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A Sourcebook for the Worldwide Discovery of a Creative Organic Universe
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III. An Organic, Conducive, Habitable MultiUniVerse

F. Systems Cosmology: Fractal SpaceTimeMatter

Fujiwara, Noboru. The Scaling Rule for Environmental Organizing Systems in a Gravitational Field. BioSystems. 73/2, 2004. A computer scientist at the Nara Women’s University in Japan finds a constant, proportional relationship from unicellular organisms to humans and on to stellar dimensions. As cosmic and planetary evolution proceeds, it can be qualified by an increase in organized information. Fujiwara’s studies have been periodically published in this journal.

The present paper examines a scaling rule for the relationship between the integrated scaled metabolic energy and the mass of a system for a wide range of masses, from animals to 4He cores of main-sequence stars, considering the effect of gravitational energy. (111)

Furtenbacher, Tibor, et al. Simple Molecules as Complex Systems. Nature Scientific Reports. 4/4654, 2014. Hungarian and German chemists, mathematicians, and physicists broach how ubiquitous network phenomena can be extrapolated even to quantum activities. In regard, one lively universe to us, cosmic to civic, similar anatomy and physiology is being deftly filled in. However in this month of May might it dawn that our planetary progeny is discovering a grand new genesis

For individual molecules, quantum mechanics (QM) offers a simple, natural and elegant way to build large-scale complex networks: quantized energy levels are the nodes, allowed transitions among the levels are the links, and transition intensities supply the weights. QM networks are intrinsic properties of molecules and they are characterized experimentally via spectroscopy; thus, realizations of QM networks are called spectroscopic networks (SN). The proposed novel view of high-resolution spectroscopy and the observed degree distributions have important implications: appearance of a core of highly interconnected hubs among the nodes, a generally disassortative connection preference, considerable robustness and error tolerance, and an “ultra-small-world” property. The network-theoretical view of spectroscopy offers a data reduction facility via a minimum-weight spanning tree approach, which can assist high-resolution spectroscopists to improve the efficiency of the assignment of their measured spectra. (Abstract)

Gaite, Jose. Scale Symmetry in the Universe. Symmetry. 12/4, 2020. As noted herein, when this section went online in 2004 only patchy inklings of self-similar cosmic structures could be found. In this essay a Polytechnic University of Madrid physicist (search) can well quantify and install a “multifractal cosmology” in extensive mathematical detail. The celestial reaches which firstly seem vastly opaque are now found to be suffused with a discernible patterning due an infinity of nested repetitions. The atomic quantum depths are likewise graced by a fractal fabric, along with the mesocosmic phases in between. Circa 2020, as a worldwise supermind proceeds to learns by itself, the ancient, tradition sense of an as above, so below correspondence is at last becomes verified and explained.

Scale symmetry is a fundamental symmetry of physics that seems however not to be fully realized in the universe. Here, we focus on the astronomical scales ruled by gravity, where scale symmetry holds and gives rise to a scale invariant distribution of matter, namely a true fractal geometry. A suitable explanation of the fractal cosmic mass distribution is provided by the nonlinear Poisson–Boltzmann–Emden equation. We study the fractal solutions of the equation and connect them with the statistical theory of random multiplicative cascades. The type of multifractal mass distributions so obtained agrees with results from the analysis of cosmological simulations and of observations of the galaxy distribution. (Abstract)

Simply put, if one finds an object of a given size, there must be similar objects of larger size. For example, take a cluster of galaxies; there must be similar superclusters of every possible size. Not surprisingly, the idea of a scale invariant structure of the universe on large scales is old, but its modern formulation had to await the advent of the appropriate description in the form of fractal geometry. Simple fractals are scale invariant and are indeed composed of clusters of clusters of down to the infinitesimally small. Naturally, in the universe, the self-similarity must stop at a scale about the size of galaxies, although it could be limitless towards the large scales, in principle. (1-2)

Gefter, Amanda. Fractal Universe. New Scientist. March 10, 2007. To the proponents of this vision such as Luciano Pietronero of the University of Rome, the latest results of the Sloan Digital Sky Survey of over 50,000 galaxies quite support a self-similar geometry across many scales. But from another mathematical approach, David Hogg of New York University finds a homogeneity across their span of 400 million light years. His worry is that relativistic physics would be in peril because it cannot explain how these iterative structures would arise. (An interesting note is included about initial indications that dark matter, which makes up some 85% of the cosmos, also appears to take on a fractal array.) These discussions reflect an inability of the 20th century particulate emphasis, albeit a necessary stage, to appreciate complex, self-organizing phenomena because they lie outside their model and survey.

