Symmetry

Symmetry, Vol. 2, Pages 272-283: Time-Symmetric Boundary Conditions and Quantum Foundations

Ken Wharton — Mon, 08 Mar 2010 00:00:00 CET

Despite the widely-held premise that initial boundary conditions (BCs) corresponding to measurements/interactions can fully specify a physical subsystem, a literal reading of Hamilton’s principle would imply that both initial and final BCs are required (or more generally, a BC on a closed hypersurface in spacetime). Such a time-symmetric perspective of BCs, as applied to classical fields, leads to interesting parallels with quantum theory. This paper will map out some of the consequences of this counter-intuitive premise, as applied to covariant classical fields. The most notable result is the contextuality of fields constrained in this manner, naturally bypassing the usual arguments against so-called “realistic” interpretations of quantum phenomena.

Symmetry, Vol. 2, Pages 230-271: Doubly-Special Relativity: Facts, Myths and Some Key Open Issues

Giovanni Amelino-Camelia — Mon, 08 Mar 2010 00:00:00 CET

I report, emphasizing some key open issues and some aspects that are particularly relevant for phenomenology, on the status of the development of “doubly-special” relativistic (“DSR”) theories with both an observer-independent high-velocity scale and an observer-independent small-length/large-momentum scale, possibly relevant for the Planck-scale/quantum-gravity realm. I also give a true/false characterization of the structure of these theories. In particular, I discuss a DSR scenario without modification of the energy-momentum dispersion relation and without the қ-Poincaré Hopf algebra, a scenario with deformed Poincaré symmetries which is not a DSR scenario, some scenarios with both an invariant length scale and an invariant velocity scale which are not DSR scenarios, and a DSR scenario in which it is easy to verify that some observable relativistic (but non-special-relativistic) features are insensitive to possible nonlinear redefinitions of symmetry generators.

Symmetry, Vol. 2, Pages 213-229: On the Physical Reasons for the Extension of Symmetry Groups in Molecular Spectroscopy

Franca Lattanzi — Thu, 25 Feb 2010 00:00:00 CET

Several situations of general interest, in which the symmetry groups usually applied to spectroscopy problems need to be extended, are reviewed. It is emphasized that any symmetry group of geometrical operations to be used in Molecular Spectroscopy should be extended for completeness by considering the time reversal operator, as far as the Hamiltonian is invariant with respect to the inversion of the direction of motion. This can explain the degeneracy of pairs of vibrational and rotational states spanning the so-called separably degenerate irreducible representations, in symmetric tops of low symmetry, and Kramers degeneracy in odd electron molecules in the absence of magnetic fields. An extension with account of time reversal is also useful to determine relative phase conventions on vibration-rotation wavefunctions, which render all vibration-rotation matrix elements real. An extension of a molecular symmetry group may be required for molecules which can attain different geometries by large amplitude periodical motions, if such motions are hindered and are not completely free. Special cases involving the internal rotation are discussed in detail. It is observed that the symmetry classification of vibrational modes involving displacements normal to the internal rotation axis is not univocal, but can be done in several ways, which actually correspond to different conventions on the separation of vibration and internal rotation in the adopted basis functions. The symmetry species of the separate vibrational and torsional factors of these functions depend on the adopted convention.

Symmetry, Vol. 2, Pages 201-212: Symmetry Analysis in Mechanistic Studies of Nucleophilic Substitution and β-Elimination Reactions

Xiaoping Sun — Thu, 25 Feb 2010 00:00:00 CET

A mechanistic study of the bimolecular nucleophilic substitution (SN2) reaction for halomethane CH3X (X = Cl, Br, or I) is approached by using symmetry principles and molecular orbital theory. The electrophilicity of the functionalized sp3–carbon is attributable to a 2p-orbital-based antibonding MO along the C–X bond. This antibonding MO, upon accepting an electron pair from a nucleophile, gives rise to dissociation of the C–X bond and formation of a new Nuc–C bond. Correlations are made between the molecular orbitals of reactants (Nuc- and CH3X) and products (NucCH3 and X-). Similar symmetry analysis has been applied to mechanistic study of the bimolecular b-elimination (E2) reactions of haloalkanes. It well explains the necessity of an anti-coplanar arrangement of the Cα–X and Cβ–H bonds for an E2 reaction (anti-elimination). Having this structural arrangement, the bonding Cα–X (σC-X) and antibonding Cβ–H (σC-H*) orbitals become symmetry–match. They can partially overlap resulting in increase in electron density in σC-H*, which weakens and polarizes the Cβ–H bond making the β-H acidic. An E2 reaction can readily take place in the presence of a base. The applications of symmetry analysis to the SN2 and E2 reactions represent a new approach to studying organic mechanisms.

