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A Sourcebook for the Worldwide Discovery of a Creative Organic Universe
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III. Ecosmos: A Revolutionary Fertile, Habitable, Solar-Bioplanet Incubator Lifescape

E. Systems Cosmology: Fractal SpaceTimeMatter

Smet, Jurgen. Wheels within Wheels. Nature. 422/391, 2003. A report on the discovery that when electrons constrained to move in a plane are exposed to a perpendicular magnetic field, known as the Quantum Hall effect, the curve of voltage vs. field strength takes on an iterative, fractal self-similarity.

Smolin, Lee. Three Roads to Quantum Gravity. New York: Basic Books, 2001. Noted earlier in Quantum Cosmology, the work entertains theories of the grainy, fractal, and holographic character of an integral universe.

We realized during that work that one way of making such a fractal spacetime is to build it up from a network of interacting loops. (124)

Sole, Ricard and A. Munteanu. The Large-Scale Organization of Chemical Reaction Networks in Astrophysics. EPL Europhysics Letters. 68/2, 2004. Universitat Pompeu, Fabra, Spain, systems theorists make an early, prescient notice that small world, scale-free topologies appear in astrochemical complexities. This reference is cited in a 2016 Nature Scientific Reports paper Multilayer Network Analysis of Nuclear Reactions (6/31882, 2016, see Liang Zhu in Systems Chemistry) whence a dozen years later this organic physiology and anatomy is robustly verified from cosmos to culture.

The large-scale organization of complex networks, both natural and artificial, has shown the existence of highly heterogeneous patterns of organization. Such patterns typically involve scale-free degree distributions and small-world, modular architectures. One example is provided by chemical reaction networks, such as the metabolic pathways. The chemical reactions of the Earth's atmosphere have also been shown to give rise to a scale-free network. Here we present novel data analysis on the structure of several astrophysical networks including the chemistry of the planetary atmospheres and the interstellar medium. Our work reveals that Earth's atmosphere displays a hierarchical organization, close to the one observed in cellular webs. Instead, the other astrophysical reaction networks reveal a much simpler pattern consistent with an equilibrium state. (Abstract)

Sowmya, G., et al. Supergranular Fractal Dimension and Solar Radiation. arXiv:2207.10490. In these 2020s, CSSS Institute of Engineering and Technology for Women, Karnataka, Poornaprajna Institute of Scientific Research, Devanahalli, and Bangalore University astrophysicists proceed to find and quantify evidence of deep self-similarities even in the sun’s rays.

In regard, see The Sun’s Supergranulation by Francois Rincon and Michel Rieutord in Living Reviews in Solar Physics (15/6, 2018) whence Supergranulation is a fluid-dynamical phenomenon in the solar photosphere in the form of a vigorous cellular flow pattern.

We present findings from an analysis of the fractal dimension of solar supergranulation as a function of latitude, cell size and solar rotation, by way of spectroheliographic data. We find that the fractal dimension tends to decrease from about 1.37 at the equator to about 1 at 20 degree latitude in either hemisphere, suggesting that solar rotation rate has the effect of augmenting the irregularity of supergranular boundaries. (Excerpt)

Sroor, Hend, et al. Fractal Light from Lasers. arXiv:1809.02501. We cite this clever entry by University of Witwatersrand and CSIR National Laser Center, Pretoria, RSA researchers including Andrew Hughes for its quantified sense of how much our natural abide seems to be innately graced and suffused by self-similar, infinitely iterating geometries, which would quite please Galileo as we come upon 400 years of his famous avowal to this effect.

