  We consider the iterated function systems (IFSs) that consist of three
general similitudes in the plane with centres at three non-collinear points,
and with a common contraction factor $\la\in(0,1)$.
  As is well known, for $\la=1/2$ the invariant set, $\S_\la$, is a fractal
called the Sierpi\'nski sieve, and for $\la<1/2$ it is also a fractal. Our goal
is to study $\S_\la$ for this IFS for $1/2<\la<2/3$, i.e., when there are
"overlaps" in $\S_\la$ as well as "holes". In this introductory paper we show
that despite the overlaps (i.e., the Open Set Condition breaking down
completely), the attractor can still be a totally self-similar fractal,
although this happens only for a very special family of algebraic $\la$'s
(so-called "multinacci numbers"). We evaluate $\dim_H(\S_\la)$ for these
special values by showing that $\S_\la$ is essentially the attractor for an
infinite IFS which does satisfy the Open Set Condition. We also show that the
set of points in the attractor with a unique ``address'' is self-similar, and
compute its dimension.
  For ``non-multinacci'' values of $\la$ we show that if $\la$ is close to 2/3,
then $\S_\la$ has a nonempty interior and that if $\la<1/\sqrt{3}$ then \S_\la$
has zero Lebesgue measure. Finally we discuss higher-dimensional analogues of
the model in question.
