Golden gaskets: variations on the Sierpiński sieve

Abstract

We consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, with a common contraction factor λ ϵ (0, 1).As is well known, for λ = ½ the attractor, , is a fractal called the Sierpiński sieve and for λ < ½ it is also a fractal. Our goal is to study for this IFS for ½ < λ < 2/3, i.e. when there are ‘overlaps’ in as well as ‘holes’. In this introductory paper we show that despite the overlaps (i.e. the breaking down of the open set condition (OSC)), the attractor can still be a totally self-similar fractal, although this happens only for a very special family of algebraic λ (so-called multinacci numbers). We evaluate for these special values by showing that is essentially the attractor for an infinite IFS that does satisfy the OSC. 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 λ we show that if λ is close to 2/3, then has a non-empty interior. Finally we discuss higher-dimensional analogues of the model in question.