  Even the most superficial glance at the vast majority of crossing-minimal
geometric drawings of $K_n$ reveals two hard-to-miss features. First, all such
drawings appear to be 3-fold symmetric (or simply {\em 3-symmetric}) . And
second, they all are {\em 3-decomposable}, that is, there is a triangle $T$
enclosing the drawing, and a balanced partition $A, B, C$ of the underlying set
of points $P$, such that the orthogonal projections of $P$ onto the sides of
$T$ show $A$ between $B$ and $C$ on one side, $B$ between $A$ and $C$ on
another side, and $C$ between $A$ and $B$ on the third side. In fact, we
conjecture that all optimal drawings are 3-decomposable, and that there are
3-symmetric optimal constructions for all $n$ multiple of 3. In this paper, we
show that any 3-decomposable geometric drawing of $K_n$ has at least
$0.380029\binom{n}{4}+\Theta(n^3)$ crossings. On the other hand, we produce
3-symmetric and 3-decomposable drawings that improve the {\em general} upper
bound for the rectilinear crossing number of $K_n$ to
$0.380488\binom{n}{4}+\Theta(n^3)$. We also give explicit 3-symmetric and
3-decomposable constructions for $n<100$ that are at least as good as those
previously known.