Symmetry, Vol. 2, Pages 184-200: Effective Supramolecular Chirogenesis in Ethane-Bridged Bis-Porphyrinoids

Victor Borovkov — Tue, 23 Feb 2010 00:00:00 CET

This feature article gives a general introduction to the phenomenon of supramolecular chirogenesis using the most representative examples of different chirogenic assemblies on the basis of ethane-bridged bis-porphyrinoids. Supramolecular chirogenesis is based upon a smart combination of supramolecular chemistry and chirality sciences and deals with various aspects of asymmetry induction, transfer, amplification, and modulation. These chiral processes are governed by numerous noncovalent supramolecular forces thus allowing a judicious, mechanistic, and dynamic control by applying a variety of internal and external influencing factors. Currently, supramolecular chirogenesis is widely used in different fields of fundamental and applied branches of science and modern technology, touching on such important issues as origin of chirality on the Earth, asymmetry sensing, enantioselective catalysis, nonlinear optics, polymer and materials science, pharmacy and medicine, nanotechnology, molecular and supramolecular devices, chiral memory, absolute configuration determination, etc.

Symmetry, Vol. 2, Pages 150-183: Engineering Life into Technology: the Application of Complexity Theory to a Potential Phase Transition in Intelligence

Melanie Swan — Tue, 23 Feb 2010 00:00:00 CET

Information optimization is a centerpiece phenomenon in the universe. It develops from simplicity, then continuously breaks symmetry and cycles through instability to progress to increasingly dense nodes of complexity and diversity. Intelligence has arisen as the information optimization node with the greatest complexity. A contemporary imbalance is presented in that exponentially growing technology could be poised as a potential sole successor to human intelligence. A complex dynamical system is emerging in response, the engineering of life into technology. Numerous network elements are developing which could self-organize into the next node of symmetry, a phase transition in intelligence.

Symmetry, Vol. 2, Pages 136-149: Asymmetry and Symmetry in the Beauty of Human Faces

Dahlia W. Zaidel — Tue, 23 Feb 2010 00:00:00 CET

The emphasis in the published literature has mostly been on symmetry as the critical source for beauty judgment. In fact, both symmetry and asymmetry serve as highly aesthetic sources of beauty, whether the context is perceptual or conceptual. The human brain is characterized by symbolic cognition and this type of cognition facilitates a range of aesthetic reactions. For example, both art and natural scenery contain asymmetrical elements, which nevertheless render the whole effect beautiful. A further good case in point is, in fact, human faces. Normally, faces are structurally left-right symmetrical content-wise but not size-wise or function-wise. Attractiveness has often been discussed in terms of content-wise full-face symmetry. To test whether or not attractiveness can be gleaned only from the presence of left-right full-faces we tested half faces. Three separate groups of participants viewed and rated the attractiveness of 56 full-faces (women’s and men’s), their 56 vertical left hemi-faces and 56 vertical right hemi-faces. We found no statistically significant differences in the attractiveness ratings of full- and hemi-faces (whether left or right). Instead, we found a strong and significant positive correlation between the ratings of the hemi- and full-faces. These results are consistent with the view that the underpinning of human facial beauty is complex and that bilateral symmetry does not constitute a principle factor in beauty assessment. We discuss that the highly evolved human brain, compared to other animals, as well as symbolic and abstract cognition in humans enable a wide variety of aesthetic reactions.

Symmetry, Vol. 2, Pages 112-135: Chiral Symmetry Breaking Phenomenon Caused by a Phase Transition

Rui Tamura — Wed, 17 Feb 2010 00:00:00 CET

We report the mechanism and scope of “preferential enrichment”, which is an unusual symmetry-breaking enantiomeric resolution phenomenon that is initiated by the solvent-assisted solid-to-solid transformation of a metastable polymorphic form into a thermodynamically stable one during crystallization from the supersaturated solution of certain kinds of racemic mixed crystals (i.e., solid solutions or pseudoracemates) composed of two enantiomers. The mechanism can well be interpreted in terms of a symmetrybreaking complexity phenomenon involving multistage processes that affect each other.