Fractals, complex shapes with structure at multiple scales, have long been observed in Nature: as symmetric fractals in plants and sea shells, and as statistical fractals in clouds, mountains and coastlines. With their highly polished spherical mirrors, laser resonators are almost the precise opposite of Nature, and so it came as a surprise when, in 1998, transverse intensity cross-sections of the eigenmodes of unstable canonical resonators were predicted to be fractals. Experimental verification has so far remained elusive. Here we observe a variety of fractal shapes in transverse intensity cross-sections through the lowest-loss eigenmodes of unstable canonical laser resonators, thereby demonstrating the controlled generation of fractal light inside a laser cavity. Our work offers a significant advance in the understanding of a fundamental symmetry of Nature as found in lasers. (Abstract)

Sylos Labini, Francesco, et al. Persistent Fluctuations in the Distributions of Galaxies from the Two-degree Field Galaxy Redshift Survey. EPL. 85/29002, 2009. A team of Italian and Russian astronomers further quantify self-similar geometries across the celestial realms.

We apply the scale-length method to several three-dimensional samples of the Two-degree Field Galaxy Redshift Survey. This method allows us to map in a quantitative and powerful way large scale structures in the distribution of galaxies controlling systematic effects. By determining the probability density function of conditional fluctuations we show that large-scale structures are quite typical and correspond to large fluctuations in the galaxy density field. We do not find a convergence to homogeneity up to the samples sizes, i.e. 75 Mpc/h. We then measure, at scales r

Tatekawa, Takayuki and Kei-chi Maeda. Primordial Fractal Density Perturbations and Structure Formation in the Universe. The Astrophysical Journal. 547/531, 2001. A technical paper on how such recurrently ordered forms appear and ramify in the developing cosmos.

One of the most plausible explanations is the nonlinear dynamics of the perturbations will provide such a scale-free structure during the evolution of the universe. (531)

Teles, Sharon, et al. Fractal Analysis of the UltraVISTA Galaxy Survey. arXiv:2013.07164. In 2000, one could find inklings that self-similar geometries might be present across the celestial reaches. Two decades later, Valongo Observatory, Universidade Federal do Rio de Janeiro Observatório Nacional astronomers add another strong confirmation that all manner of spacescape phenomena is indeed arrayed in and sustained by fractal scales.

This paper seeks to test if the large-scale galaxy distribution can be characterized as fractal system. Tools appropriate for describing galaxy fractal structures with a single fractal dimension in relativistic settings are developed and applied to the UltraVISTA galaxy survey. A graph of volume-limited samples corresponding to the redshift limits in each redshift bins for absolute magnitude is presented. The results show two consecutive and distinct redshift ranges in both the reduced and complete samples where the data behave as a single fractal galaxy structure. (Abstract excerpt)

The fractal galaxy distribution hypothesis is an approach for the description of the large-scale structure of the Universe which assumes that this distribution is formed by a fractal system. This approach characterizes the system by means of its key feature, thefractal dimension, which is basically a way of quantifying the irregularity of the distribution. In the context of the large-scale structure of the Universe, essentially measures galactic clustering sparsity or, complementary, the dominance of voids. (1)

Differently from the single fractal approach, the multifractal one characterizes the fractal system by several fractal dimensions in the same scaling range, that is, a whole spectrum of dimensions whose maximum value corresponds to the single fractal dimension the structure would have if it were treated as a single fractal. The multifractal approach is applied when quantities like galactic luminosity or mass have a distribution, that is, they range between very different values. (1)

Theel, Friethjof, et al. The Fractal Geometry of Hartree-Fock. Chaos. 27/12, 2017. When this section went online in 2004, scientific perceptions of a natural self-similarity from atomic depths to cosmic breadth were spurious and rudimentary, with a smattering of evidence. A decade and a half later, as this entry by University of Hamburg physicists, and many citations herein now testify, iterative fractal forms are quantified and known to array across these reaches, and everywhere in between. Circa 2018, by a natural philosophy view, our worldwide humankinder seems to be well finding a new genesis universe graced by these intrinsic phenomenal qualities. OK

