Golden gaskets: variations on the Sierpiński sieve
By David S. Broomhead, James Montaldi, Nikita Sidorov. 2003
53 citations recorded by the source corpus.
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 in (0, 1). As is well known, for λ = 1/2 the attractor, S_λ, is a fractal called the Sierpinski sieve and for λ < 1/2 it is also a fractal. Our goal is to study S_λ for this IFS for 1/2 < λ < 2/3 , i.e. when there are ‘overlaps’ in S_λ 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 the ausdorff dimension of S_λ for these special values by showing that S_λ 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 S_λ has a non-empty interior. Finally we discuss higher-dimensional analogues of the model in question.
Subfields: Cell Biology, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/17480a53d7bee07716a538b354631aca626f48a9
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Cell Biology
In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Quantization (2019, unknown): “We write Let θ denote the empty word and A 0 := { θ } . we σ We define We need the total self-similarity which is proposed by Broomhead, Montaldi and Sidorov (see [1]): ∩ ∈ A [1, 2] for some interesting results and remarks on the total self-similarity.”
- Hausdorff Dimension for Randomly Perturbed Self Affine Attractors (2007, methodology): “expected dimension from Theorem 1 is 2 but was computed in [1] as 1.”