{N}_0 : k+m\le |V|\} $, we show that
every strategy-proof, anonymous social choice function can be described
geometrically by listing, in a sequential manner, groups of segments of G, of
equal (maximum possible) length, alternately horizontal and vertical,
representative of preference profiles that determine the collective choice of
one of the two alternatives. Indeed, we show that every function which is
anonymous and strategy-proof can be described in terms of a sequence of
nonnegative integers $(q_1, q_2, \cdots, q_s)$ corresponding to the
cardinalities of the mentioned groups of segments. We also analyze the
connections between our present representation with another of our earlier
representations involving sequences of majority quotas.
  A Python code is available with the authors for the implementation of any
such social choice function.
