Geometric State Sum Models from Quasicrystals

Amaral, Marcelo M.; Fang, Fang; Hammock, Dugan; Irwin, Klee

doi:10.3390/foundations1020011

Cited by 3 publications

(6 citation statements)

References 28 publications

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“…As a final note, we refer to the paper [34,35] in the domain of quantum gravity, where the ordering rules with syntactical freedom are those of quasicrystals instead of those of the biological crystal structures.…”

Section: Discussionmentioning

confidence: 99%

Group Theory of Syntactical Freedom in DNA Transcription and Genome Decoding

Planat

Amaral

Fang

et al. 2022

CIMB

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Transcription factors (TFs) are proteins that recognize specific DNA fragments in order to decode the genome and ensure its optimal functioning. TFs work at the local and global scales by specifying cell type, cell growth and death, cell migration, organization and timely tasks. We investigate the structure of DNA-binding motifs with the theory of finitely generated groups. The DNA ‘word’ in the binding domain—the motif—may be seen as the generator of a finitely generated group Fdna on four letters, the bases A, T, G and C. It is shown that, most of the time, the DNA-binding motifs have subgroup structures close to free groups of rank three or less, a property that we call ‘syntactical freedom’. Such a property is associated with the aperiodicity of the motif when it is seen as a substitution sequence. Examples are provided for the major families of TFs, such as leucine zipper factors, zinc finger factors, homeo-domain factors, etc. We also discuss the exceptions to the existence of such DNA syntactical rules and their functional roles. This includes the TATA box in the promoter region of some genes, the single-nucleotide markers (SNP) and the motifs of some genes of ubiquitous roles in transcription and regulation.

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Section: Discussionmentioning

confidence: 99%

Group Theory of Syntactical Freedom in DNA Transcription and Genome Decoding

Planat

Amaral

Fang

et al. 2022

CIMB

Self Cite

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“…The basic idea that appears in many state sum constructions is to have a labeling scheme for some discretization, leading to a number which is shown to be an invariant, independent of the original discretization. In [31] we proposed the idea of labeling schemes for discrete models based on the discrete geometry of the discretization itself, which is straightforward to implement for general lattice-like models, in particular quasicrystals. At first glance, a labeling scheme based on the discretization itself seems to conflict with the desired goal of discretization independence, but this may be resolvable in structures that possess 'self' properties such as being self-modeling or self-contained.…”

Section: Geometric State Sum Models From Model Sets and Expectation V...mentioning

confidence: 99%

“…3-Geometric realism: The symmetry of the local weights should reflect the symmetry of the underlying discrete geometry as discussed above (see also [31,40]). Implementing the symmetry at the level of the weight itself has the bonus that the implementation of superposition of local configurations by the imposition of the first principle above must take into account the admissible geometric configurations of the tiling configuration space.…”

Section: Geometric State Sum Models From Model Sets and Expectation V...mentioning

confidence: 99%

“…4-Principle of efficient language (PEL) [41][42][43]: This aims to implement a notion of efficiency in discrete or computational systems, which are codes or languages. The random walks with their associated animation sets A κ i t above can be interpreted as a form of look-ahead algorithm (so-called look-savings-ahead algorithm in [31]) that allows us to define two computational functions that can be coupled to the geometric state sum model. First we consider the so-called hit potential, Y, which takes values on the natural numbers including zero, Y : T λ → N. We start with the initial condition of a tiling at x i with κ i at the origin, which generates the subset of translations T κ i ⊂ T from which each point of one animation will be sampled.…”

Section: Dynamics-computational Aspects Of Tiling Spacesmentioning

confidence: 99%

“…We focus on the former situation of quasicrystals considered at small scales looking for some understanding of large scale physics in classical/continuous limits. This more general and less explored situation allows us to relax constraints on the construction of the partition function (and associated probabilistic measure) in terms of the underlying Hamiltonian or Lagrangian and to propose directly the partition function in terms of a state sum model [30,31]. State sums allow for constructing models which are an intermediate tool between the path integral for a continuous theory in quantum gravity and a lattice approximation of the same theory, because of the state sums independence of the discretization.…”

Section: Introductionmentioning

confidence: 99%

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On the Emergence of Spacetime and Matter from Model Sets

Amaral¹,

Fang²,

Aschheim³

et al. 2022

Preprint

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We consider partition functions, in the form of state sums, and associated probabilistic measures for aperiodic substrates described by model sets and their associated tiling spaces. We propose model set tiling spaces as microscopic models for small scales in the context of quantum gravity. Model sets possess special self-similarity properties that allow us to consider implications on large and observable scales from the underlying (non-ergodic) dynamics. In particular we consider the implication of the underlying aperiodic substrate for the well known problem of time in quantum gravity, and propose a correspondence between small and large scales, the so-called ergodic correspondence, that addresses the emergence of matter properties and spacetime structure. In the process we find a possible bound in the mass spectrum of fundamental particles.

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From the Fibonacci Icosagrid to E8 (Part I): The Fibonacci Icosagrid, an H3 Quasicrystal

Fang,

Irwin

2024

Crystals

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This paper introduces a new kind of quasicrystal by Fibonacci-spacing a multigrid of a certain symmetry, like H2, H3, T3, etc. Multigrids of a certain symmetry can be used to generate quasicrystals, but multigrid vertices are not a quasicrystal due to arbitrary closeness. By Fibonacci-spacing the grids, the structure transit into an aperiodic order becomes a quasicrystal itself. Unlike the quasicrystal generated by the dual-grid method, this kind of quasicrystal does not live in the dual space of the grid space. It is the grid space itself and possesses quasicrystal properties, except that its total number of vertex types are not finite and fixed for the infinite size of the quasicrystal but bounded by a slowly logarithmic growing number. A 2D example, the Fibonacci pentagrid, is given. A 3D example, the Fibonacci icosagrid (FIG), is also introduced, as well as its subsets, the Fibonacci tetragrid (FTG). The FIG can be thought of as a golden composition of five sets of FTGs. The golden composition procedure is another way to transit a random structure into aperiodic order, and the associated rotational angle is the same as the angle that resolves the geometric frustration for the H3 tetrahedral clusters. The FIG resembles another quasicrystal that is the same golden composition of five quasicrystals that are cut and projected and sliced from the E8 lattice. This leads to further exploration in mapping the FIG to the E8 lattice, and the results will be published following this paper.

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Geometric State Sum Models from Quasicrystals

Abstract: In light of the self-simulation hypothesis, a simple form of implementation of the principle of efficient language is discussed in a self-referential geometric quasicrystalline state sum model in three dimensions. Emergence is discussed in the context of geometric state sum models.

Cited by 3 publications

References 28 publications

Group Theory of Syntactical Freedom in DNA Transcription and Genome Decoding

Group Theory of Syntactical Freedom in DNA Transcription and Genome Decoding

On the Emergence of Spacetime and Matter from Model Sets

From the Fibonacci Icosagrid to E8 (Part I): The Fibonacci Icosagrid, an H3 Quasicrystal

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