Symmetry, Vol. 2, Pages 98-111: Symmetries and (Related) Recursion Operators of Linear Evolution Equations

Giampaolo Cicogna — Fri, 05 Feb 2010 00:00:00 CET

Significant cases of time-evolution equations, the linear Schr¨odinger and the Fokker–Planck equation are considered. It is known that equations of this type can be transformed, in some cases, into a highly simplified form. The properties of these equations in their initial and their simplified form are compared, showing in particular that this transformation partially prevents a clear understanding and a full application of the (physically relevant) notion of the so-called step up/down operators. These operators are shown to be recursion operators, related to the Lie point symmetries of the equations, which are also carefully discussed.

Symmetry, Vol. 2, Pages 76-97: Recent Studies on the Aromaticity and Antiaromaticity of Planar Cyclooctatetraene

Tohru Nishinaga — Fri, 05 Feb 2010 00:00:00 CET

Cyclooctatetraene (COT), the first 4nπ-electron system to be studied, adopts an inherently nonplanar tub-shaped geometry of D2d symmetry with alternating single and double bonds, and hence behaves as a nonaromatic polyene rather than an antiaromatic compound. Recently, however, considerable 8π-antiaromatic paratropicity has been shown to be generated in planar COT rings even with the bond alternated D4h structure. In this review, we highlight recent theoretical and experimental studies on the antiaromaticity of hypothetical and actual planar COT. In addition, theoretically predicted triplet aromaticity and stacked aromaticity of planar COT are also briefly described.

Symmetry, Vol. 2, Pages 69-75: The Number of Symmetric Colorings of the Quaternion Group

Yuliya Zelenyuk — Wed, 20 Jan 2010 00:00:00 CET

We compute the number of symmetric r-colorings and the number of equivalence classes of symmetric r-colorings of the quaternion group.

Symmetry, Vol. 2, Pages 40-68: Systems with Symmetry Breaking and Restoration

Vyacheslav I. Yukalov — Mon, 11 Jan 2010 00:00:00 CET

Statistical systems, in which spontaneous symmetry breaking can be accompanied by spontaneous local symmetry restoration, are considered. A general approach to describing such systems is formulated, based on the notion of weighted Hilbert spaces and configuration averaging. The approach is illustrated by the example of a ferroelectric with mesoscopic fluctuations of paraelectric phase. The influence of the local symmetry restoration on the system characteristics, such as sound velocity and Debye-Waller factor, is discussed.

Symmetry, Vol. 2, Pages 15-39: Dual Symmetry in Bent-Core Liquid Crystals and Unconventional Superconductors

Bruno Mettout — Mon, 11 Jan 2010 00:00:00 CET

We extend the Landau theory of bent-core mesophases and d-wave high-Tc superconductors by considering additional secondary pseudo-proper order parameters. These systems exhibit a remarkable analogy relating their symmetry groups, lists of phases, and an infinite set of physical tensors. This analogy lies upon an internal dual structure shared by the two theories. We study the dual operator transforming rotations into translations in liquid crystals, and gauge symmetries into rotations in superconductors. It is used to classify the bent-core line defects, and to analyze the electronic gap structure of lamellar d-wave superfluids.

Symmetry, Vol. 2, Pages 1-14: Symmetry, Optical Properties and Thermodynamics of Neptunium(V) Complexes

Linfeng Rao — Mon, 28 Dec 2009 00:00:00 CET

Recent results on the optical absorption and symmetry of the Np(V) complexes with dicarboxylate and diamide ligands are reviewed. The importance of recognizing the “silent” feature of centrosymmetric Np(V) species in analyzing the absorption spectra and calculating the thermodynamic constants of Np(V) complexes is emphasized.

Symmetry, Vol. 1, Pages 240-251: Phase Diffusion of a q-Deformed Oscillator

Turan Birol — Mon, 21 Dec 2009 00:00:00 CET

We examine the problem of phase diffusion rate in a U(1) global phase symmetry broken system, from the perspective of q-deformed oscillators where the deformation parameter represents the anharmonicity. It is shown that broken phase symmetry states, described by deformed coherent states, suffer phase diffusion at a rate determined by the deformation parameter. Analytical discussions are given for the case of weak deformations, while detailed numerical results are presented when strong anharmonicity is present in the system.

