Symmetry Elements Research Articles

The Symmetrical Foundation of Measure, Probability, and Quantum Theories

AbstractQuantification starts with sum and product rules that express combination and partition. These rules rest on elementary symmetries that have wide applicability, which explains why arithmetical adding up and splitting into proportions are ubiquitous. Specifically, measure theory formalizes addition, and probability theory formalizes inference in terms of proportions. Quantum theory rests on the same simple symmetries, but is formalized in two dimensions, not just one, in order to track an object through its binary interactions with other objects. The symmetries still require sum and product rules (here known as the Feynman rules), but they apply to complex numbers instead of real scalars, with observable probabilities being modulus squared (known as the Born rule). The standard quantum formalism follows. There is no mystery or weirdness, just ordinary probabilistic inference.

Connecting bright and dark states through accidental degeneracy caused by lack of symmetry.

Coupled excitonic structures are found in natural and artificial light harvesting systems where optical transitions link different excitation manifolds. In systems with symmetry, some optical transitions are allowed, while others are forbidden. Here we examine an excitonic ring structure and identify an accidental degeneracy between two categories of double-excitation eigenstates with distinct symmetries and optical transition properties. To understand the accidental degeneracy, a complete selection rule between two arbitrary excitation manifolds is derived with a physically motivated proof. Remarkably, symmetry analysis shows that the lack of certain symmetry elements in the Hamiltonian is responsible for this degeneracy, which is unique to rings with size N = 4l + 2 (l being an integer).

Using SMILES strings for the description of chemical connectivity in the Crystallography Open Database

Computer descriptions of chemical molecular connectivity are necessary for searching chemical databases and for predicting chemical properties from molecular structure. In this article, the ongoing work to describe the chemical connectivity of entries contained in the Crystallography Open Database (COD) in SMILES format is reported. This collection of SMILES is publicly available for chemical (substructure) search or for any other purpose on an open-access basis, as is the COD itself. The conventions that have been followed for the representation of compounds that do not fit into the valence bond theory are outlined for the most frequently found cases. The procedure for getting the SMILES out of the CIF files starts with checking whether the atoms in the asymmetric unit are a chemically acceptable image of the compound. When they are not (molecule in a symmetry element, disorder, polymeric species,etc.), the previously published cif_molecule program is used to get such image in many cases. The program package Open Babel is then applied to get SMILES strings from the CIF files (either those directly taken from the COD or those produced by cif_molecule when applicable). The results are then checked and/or fixed by a human editor, in a computer-aided task that at present still consumes a great deal of human time. Even if the procedure still needs to be improved to make it more automatic (and hence faster), it has already yielded more than 160,000 curated chemical structures and the purpose of this article is to announce the existence of this work to the chemical community as well as to spread the use of its results.

Using virtual laboratories in chemistry classrooms as interactive tools towards modifying alternate conceptions in molecular symmetry

Molecular symmetry plays a central role in chemistry education with regard to predicting chemical properties such as bonding and spectroscopic transitions. Better understanding of the symmetry of molecules requires high visual-spatial thinking ability. Conventional teaching methodologies, with limited teaching aides, fall short in providing a detailed understanding of scientific theories and related concepts. Incorrect understanding has been known to perpetrate concepts that are not consistent with the consensus of the research community or alternate conceptions. This work elaborates a methodology designed to discover the alternate conceptions stemming from teaching molecular symmetry in a typical classroom environment and the impact of the virtual laboratory (VL) environment in correcting these misconceptions. Three significant contributions presented in this paper include: (1) the development of a media and information-intense VL experiment platform designed to enhance understanding of symmetry elements and point groups of molecules with diverse structural geometries. (2) the development of an instrument, Molecular symmetry Alternate Conception Test (MACT), designed to capture and estimate the extent of alternate conceptions. (3) the successful identification of typical alternate conceptions amongst students in the context of molecular symmetry. In addition to perceived alternate concepts in symmetry education, the results indicate a significant statistical improvement of 156% in understanding of molecular symmetry concepts (p < 0.05) after subjecting students to the interactive VL platform. This study also shows identifying bond angles and planarity as concepts crucial for students. It is also implicit that estimations of discrimination skills related to identifying concept-based learning may be relevant for perceiving alternate concepts among learners.

Chiral Monomers Ensure Orientational Specificity of Monomer Binding During Polymer Self-Replication.