Guszejnov, David, et al. Universal Scaling Relations in Scale-Free Structure Formation. arXiv:1707.05799. Reviewed more in Mid 2010s Universality, whereof a sophisticated cosmic science by Cal Tech astrophysicists affirms a pervasive, natural interstellar self-similarity.

Halasz, Gabor, et al. Fracton Topological Phases from Strongly Coupled Spin Chains. arXiv:1707.02308. UC Santa Barbara, Kavli Institute, theorists press this intricate mathematical discernment of quantum structures. We quote the opening paragraph as to convey this initial phase of mapping a frontier expanse which is just opening. See also, e.g., Fracton Models on General Three-Dimensional Manifolds at 1712.05892 for more exposition.

Fracton topological phases are topologically ordered phases in three dimensions with a particularly extreme form of fractionalization. In these phases, there are point-like excitations that are either completely immobile or only mobile in a lower-dimensional subsystem, such as an appropriate line or plane. Remarkably, the restricted mobility of excitations has a purely topological origin and appears in translation-invariant systems without any disorder. In addition to being of fundamental interest from the perspective of topological phases, and providing an exciting disorder-free alternative to many-body localization, this phenomenology has important implications for quantum-information storage. Indeed, the immobility of excitations makes encoded quantum information more stable at finite temperature than in conventional topologically ordered phases. (1)

Hnat, Bogdan, et al. Scale-Free Texture of the Fast Solar Wind. Physical Review E. 84/065401, 2011. University of Warwick, Coventry, Ilia State University, Tbilisi, and Imperial College, London, astrophysicists further quantify the nebulous realms of stellar fractal galaxies. See also “On the Fractal Nature of the Magnetic Field Energy Density in the Solar Wind” in Geophysical Research Letters (34/L15108, 2007).

The higher-order statistics of magnetic field magnitude fluctuations in the fast quiet solar wind are quantified systematically, scale by scale. We find a single global non-Gaussian scale-free behavior from minutes to over 5 h. This spans the signature of an inertial range of magnetohydrodynamic turbulence and a ~ 1/f range in magnetic field components. This global scaling in field magnitude fluctuations is an intrinsic component of the underlying texture of the solar wind and puts a strong constraint on any theory of solar corona and the heliosphere. Intriguingly, the magnetic field and velocity components show scale-dependent dynamic alignment outside of the inertial range. (065401)

Ijjas, Anna, et al. Scale-Free Primordial Cosmology. arXiv:1309.4480. Online September 2013, Iggas, Max Planck Institute for Gravitational Physics, with Paul Steinhardt, Princeton University, and Abraham Loeb, Harvard-Smithsonian Center for Astrophysics, contend that European Space Agency, Planck microwave satellite, 2013 measurements “…showed with high precision that the spectrum of primordial density fluctuations is nearly scale-invariant.” These results are seen to confirm a “scale-freeness” that has been predicted for some forty years. A popular piece “Pop-Up Universe” in New Scientist (October 5, 2013) notes that these results call the various theories of an inflationary burst of the initial cosmos into some question. A rebuttal by original theorists Alan Guth, et al was posted at arXiv:1312.7619 in December 2013.

The large-scale structure of the universe suggests that the physics underlying its early evolution is scale-free. This was the historic motivation for the Harrison-Zel'dovich-Peebles spectrum and for inflation. Based on a hydrodynamical approach, we identify scale-free forms for the background equation-of-state for both inflationary and cyclic scenarios and use these forms to derive predictions for the spectral tilt and tensor-to-scalar ratio of primordial density perturbations. For the case of inflation, we find three classes of scale-free models with distinct predictions. Including all classes, we show that scale-free inflation predicts tensor-to-scalar ratio r > 10-4. We show that the observationally favored class is theoretically disfavored because it suffers from an initial conditions problem and the hydrodynamical form of an unlikeliness problem similar to that identified recently for certain inflation potentials. We contrast these results with those for scale-free cyclic models. (Abstract)

Iovane, G., et al. Stochastic Self-Similar and Fractal Universe. Chaos, Solitons & Fractals. 20/3, 2004. The authors provide a detailed mathematical analysis said to explain a universally recurrent pattern and dynamics at every emergent domain.