Symmetry, Vol. 1, Pages 226-239: Polyhedral Phenylacetylenes: The Interplay of Aromaticity and Antiaromaticity in Convex Graphyne Substructures

Daniel Sebastiani — Fri, 11 Dec 2009 00:00:00 CET

We have studied a series of bridged phenylacetylene macrocycles with topologies based on Platonic and Archimedean polyhedra, using density functional calculations to determine both their molecular structure and their electronic response to external magnetic fields (NICS maps). We are able to elucidate the interplay of aromaticity and anti-aromaticity as a function of structural parameters, in particular the symmetry properties of the intramolecular bond connectivities, in these compounds.

Symmetry, Vol. 1, Pages 215-225: Symmetry in Infancy: Analysis of Motor Development in Autism Spectrum Disorders

Gianluca Esposito — Thu, 10 Dec 2009 00:00:00 CET

Motor asymmetry, defined as the lack of symmetry in movements or postures, is often observed briefly in many typically developing children. However, if such asymmetry persists, it may be a sign of neurological disease. Recent studies have suggested that motor asymmetries may be an early symptom of Autism Spectrum Disorders (ASD). ASD involve a range of social, cognitive, and behavioral problems, at different degrees of functioning, which are thought to be the final common pathway of multiple etiological mechanisms. Furthermore, early identification of ASD has been recognized as a critical aspect for treatment. Our study aims to analyze symmetry in the motor milestones of infants with ASD compared with typically developing infants (TD) or infants with other developmental delay (DD) during the first year of life. Our results highlight that there are different patterns of motor symmetry in the groups. In particular, infants with ASD scored significantly poorer (higher levels of asymmetry) then the TD and DD infants. We also identified two subgroups of infants with ASD, one with a typical level and the other with a lower level of motor functioning. Implications of the study for diagnosis and treatment are described.

Symmetry, Vol. 1, Pages 201-214: On a Symmetric, Nonlinear Birth-Death Process with Bimodal Transition Probabilities

Antonio Di Crescenzo — Thu, 26 Nov 2009 00:00:00 CET

We consider a bilateral birth-death process having sigmoidal-type rates. A thorough discussion on its transient behaviour is given, which includes studying symmetry properties of the transition probabilities, finding conditions leading to their bimodality, determining mean and variance of the process, and analyzing absorption problems in the presence of 1 or 2 boundaries. In particular, thanks to the symmetry properties we obtain the avoiding transition probabilities in the presence of a pair of absorbing boundaries, expressed as a series.

Symmetry, Vol. 1, Pages 180-200: Testing Group Symmetry of a Multivariate Distribution

Lyudmila Sakhanenko — Thu, 26 Nov 2009 00:00:00 CET

We propose and study a general class of tests for group symmetry of a multivariate distribution, which encompasses different types of symmetry, such as ellipsoidal and permutation symmetries among others. Our approach is based on supremum norms of special empirical processes combined with bootstrap. We show that these tests are consistent against any fixed alternative. This work generalizes the methodology of Koltchinskii and Sakhanenko [7], developed for ellipsoidal symmetry to the case of group symmetry. It also provides a unified approach to testing different types of symmetry of a multivariate distribution.

Symmetry, Vol. 1, Pages 155-179: Tetraquark Spectroscopy: A Symmetry Analysis

Javier Vijande — Mon, 23 Nov 2009 00:00:00 CET

We present a detailed analysis of the symmetry properties of a four-quark wave function and its solution by means of a variational approach for simple Hamiltonians. We discuss several examples in the light and heavy-light meson sector.

Symmetry, Vol. 1, Pages 153-154: Visual Symmetry. By Magdolna Hargittai and István Hargittai, World Scientific Publishing, 2009; 224 pages. Price: US$ 48 / £ 36 ISBN 978-981-283-531-4

Shu-Kun Lin — Fri, 23 Oct 2009 00:00:00 CEST

Symmetry is as simple or as complicated as we are ready to absorb it in everything around us. From flowers to bridges, buildings, coke machines, and snowflakes; from molecules to walnuts, fences, pine cones, and sunflowers; from music to children's drawings; from hubcaps to bank logos, propellers, wallpaper decorations, and pavements, we recognize it if we walk around with open eyes and an open mind. This book provides aesthetic pleasure and covert education, immersing the reader in both the familiar and the unknown and leading always to unexpected discoveries.