Biomolecular homochirality is universally observed in living systems but the molecular and evolutionary dynamics that led to its emergence are unknown. In fact, there are significant disadvantages in using chiral monomers for polymerization, which include enantiomeric cross-inhibition in racemic medium and under-utilization of available resources for self-replication in the primordial environment. Nevertheless, most investigations of homochirality in living systems assume that the individual primordial monomers were chiral prior to the formation of self-replicating polymer and therefore focus on identifying a symmetry-breaking mechanism that might choose one enantiomer over the other in a racemic medium. Within the premise that the extant biomolecules are products of molecular evolution, we ask a related but distinct question: why is an achiral monomer molecule disfavored? Here we identify an evolutionary advantage for molecular evolution to choose chiral over achiral monomers to construct primordial self-replicating polymers. We argue that when polymerization is constrained to proceed in only one direction along the template, as in DNA, evolution favors chiral monomers and homochiral polymers. This evolutionary advantage stems from the ability of a chiral monomer to bond with the template in only one orientation relative to the template monomer, along the direction of polymerization. An achiral monomer, on the other hand, offers more than one possible orientation for bonding with the template monomer, due to the presence of symmetry elements in its structure, which would lead to inhibition of polymerization. We show that the requirement of orientational specificity leads to monomer chirality, by using a known relationship between rotational and reflection symmetry elements, within the constraint that the resultant polymers are helical.

Side Methyl Groups Control the Conformation and Contribute to Symmetry Breaking of Isoprenoid Chromophores.

Ab initio DFT computations reveal that the essential structural and photophysical features of the conjugated π-electron system of retinal and carotenoids are dictated by "innocent" methyl substituents. These methyl groups shape the conformation and symmetry of the isoprenoid chromophores by causing a sigmoidal distortion of the polyene skeleton and increasing its flexibility, which facilitates fitting to their binding pockets in proteins. Comparison of in vacuo conformations of the chromophores with their native (protein-bound) conformations showed, surprisingly, that the peripheral groups and interactions with the protein environment are much less significant than the methyl side groups in tuning their structural features. The methyl side groups also contribute to a loss of symmetry elements specific to linear polyenes. In effect, the symmetry-imposed restrictions on the chromophore electronic properties are disabled, which is of tremendous relevance to their photophysics. This is evidenced by their non-negligible permanent dipole moments and by the simulated and experimentally measured circular dichroism spectra, which necessarily reflect the chirality of the conjugated π-electron system.

Side Methyl Groups Control the Conformation and Contribute to Symmetry Breaking of Isoprenoid Chromophores

AbstractAb initio DFT computations reveal that the essential structural and photophysical features of the conjugated π‐electron system of retinal and carotenoids are dictated by “innocent” methyl substituents. These methyl groups shape the conformation and symmetry of the isoprenoid chromophores by causing a sigmoidal distortion of the polyene skeleton and increasing its flexibility, which facilitates fitting to their binding pockets in proteins. Comparison of in vacuo conformations of the chromophores with their native (protein‐bound) conformations showed, surprisingly, that the peripheral groups and interactions with the protein environment are much less significant than the methyl side groups in tuning their structural features. The methyl side groups also contribute to a loss of symmetry elements specific to linear polyenes. In effect, the symmetry‐imposed restrictions on the chromophore electronic properties are disabled, which is of tremendous relevance to their photophysics. This is evidenced by their non‐negligible permanent dipole moments and by the simulated and experimentally measured circular dichroism spectra, which necessarily reflect the chirality of the conjugated π‐electron system.

Bowl Shaped Monodentate Phosphines by Condensation of Imidazole Units with Boranes

The condensation of imidazole moieties with boranes leading to molecular units with N–BR2–N bridges was accomplished by attaching three imidazole units to a single phosphorus atom. In this way, boron bridged trisimidazolylphosphines 3–7 were synthesized in good yields. The chiral derivatives 5a, 5d, 6, and 7 are C3 symmetric, whereas 5b and 5c carry no symmetry element (C1). The steric bulk of the ligands can easily be tuned by introduction of different substituents at the three bridging boron atoms. All phosphines feature small CPC angles between 91–93° leading to a s‐type lone pair donor at the phosphorus atom combined with reasonable π‐accepting properties. Ni (8) and Au metal complexes (9, 10) with very short metal phosphorus bonds were synthesized. The short metal–phosphorus distance is a consequence of the pronounced s‐character of the phosphine lone pair.