The Hartree-Fock method is an important approximation for the ground-state electronic wave function of atoms and molecules so that its usage is widespread in computational chemistry and physics. The Hartree-Fock method is an iterative procedure in which the electronic wave functions of the occupied orbitals are determined. The set of functions found in one step builds the basis for the next iteration step. In this work, we interpret the Hartree-Fock method as a dynamical system since dynamical systems are iterations where iteration steps represent the time development of the system, as encountered in the theory of fractals. The focus is put on the convergence behavior of the dynamical system as a function of a suitable control parameter. An investigation of the convergence behavior depending on the parameter λ is performed for helium, neon, and argon. We observe fractal structures in the complex λ-plane, which resemble the well-known Mandelbrot set, determine their fractal dimension, and find that with increasing nuclear charge, the fragmentation increases as well. (Abstract)

Toriumi, Shin and Vladimir Airapelian. Universal Scaling Laws for the Solar and Stellar Atmospheric Heating. arXiv:2202.01232. We cite this entry by Japan Aerospace Exploration Agency and NASA Goddard Space Flight Center researchers as a typical instance nowadays of how a global science is finding an infinite recurrence in kind across every celestial, and cultural scale and realm. See also Universal Scaling Law of Glass Rheology by Shuangxi, Song, et al. in Nature Materials (February 2022).

The Sun-like stars commonly host multi-million-Kelvin coronae an 10,000-Kelvin chromospheres. These hot gases generate X-ray and Ultraviolet emissions that may influence the chemistry of (exo)planetary atmospheres, along the climate and habitability. However, the mechanism of coronal and chromospheric heating is poorly understood. To this end, we report on a systematic survey of the responses of solar and stellar atmospheres to surface magnetic flux over a wide range of temperatures. We reveal that the irradiance and magnetic flux show power-law relations with an exponent from above- to sub-unity as the temperature decreases from the corona to the chromosphere. Our study provides observational evidence that the mechanism of atmospheric heating is universal among the Sun and sun-like stars, regardless of age or activity. (Abstract excerpt)

The similarity in atomic/molecular structure between liquids and glasses has stimulated a long-standing view that glasses may be more fluid-like than the apparent solid. Here we report the dynamic response of shear stress to the strain rate of metallic glasses over nine orders of magnitude. The dynamic response follows a universal scaling law within the framework of fluid dynamics. (S. Song)

Venkataramani, Shankar and Alan Newell. Pattern Dark Matter and Galaxy Scaling Relations. European Physical Journal Special Topics. June, 2021. University of Arizona astrophysicists press on through 40 pages to express the latest mathematic verities of “universal equations” that distinguish and arrange these celestial reaches. Once again, we ought to reflect on how incredible it is that fraught human beings can yet achieve such cosmic quantifications, as if we are meant to perform necessary function of natural self-description.

University of Arizona astrophysicists press on through 40 pages to express the latest mathematic verities of “universal equations” that distinguish and arrange these celestial reaches. Once again, we ought to reflect on how incredible it is that fraught human beings can yet achieve such cosmic quantifications, as if we are meant to perform necessary function of natural self-description.

Vijar, Sagar, et al. A New Kind of Topological Quantum Order. arXiv:1505.02576. With a Dimensional Hierarchy of Quasiparticles Built from Stationary Excitations subtitle, by way of clever mathematics, MIT physicists SV, Jeongwan Haah, and Liang Fu advance deep understandings about how cosmic nature’s is actually suffused by intrinsic structural geometries. See also Haah’s 2011 original paper Local Stabilizer Codes in Three Dimensions without String Logical Operators at 1101.1962 with much set off this quest.

We introduce exactly solvable models of interacting (Majorana) fermions in d≥3 spatial dimensions that realize a new kind of topological quantum order, building on a model presented in ref. [1]. These models have extensive topological ground-state degeneracy and a hierarchy of point-like, topological excitations that are only free to move within sub-manifolds of the lattice. In particular, one of our models has fundamental excitations that are completely stationary. To demonstrate these results, we introduce a powerful polynomial representation of commuting Majorana Hamiltonians. Remarkably, the physical properties of the topologically-ordered state are encoded in an algebraic variety, defined by the common zeros of a set of polynomials over a finite field. This provides a "geometric" framework for the emergence of topological order. (Abstract)

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