Consequently the Universe, with its structures at all scales (atomic nucleus, organic cell, human, planet, solar system, galaxy, clusters of galaxy, super clusters of galaxy), could have a fundamental quantum reason. (415) This means that Nature manifests itself through its relativistic quantum fractal geometry aspects. (424) Our conclusion is that the fractal power law suggests a fractal Universe. Therefore we can state that nature uses the language of a relativistic, quantum and fractal geometry. (425)

Jarboe, Thomas, et al. Self-Organization of Solar Magnetic Fields. arXiv:1807.09593. We cite this entry by nine University of Washington astrophysicists for its content, and also to make a point, which ought to be strongly put, that these formative stellar forces come from the same natural, independent source as those which organize life’s origin, cellular activities, our brains and linguistic societies. They strongly imply, as this universality becomes filled in, a single, common program, aka a uniVerse to human epitome cosmome code.

Self-organization properties of sustained magnetized plasma are applied to solar data to understand solar magnetic fields. Torsional oscillations are speed-up and slow-down bands of the azimuthal flow that correlate with the solar cycle, and they imply the existence of a symmetric solar dynamo. The dynamo has enough power to heat the chromosphere and to power the corona and the solar wind. (Abstract brief)

Jones, Bernard, et al. Scaling Laws in the Distribution of Galaxies. Reviews of Modern Physics. 76/4, 2004. An extensive technical paper with co-authors Vicent Martinez, Enn Saar and Virginia Trimble. The phenomenon of a worldwide humankind explores and discovers an inherently variegated, clustered, self-similar universe from which it arose.

In describing scaling laws it is helpful to make analogies with fractals, mathematical constructs that can possess a wide variety of scaling properties. We must beware, however, of saying that the universe is a fractal on some range of scales: it merely exhibits a specific kind of fractal-like behavior on those scales. The richness of fractal scaling behavior is an important supplement to the usual battery of statistical descriptors. (Abstract, 1211) The aim of this article is to show how the paradigm of a homogeneous and isotropic universe with a hot singular origin has emerged, and to explain how, within this framework, we can quantify and understand the growth of the large scale cosmic structure. (1213) This scaling is almost certainly a consequence of two factors: the nature of the initial conditions for cosmic structure formation and the fact that the gravitational force law is itself scale-free. (1215)

Kempkes, Sander, et al. Design and Characterization of Electrons in a Fractal Geometry. Nature Physics. 17/2, 2019. As the Abstract details, Utrecht University physicists deftly show how even atoms and electrons, in their dynamic forms, naturally take on this iterative patterning. We offer two comments. When this section was first posted in 2004, the presence of a common, natural self-similarity was spurious and patchy. Fifteen years later it has become robustly evident that every universal, atomic, and animate complexity is graced by this infinite iteration. Whomever in the cosmos are we peoples to consider and begin a second materiality by way of “artificial atoms.” See also in the same issue Quantum Fractals by Dario Bercioux and Ainhoa Iriguez.

Here, we show how arrays of artificial atoms can be defined by controlled positioning of CO molecules on a Cu (111) surface, and how these sites couple to form electronic Sierpiński fractals. We characterize the electron wavefunctions at different energies with scanning tunnelling microscopy and spectroscopy, and show that they inherit the fractional dimension. Wavefunctions delocalized over the Sierpiński structure decompose into self-similar parts at higher energy, and this scale invariance can also be retrieved in reciprocal space. Our results show that electronic quantum fractals can be artificially created by atomic manipulation in a scanning tunnelling microscope. Moreover, the rational concept of artificial atoms can readily be transferred to planar semiconductor electronics, allowing for the exploration of electrons in a well-defined fractal geometry, including interactions and external fields. (Abstract)

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