Temporal Coupled-Mode Theory of Electromagnetic Components Described by Magnetic Groups of Symmetry

We consider peculiarities of application of the temporal coupled-mode theory to the electromagnetic components with magnetic symmetry. The coupled-mode theory is widely used for analysis of reciprocal devices described by symmetrical scattering matrices and by unitary elements of symmetry, such as for example, a plane of symmetry. However, nonreciprocity and low symmetry of components described by antiunitary elements lead to a necessity of modification of the method. Using as an example a photonic crystal-based W-circulator described by an antiplane of symmetry, we show applicability of the method to the components with antiunitary elements. The circulator characteristics are calculated and compared with those obtained by a numerical method, demonstrating good agreement between the two methods.

Symplectic Field Theories: Scalar and Spinor Representations

Using elements of symmetry, as gauge invariance, aspects of field theories represented in symplectic space are introduced and analyzed under physical bases. The states of a system are described by symplectic wave functions, which are associated with the Wigner function. Such wave functions are vectors in a Hilbert space introduced from the cotangent-bundle of the Minkowski space. The symplectic Klein–Gordon and the Dirac equations are derived, and a minimum coupling is considered in order to analyze the Landau problem in phase space.

Structure and vibrational spectroscopic study of phthalimido-functionalized N-heterocyclic palladium complexes. Correlations between structure and catalytic activity

New complexes of Pd(II) with cationic phthalimido-functionalized N-heterocyclic carbene (NHC), the cis-Diacetonitrile(chloro)[1-(2′-phthalamidoethyl)-3-methylimidazolin-ylidene]palladium(II) Hexafluorophosphate (and its CD3CN derivative) had been successfully used as catalyst in the Suzuki–Miyaura cross-coupling reaction. This complex have been studied by structural (XRD), spectroscopic (infrared, far-infrared and Raman) and theoretical (DFT and normal coordinate calculations) methods. Palladium is bound to two acetonitrile ligands and one-one chloride ion and phthalimido-functionalized imidazolium salt, [3-methyl-1-(2′-phthalimidoethyl) imidazolium] (L = ligand) in a square–planar cis arrangement. The molecular structure of the [PdCl(NCCH3)2L]+ was obtained by XRD and DFT geometry optimization. The two acetonitrile ligands exhibited two different Pd-N bonds a shorter one 2.080 Å (opposite to Cl ion) and a longer one 2.120 Å (trans to heterocyclic ligand). The cationic [PdCl(NCCH3)2L+] complex has as much as 132 fundamental modes and practically all of them have been assigned. The complex has no symmetry elements, but interpretation has been performed on the basis of local symmetries of its fragments. Tans-conformation of the bridging –CH2-CH2– group has been established from vibrational spectra, and this local structure avoided interaction of phthalimido moiety with the palladium center. Special attention has been paid for understanding and interpretation of the characteristic imidazolium ring vibrations connected to the strong electron donor phthalimido ring at 1553, 1397, 1347, 1218, 1149, 1118, 845, 748, 666, 629, 471, 245, 212, 124 and 104 cm−1 which bands are up shifted comparing to those of 2,3-dialkyl substituted ring modes. Clear differences were established between the fundamentals and force constants of the two differently coordinated acetonitrile. The more strongly coordinated CH3CN(1) has Raman bands at 2336, 958 and 279 cm−1 while the other CH3CN(2) has at 2321, 936 and 266 cm−1 like antisymmetric CN, CC and Pd-N stretching modes, respectively. The Pd-N stretching modes were assigned with the help of spectral and calculated data of CD3CN complex. For acetonitrile (1) the light and heavy stretching modes were 279 and 275 cm−1, respectively exhibiting 4 cm−1 isotope shift whyle those for acetonitrile (2) were as 266 and 264 cm−1 with less isotope shift. Rather small but different 1.630 and 1.493 Ncm−1 Pd-N stretching force constants were fitted for stronger and weaker bonded acetonitrile as compared to those of 3.931 Ncm−1 for [Pd(NCCH3)2]2+, and 4.050 Ncm−1 for cis-PtCl2(NCCH3)2. In contrast, the strongly weakened Pd-N bonds the Pd-Cl bond is enhanced and exhibiting relative high 2.03 Ncm−1 force constants in comparison with 1.53 Ncm−1 of [PdCl4]2- anion. The enhanced Pd-Cl bond strength is escorted by strong weakening of Pd-acetonitrile coordination. Surprisingly, from DFT calculation and spectroscopic data higher Pd-N force constant was obtained for the Pd-N bond with longer bond length and opposite, lower force constant for shorter Pd-N bond. These unusual discrepancies between bond lengths and bond force constants can be explained with the presence of electron-rich σ-donating ligand opposite to acetonitrile (1) which lead to enhancement of Pd-N bond. These specific properties of coordination sphere of [PdCl(NCCH3)2L]+ complex providing possibilities for easily releasing acetonitrile ligands during catalytic processes and give reasonable structural explanation of its high catalytic activity.