Symmetry, Vol. 1, Pages 145-152: On the Symmetry of a Zig-Zag and an Armchair Polyhex Carbon Nanotorus

Morteza Yavari — Thu, 08 Oct 2009 00:00:00 CEST

A Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [dij], where for i≠j, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for distinct nuclei. The aim of this paper is to compute the automorphism group of the Euclidean graph of a carbon nanotorus. We prove that this group is a semidirect product of a dihedral group by a group of order 2.

Symmetry, Vol. 1, Pages 115-144: Supersymmetry of Generalized Linear Schrödinger Equations in (1+1) Dimensions

Axel Schulze-Halberg — Tue, 06 Oct 2009 00:00:00 CEST

We review recent results on how to extend the supersymmetry SUSY normalism in Quantum Mechanics to linear generalizations of the time-dependent Schrödinger equation in (1+1) dimensions. The class of equations we consider contains many known cases, such as the Schrödinger equation for position-dependent mass. By evaluating intertwining relations, we obtain explicit formulas for the interrelations between the supersymmetric partner potentials and their corresponding solutions. We review reality conditions for the partner potentials and show how our SUSY formalism can be extended to the Fokker-Planck and thenonhomogeneous Burgers equation.

Symmetry, Vol. 1, Pages 106-114: The Relationship Between Morphological Symmetry and Immune Response in Wild-Caught Adult Bush-Crickets

Åsa Berggren — Mon, 28 Sep 2009 00:00:00 CEST

Despite interest in the relationship between fluctuating asymmetry (FA), immune response and ecological factors in insects, little data are available from wild populations. In this study we measured FA and immune response in 370 wild-caught male bush-crickets, Metrioptera roeseli, from 20 experimentally introduced populations in southern-central Sweden. Individuals with more-symmetric wings had a higher immune response as measured by the cellular encapsulation of a surgically-implanted nylon monofilament. However, we found no relationship between measures of FA in other organs (i.e. tibia and maxillary palp) and immune response, suggesting that this pattern may reflect differing selection pressures.

Symmetry, Vol. 1, Pages 64-105: Nuclei, Primes and the Random Matrix Connection

Frank W. K. Firk — Sun, 20 Sep 2009 00:00:00 CEST

In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe the essential aspects of these fields, the quantities studied, and how insights in one have been fruitfully applied in the other. The exciting branch of modern mathematics – random matrix theory – provides the connection between the two fields. We assume no detailed knowledge of number theory, nuclear physics, or random matrix theory; all that is required is some familiarity with linear algebra and probability theory, as well as some results from complex analysis. Our goal is to provide the inquisitive reader with a sound overview of the subjects, placing them in their historical context in a way that is not traditionally given in the popular and technical surveys.

Symmetry, Vol. 1, Pages 55-63: A Stochastic Poisson Structure

Rémi Léandre — Thu, 20 Aug 2009 00:00:00 CEST

We define a Poisson structure on the Nualart-Pardoux test algebra associated to the path space of a finite dimensional Lie algebra.

