Topological structure of fractal squares

Lau, Ka–Sing; Luo, Jun; Rao, Hui

doi:10.1017/s0305004112000692

Cited by 38 publications

(23 citation statements)

References 17 publications

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“…We remind that generalised Sierpiński carpets [4,5] studied some years ago were also defined with the help of patterns, and are in general not self-similar. There are recent results [11] on the topology of a class of self-similar Sierpiński carpets called fractal squares. In the case of the mixed labyrinth fractals, there are special restrictions on the patterns, that correspond to the properties of labyrinth sets [2,3].…”

Section: Introductionmentioning

confidence: 99%

Mixed labyrinth fractals

Cristea

Steinsky

2017

Topology and its Applications

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Labyrinth fractals are self-similar fractals that were introduced and studied in recent work [2, 3]. In the present paper we define and study more general objects, called mixed labyrinth fractals, that are in general not self-similar and are constructed by using sequences of labyrinth patterns. We show that mixed labyrinth fractals are dendrites and study properties of the paths in the graphs associated to prefractals, and of arcs in the fractal, e.g., the path length and the box counting dimension and length of arcs. We also consider more general objects related to mixed labyrinth fractals, formulate two conjectures about arc lengths, and establish connections to recent results on generalised Sierpiński carpets.

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Section: Introductionmentioning

confidence: 99%

Mixed labyrinth fractals

Cristea

Steinsky

2017

Topology and its Applications

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“…Many examples of dendrites are defined as fractal squares [14]. A special case of such squares, fractal labyrinths, were studied in [5,6,7].…”

Section: Introductionmentioning

confidence: 99%

On deformation of polygonal dendrites preserving the intersection graph

Drozdov

Samuel²,

Tetenov

2020

Art Discrete Appl. Math.

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Let S = S 1 , ..., S m be a system of contracting similarities of R 2 . The attractor K(S) of the system S is a non-empty compact set satisfying K = S 1 (K) ∪ ... ∪ S m (K). We consider contractible polygonal systems S which are defined by a finite family of polygons whose intersection graph is a tree and therefore the attractor K(S) is a dendrite. We find conditions under which a deformation S of a contractible polygonal system S has the same intersection graph and therefore the attractor K(S ) is a self-similar dendrite which is isomorphic to the attractor K of the system S.

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“…al. [13], who concentrated on the topological structure (connectedness properties). In the present article we adopt the concept from [3,4] and introduce triangular labyrinth fractals.…”

Section: Introductionmentioning

confidence: 99%

Triangular Labyrinth Fractals

Cristea

Surer

2019

Fractals

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Labyrinth fractals are dendrites in the unit square. They were introduced and studied in the last decade first in the self-similar case [3,4], then in the mixed case [5,6]. Supermixed fractals constitute a significant generalisation of mixed labyrinth fractals: each step of * L.L. the iterative construction is done according to not just one labyrinth pattern, but possibly to several different patterns. In this paper we introduce and study supermixed labyrinth fractals and the corresponding prefractals, called supermixed labyrinth sets, with focus on the aspects that were previously studied for the self-similar and mixed case: topological properties and properties of the arcs between points in the fractal. The facts and formulae found here extend results proven in the above mentioned cases. One of the main results is a sufficient condition for infinite length of arcs in mixed labyrinth fractals.

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Topological structure of fractal squares

Cited by 38 publications

References 17 publications

Mixed labyrinth fractals

Mixed labyrinth fractals

On deformation of polygonal dendrites preserving the intersection graph

Triangular Labyrinth Fractals

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