Noncommutative Quantum Mechanics based on Representations of Exotic Galilei Group

Using elements of symmetry, we constructed the Noncommutative Schrödinger Equation from a representation of Exotic Galilei Group. As a consequence, we derive the Ehrenfest theorem using noncommutative coordinates. We also have showed others features of quantum mechanics in such a manifold. As an important result, we find out that a linear potential in the noncommutative Schrödinger equation is completely analogous to the ordinary case. We also worked with harmonic and anharmonic oscillators, giving corrections in the energy for each one.

Tetraphenylphosphonium tetrakis(trimethylsilanolato)ferrate(III)

The structure of tetraphenylphosphonium tetrakis(trimethylsilanolato)ferrate(III), [(C6H5)4P][Fe(OSi(CH3)3)4], has tetragonal (I-4) symmetry, and was refined as an inversion twin. It is an ionic compound consisting of a tetraphenylphosphonium cation and a tetrakis(trimethylsilanolato)ferrate(III) anion. The crystal structure comprises the two ionic species each centered on a -4 symmetry element and contributing a fourth of its structure to the asymmetric unit. Each is surrounded by counter-ions on all sides. The cation contains a central phosphorous atom bound to four phenyl groups in a tetrahedral arrangement, while the anion contains a central iron(III) atom tetrahedrally coordinated by four trimethylsilanolato ligands.

Asymmetry and symmetry of valley-river of landscape and technical systems

The aim of the article is to consider the manifestation of landscape asymmetry and symmetry within the valley-river landscape and technical systems, which are confined to the flat part of Right-Bank Ukraine, on the basis of a generalized spatial and temporal analysis. Methods. The study is based on the ideas of F. M. Milkov on anthropogenic landscapes and «Pierre Curie principles»: the universality of symmetry, the forced symmetry and dynamism of symmetry, which were adapted in physical geography. Scientific novelty. In practice, application of the principles of symmetry and asymmetry has been widely implemented in landscape design and architecture in the design and construction of gardening landscape. In modern Ukrainian landscape science, this issue is only beginning to be considered at both theoretical, and practical levels. Individual publications are devoted to this problem, but there is no detailed research where asymmetry and symmetry of valley-river landscape and technical systems would be analyzed. Practical value is determined by the introduction into the practice of working with landscape and technical systems of universal principles of symmetry and asymmetry, which will make it possible to find effective ways of rational use of natural resources. Research results. It was noted that in the study of landscape and technical systems in them it is worth to distinguish elements of symmetry and asymmetry – auxiliary geometric images (points, lines, planes) in respect of which equal parts of geographical objects are repeated in a certain way. In complex landscape and technical systems basic engineering structures often play the role of such geometric images. It is noted that the leading role in the formation of the asymmetry of valley-river landscapes is played by the anthropogenic factor. Under its influence, natural landscape asymmetry can radically change. Depending on the development stages of landscape and technical systems, possible variants of landscape asymmetry and symmetry transformation from natural to anthropogenic are analyzed. On the example of valley-river landscape and technical systems of Right-Bank Ukraine (reservoirs, ponds, canals and bridges), manifestations of anthropogenic landscape asymmetry and symmetry are considered. Based on our own field research and spatial and temporal analysis of the economic development of river valleys of Right-Bank Ukraine, a model for the development of a typical pond valley-river landscape and technical system in the Southern Bug basin was developed, where one can observe the process of landscape asymmetry complication in various respects. The most typical variant of transformation is the one in which the natural landscape asymmetry changes into anthropogenic and manifests itself at all stages of development. It is concluded that the violation of geometric images of valley-river landscape and technical systems occurs as a result of the ineffective activity of the control unit. Consideration of individual properties of asymmetry and symmetry of valley-river landscape and technical systems is necessary for solving many problems associated with environmental management and the optimization of such systems. To this end, it is necessary to increase the attention of modern geographers and landscape scholars to study the problems of the influence of asymmetry and symmetry on the structure, dynamics and stability of landscape and technical systems.

