III. Ecosmos: A Revolutionary Fertile, Habitable, Solar-Bioplanet, Incubator Lifescape

E. Systems Cosmology: Fractal SpaceTimeMatter

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.
See also Pattern universes by these authors in Comptes Rendus Mécanique (347/4, 2019).

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)

Von Korff, Modest and Thomas Sander. Molecular Complexity Calculated by Fractal Dimension. Nature Scientific Reports. 9/967, 2019. Scientific Computing Drug Discovery, Idorsia Pharmaceuticals, Switzerland researchers achieve another novel recognition that nature’s proclivity to adopt and display a self-similar, iterative essence can be traced even to molecular and atomic forms and sub-structures.

Molecular complexity is an important characteristic of organic molecules for drug discovery. How to calculate molecular complexity has been discussed in the scientific literature for decades. It was known from early on that the numbers of substructures that can be cut out of a molecular graph are of importance. However, it was never realized that the cut-out substructures show self-similarity to the parent structures. Such a series shows self-similarity similar to fractal objects. The fractal dimension of a molecule is a new matter constant that incorporates all features that are currently known to be important for describing molecular complexity.(Abstract)

During our work on repetitive molecular features we realized that every molecular substructure displays self-similarity to its parent structure. Self-similarity means that an object is similar to a part of itself. Let us derive the concept of self-similarity for organic molecules at the example of n-hexane, a linear alkane. A linear alkane consists of a chain of carbon atoms saturated with hydrogen atoms. Removing one of the two outmost carbon-atoms and the connecting bond creates a new chain with one bond less. To the chain carbon-atom from where the bond was removed, a hydrogen atom has to be added for completing the saturation of the chain. (2)

Vrobel, Susie, et al, eds. Simultaneity: Temporal Structures and Observer Perspectives. Hackensack, NJ: World Scientific, 2008. With co-editors Otto Rossler and Terry Marks-Tarlow, a volume emanating from an Institute for Fractal Research which seems to get wound up in its own vernacular claiming “the next revolution in physics” due to novel appreciations of human observership and the inherent “fractality of space and time.” An eclectic collection, with various contributions by the editors, Peter Allen on a hierarchy of evolutionary systems, Uri Fidelman on complementary brain hemispheres whereof the right side achieves its simultaneous synthesis, and on to a brush with econophysics. In any event, still another sign of coming universe change.

Walcher,, C. J., et al. Self-Similarity in the Chemical Evolution of Galaxies. arXiv:1607.00015. A ten person team from Germany and South America report signs of a universal geometric repetition across material, stellar and galactic realms that is “more than just a coincidence.”

Recent improvements in the age dating of stellar populations and single stars allow us to study the ages and abundance of stars and galaxies with unprecedented accuracy. We here compare the relation between age and \alpha-element abundances for stars in the solar neighborhood to that of local, early-type galaxies. This quantitative similarity seems surprising, given the different types of galaxies and scales involved. The data are consistent with a power law delay time distribution. We thus confirm that the delay time distribution inferred for the Milky Way from chemical evolution arguments also must apply to massive early-type galaxies.

Wang, Xin and Alex Szalay. On the Nonlinear Evolution of Cosmic Web. arXiv.1411.4117. Johns Hopkins University astrophysicists explain “cosmic morphologies of the large-structure” by way of Lagrangian dynamics, a technical finesse of statistical mechanics. We cite because by natural philosophy wonder, how fantastic is it that human folks can altogether suddenly traverse and quantify such infinite vistas. You might read Johns Hopkins (1795-1873) biography on Wikipedia for some context. Surely we peoples ought to grant ourselves a central significance to the course and fate of this genesis universe.

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