THE GENERALIZATION OF SIERPINSKI CARPET AND MENGER SPONGE IN n-DIMENSIONAL SPACE

Yang, Yun; Feng, Yuting; Yu, Yao

doi:10.1142/s0218348x17500402

Cited by 6 publications

(4 citation statements)

References 25 publications

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“…There are two kinds of generalization of the standard Sierpi ński carpets in three dimensions-the Sierpi ński cubes and the Menger sponges (see Refs. [141][142][143][144][145][146][147][148][149][150] and references therein). The peculiar properties of standard Sierpi ński carpets and cubes and Menger sponges remain an active topic of research (see Refs.…”

Section: The Standard Sierpiński Carpetsmentioning

confidence: 99%

A Brief Survey of Paradigmatic Fractals from a Topological Perspective

Ortíz

Martínez-Cruz

et al. 2023

Fractal Fract

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The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension D which exceeds the topological dimension d. In this regard, we point out that the constitutive inequality D>d can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined.

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Section: The Standard Sierpiński Carpetsmentioning

confidence: 99%

A Brief Survey of Paradigmatic Fractals from a Topological Perspective

Ortíz

Martínez-Cruz

et al. 2023

Fractal Fract

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“…The resulting fractal arrangement of copies of M will be denoted by P M . For a higher dimensional simplex Δ n generalizations of the Sierpinski construction have been considered, for instance, in [64].…”

Section: Fractal Structures On Manifoldsmentioning

confidence: 99%

Fractality in cosmic topology models with spectral action gravity

Guicardi¹,

Marcolli²

2022

Class. Quantum Grav.

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We consider cosmological models based on the spectral action formulation of (modiﬁed) gravity. We analyze the coupled eﬀects, in this model, of the presence of nontrivial cosmic topology and of fractality in the large scale structure of spacetime. We show that the topology constrains the possible fractal structures, and in turn the correction terms to the spectral action due to fractality distinguish the various cosmic topology candidates, with eﬀects detectable in a slow-roll inﬂation scenario, through the power spectra of the scalar and tensor ﬂuctuations. We also discuss explicit eﬀects of the presence of fractal structures on the gravitational waves equations.

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“…The resulting fractal arrangement of copies of M will be denoted by P M . For a higher dimensional simplex ∆ n generalizations of the Sierpinski construction have been considered, for instance, in [65].…”

Section: R×gamentioning

confidence: 99%

Fractality in Cosmic Topology Models with Spectral Action Gravity

Guicardi,

Marcolli

2021

Preprint

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We consider cosmological models based on the spectral action formulation of (modified) gravity. We analyze the coupled effects, in this model, of the presence of nontrivial cosmic topology and of fractality in the large scale structure of spacetime. We show that the topology constrains the possible fractal structures, and in turn the correction terms to the spectral action due to fractality distinguish the various cosmic topology candidates, with effects detectable in a slow-roll inflation scenario, through the power spectra of the scalar and tensor fluctuations. We also discuss explicit effects of the presence of fractal structures on the gravitational waves equations.

show abstract

THE GENERALIZATION OF SIERPINSKI CARPET AND MENGER SPONGE IN n-DIMENSIONAL SPACE

Cited by 6 publications

References 25 publications

A Brief Survey of Paradigmatic Fractals from a Topological Perspective

A Brief Survey of Paradigmatic Fractals from a Topological Perspective

Fractality in cosmic topology models with spectral action gravity

Fractality in Cosmic Topology Models with Spectral Action Gravity

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