Defect Structures of La1 – ySr y F3 – y, La1 – yBa y F3 – y, and Nd1 – yCa y F3 – y (y = 0.05, 0.10) Nonstoichiometric Tysonite Phases

Characteristic features of defect structures of La1 – ySr y F3 – y, La1 – yBa y F3 – y, and Nd1 – yCa y F3 – y (y = 0.05, 0.10) nonstoichiometric phases of different compositions are determined from X-ray diffraction data. Interest in subtle details of their structure is determined by the possibility of ion transport over fluorine vacancies and by a strong compositional dependence of the ionic conductivity. The La0.95Sr0.05F2.95, La0.95Ba0.05F2.95, and Nd0.95Ca0.05F2.95 phases, as well as the La0.9Ba0.1F2.9 phase, crystallize as β-LaF3 (sp. gr. P3c1, Z = 6). The La0.9Sr0.1F2.9 and Nd0.9Ca0.1F2.9 phases lose their superstructure and are described by a cell whose volume is three times smaller (sp. gr. P63/mmc, Z = 2). Defects of crystal structure R1 – yM y F3 – y are not exhausted by vacancies in fluorine positions. All crystals with a “large” cell are twinned according to the merohedral twin law. The majority of atomic positions in models with a “small” cell are split by group symmetry elements and are occupied statistically.

A Mechanism for Symmetry Breaking and Shape Control in Single-Crystal Gold Nanorods.

The phenomenon of symmetry breaking-in which the order of symmetry of a system is reduced despite manifest higher-order symmetry in the underlying fundamental laws-is pervasive throughout science and nature, playing a critical role in fields ranging from particle physics and quantum theory to cosmology and general relativity. For the growth of crystals, symmetry breaking is the crucial step required to generate a macroscopic shape that has fewer symmetry elements than the unit cell and/or seed crystal from which it grew. Advances in colloid synthesis have enabled a wide variety of nanocrystal morphologies to be achieved, albeit empirically. Of the various nanoparticle morphologies synthesized, gold nanorods have perhaps been the most intensely studied, thanks largely to their unique morphology-dependent optical properties and exciting application potential. However, despite intense research efforts, an understanding of the mechanism by which a single crystal breaks symmetry and grows anisotropically has remained elusive, with many reports presenting seemingly conflicting data and theories. A fundamental understanding of the symmetry breaking process is needed to provide a rational framework upon which future synthetic approaches can be built. Inspired by recent experimental results and drawing upon the wider literature, we present a mechanism for gold nanorod growth from the moments prior to symmetry breaking to the final product. In particular, we describe the steps by which a cuboctahedral seed particle breaks symmetry and undergoes anisotropic growth to form a nanorod. With an emphasis on the evolving crystal structure, we highlight the key geometrical and chemical drivers behind the symmetry breaking process and factors that govern the formation and growth of nanorods, including control over the crystal width, length, and surface faceting. We propose that symmetry breaking is induced by an initial formation of a new surface structure that is stabilized by the deposition of silver, thus preserving this facet in the embryonic nanorod. These new surfaces initially form stochastically as truncations that remove high-energy edge atoms at the intersection of existing {111} facets and represent the beginnings of a {011}-type surface. Crucially, the finely tuned [HAuCl4]:[AgNO3] ratio and reduction potential of the system mean that silver deposition can occur on the more atomically open surface but not on the pre-existing lower-index facets. The stabilized surfaces develop into side facets of the nascent nanorod, while the largely unpassivated {111} facets are the predominant site of Au atom deposition. Growth in the width direction is tightly controlled by a self-sustaining cycle of galvanic replacement and silver deposition. It is the [HAuCl4]:[AgNO3] ratio that directly determines the particle size at which the more open atomic surfaces can be stabilized by silver and the rate of growth in the width direction following symmetry breaking, thus explaining the known aspect ratio control with Ag ion concentration. We describe the evolving surface faceting of the nanorod and the emergence of higher-index facets. Collectively, these observations allow us to identify facet-size and edge-atom effects as a simple fundamental driver of symmetry breaking and the subsequent development of new surfaces in the presence of adsorbates.