Symmetry, Vol. 1, Pages 21-54: Symmetry-Break in Voronoi Tessellations

Valerio Lucarini — Thu, 20 Aug 2009 00:00:00 CEST

We analyse in a common framework the properties of the Voronoi tessellations resulting from regular 2D and 3D crystals and those of tessellations generated by Poisson distributions of points, thus joining on symmetry breaking processes and the approach to uniform random distributions of seeds. We perturb crystalline structures in 2D and 3D with a spatial Gaussian noise whose adimensional strength is α and analyse the statistical properties of the cells of the resulting Voronoi tessellations using an ensemble approach. In 2D we consider triangular, square and hexagonal regular lattices, resulting into hexagonal, square and triangular tessellations, respectively. In 3D we consider the simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC) crystals, whose corresponding Voronoi cells are the cube, the truncated octahedron, and the rhombic dodecahedron, respectively. In 2D, for all values α>0, hexagons constitute the most common class of cells. Noise destroys the triangular and square tessellations, which are structurally unstable, as their topological properties are discontinuous in α=0. On the contrary, the honeycomb hexagonal tessellation is topologically stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise with α<0.12. Basically, the same happens in the 3D case, where only the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. In both 2D and 3D cases, already for a moderate amount of Gaussian noise (α>0.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations are indistinguishable. When α>2, results converge to those of Poisson-Voronoi tessellations. In 2D, while the isoperimetric ratio increases with noise for the perturbed hexagonal tessellation, for the perturbed triangular and square tessellations it is optimised for specific value of noise intensity. The same applies in 3D, where noise degrades the isoperimetric ratio for perturbed FCC and BCC lattices, whereas the opposite holds for perturbed SCC lattices. This allows for formulating a weaker form of the Kelvin conjecture. By analysing jointly the statistical properties of the area and of the volume of the cells, we discover that also the cells shape heavily fluctuates when noise is introduced in the system. In 2D, the geometrical properties of n-sided cells change with α until the Poisson-Voronoi limit is reached for α>2; in this limit the Desch law for perimeters is shown to be not valid and a square root dependence on n is established, which agrees with exact asymptotic results. Anomalous scaling relations are observed between the perimeter and the area in the 2D and between the areas and the volumes of the cells in 3D: except for the hexagonal (2D) and FCC structure (3D), this applies also for infinitesimal noise. In the Poisson-Voronoi limit, the anomalous exponent is about 0.17 in both the 2D and 3D case. A positive anomaly in the scaling indicates that large cells preferentially feature large isoperimetric quotients. As the number of faces is strongly correlated with the sphericity (cells with more faces are bulkier), in 3D it is shown that the anomalous scaling is heavily reduced when we perform power law fits separately on cells with a specific number of faces.

Symmetry, Vol. 1, Pages 10-20: Using Group Theory to Obtain Eigenvalues of Nonsymmetric Systems by Symmetry Averaging

Marion L. Ellzey — Thu, 06 Aug 2009 00:00:00 CEST

If the Hamiltonian in the time independent Schrödinger equation, HΨ = EΨ, is invariant under a group of symmetry transformations, the theory of group representations can help obtain the eigenvalues and eigenvectors of H. A finite group that is not a symmetry group of H is nevertheless a symmetry group of an operator Hsym projected from H by the process of symmetry averaging. In this case H = Hsym + HR where HR is the nonsymmetric remainder. Depending on the nature of the remainder, the solutions for the full operator may be obtained by perturbation theory. It is shown here that when H is represented as a matrix [H] over a basis symmetry adapted to the group, the reduced matrix elements of [Hsym] are simple averages of certain elements of [H], providing a substantial enhancement in computational efficiency. A series of examples are given for the smallest molecular graphs. The first is a two vertex graph corresponding to a heteronuclear diatomic molecule. The symmetrized component then corresponds to a homonuclear system. A three vertex system is symmetry averaged in the first case to Cs and in the second case to the nonabelian C3v. These examples illustrate key aspects of the symmetry-averaging process.

Symmetry, Vol. 1, Pages 3-9: Symmetry at the Foundation of Science and Nature

Joe Rosen — Fri, 05 Jun 2009 00:00:00 CEST

This article demonstrates that science is founded on symmetry and that Nature must have symmetry at its foundation. Full details are given in the book: Rosen, J. Symmetry Rules: How Science and Nature Are Founded on Symmetry; Springer-Verlag: Berlin, Germany, 2008.

Symmetry, Vol. 1, Pages 1-2: Symmetry – An International and Interdisciplinary Scientific Open Access Journal

Shu-Kun Lin — Fri, 05 Jun 2009 00:00:00 CEST

As the publisher of MDPI journals, I am pleased to launch Symmetry (ISSN 2073-8994), an international and interdisciplinary open access scientific journal. Twenty years ago, a journal entitled Symmetry – An Interdisciplinary and International Journal was launched by VCH Publishers, Inc. in New York, with Professor Istvan Hargittai as Editor-in-Chief. I submitted a paper which was processed by Professor Sven J. Cyvin from The University of Trondheim – The Norwegian Institute of Technology. The paper was accepted and scheduled for publication in the printed issue 4 of volume 1, 1990. I still keep a copy of the galley proofs. However, the publication of this journal was terminated after just the release of the first issue of volume 1, and this paper was finally published elsewhere [1]. [...]