Description of the atomic disorder (local order) in crystals by the mixed-symmetry method

An approach to the description of local atomic disorder (short-range order) in single crystals by the mixed-symmetry method based on Bragg scattering data is proposed, and the corresponding software is developed. In defect-containing crystals, each atom in the unit cell can be described by its own symmetry space group. The expression for the calculated structural factor includes summation over different sets of symmetry operations for different atoms. To facilitate the search for new symmetry elements, an “atomic disorder expert” was developed, which estimates the significance of tested models. It is shown that the symmetry lowering for some atoms correlates with the existence of phase transitions (in langasite family crystals) and the anisotropy of physical properties (in rare-earth dodecaborides RB12).

On motion decomposition and constitutive relations in geometrically nonlinear elastoviscoplasticity of crystallites

The formulation of geometrically nonlinear boundary value problems is a necessary component in modeling real technological processes of thermomechanical material processing, wherein the main issues are the description of nonlinear kinematics and the formulation of constitutive relations. Most of the existing works do not explicitly consider the used method of motion decomposition that implies the extraction of the part responsible for quasi-rigid motion from the motion of a deformed solid. On our opinion, when choosing a method of motion decomposition and the corresponding corotational frame, the latter must be associated with the material to correctly describe the loading history and symmetry properties of the modeled body. Note that crystalline materials (including metals and alloys) at different scale levels are always anisotropic to a certain extent, and their properties can differ significantly in different directions. Even initially isotropic (at the level of the representative macrovolume) polycrystalline materials also become anisotropic under severe plastic deformation due to texturing. A multilevel approach allows the determination of symmetry elements (planes and axes of symmetry) at the crystallite level in metals, and we propose to relate these elements to the axes of the corotational frame that defines quasi-rigid motion. A new way of motion decomposition is proposed, namely, a multiplicative representation of the deformation gradient with an explicit extraction of the corotational frame motion. Elastoviscoplastic constitutive relations are formulated in terms of a stress free configuration. As the used stresses and strains are energetically conjugate, the requirements of no stress hysteresis and no energy dissipation in arbitrary closed elastic strain cycles are automatically satisfied, which is exemplified for anisotropic crystals. ©hus, an approach is proposed for the construction of geometrically nonlinear kinematic and constitutive relations for metal crystallites using a physically justified method of motion decomposition, with taking into account the symmetry properties of materials.

Material rotating frame, rate-independent plasticity with regularization of Schmid law and study of channel-die compression

Metals often exhibit several crystallographic components that rotate when subjected to large plastic deformation. Studying their evolution constitutes a prospect to apply the finite plastic strain theory. In this respect, this research paper formulates a three-dimensional extension of Dogui’s plane kinematics to choose a material rotating frame in which deformation and velocity tensor gradients are represented by upper triangular matrices. This technique facilitates numerical calculation and renders it faster. This can prove to be useful if used for crystalline calculation. We study the kinematics of a channel-die loading to implement the material rotating frame model. The rate-independent formulation, which uses the multi-surface theory to represent the plastic flow of crystals, often leads to slip indeterminacy. Hence, we have developed a phenomenological method to solve such a difficulty using the regularization of Schmid’s yield law. As a significant application of this proposed phenomenological approach, aluminium crystal flow in channel-die compression is numerically simulated for three cases of loadings: (i) classical rolling components Cube, Goss, Brass, Copper and Dillamore orientations; (ii) {110} compression; and (iii) Strange orientation, which has no common element of symmetry with the channel-die. A good correlation between the simulation results and the available experimental ones is observed.

FUZZY COMPREHENSIVE EVALUATION METHOD FOR SYMMETRY DEGREE OF MECHANICAL STRUCTURE SYMMETRY

This paper proposes a fuzzy comprehensive evaluation method to determine the symmetry degree of three types of mechanical structure symmetries. In the proposed method, factors are set based on four symmetry elements; the analytic hierarchy process is used to determine the weight set, and an experiment was developed to determine the weight values. Based on the maximum membership rule, the type of symmetry of the evaluation object can be determined based on the evaluation result. Finally, instances were applied to illustrate the proposed evaluation method, which validated the proposed method is successful.