Average Symmetry Research Articles

Protect Measurement-Induced Phase Transition from Noise.

Scrambling dynamics induced by random unitary gates can protect information from low-rate measurements, which underpins the phenomenon known as the measurement-induced phase transition (MIPT). However, typical decoherence noise disrupts the volume-law phase, complicating the observation of the MIPT on noisy intermediate-scale quantum devices. Here, we demonstrate that incorporating quantum-enhanced operations can effectively protect the MIPT from environmental noise, thereby enabling its detection in experiment. The transition is characterized by conditional entanglement entropy, which is associated with a statistical mechanics model wherein noise and quantum-enhanced operations act as competing external random fields. When the net external field is zero, a ferromagnetic-paramagnetic phase transition is expected, resulting in the MIPT. This zero-field condition also ensures an average apparatus-environment symmetry, making conditional entanglement entropy a valid probe of entanglement and establishing the transition as a genuine entanglement phase transition. Additionally, we provide numerical results that demonstrate the MIPT in a (2+1)-dimensional quantum circuit under dephasing noise. We also propose a method to estimate the noise rate, enabling the zero-field condition to be achieved experimentally and ensuring the feasibility of our protocol. Our result serves as a concrete example of the power of quantum enhancement in combating noise.

S-d Coupling-Induced Dynamic Off-Centering of Cu Drives High Thermoelectric Performance in TlCuS.

Seeking new and efficient thermoelectric materials requires a detailed comprehension of chemical bonding and structure in solids at microscopic levels, which dictates their intriguing physical and chemical properties. Herein, we investigate the influence of local structural distortion on the thermoelectric properties of TlCuS, a layered metal sulfide featuring edge-shared Cu-S tetrahedra within Cu2S2 layers. While powder X-ray diffraction suggests average crystallographic symmetry with no distortion in CuS4 tetrahedra, the synchrotron X-ray pair distribution function experiment exposes concealed local symmetry breaking, with dynamic off-centering distortions of the CuS4 tetrahedra. The Cu off-centering is driven by the on-site coupling of filled 3d and unoccupied 4s orbitals (s-d) of Cu through a second-order Jahn-Teller mechanism. Bond softening by the p-d* (S-3p and Cu-3d) antibonding interaction coupled with dynamic off-centering distortions causes significant anharmonicity in the lattice, which acts as a phonon-blocking mechanism conducive to ultralow lattice thermal conductivity (κlat) (∼0.35-0.23 W m-1 K-1 in the ∼296-573 K temperature range) in TlCuS. The synergy between low κlat and multiband electronic structure leads to thermoelectric figure-of-merits (zT) of ∼1 and ∼1.3 at 573 K for pristine TlCuS and TlCu0.98S, respectively, underscoring the potential of metal sulfides as promising high-performance thermoelectrics.

Anomalies of average symmetries: entanglement and open quantum systems

Symmetries and their anomalies are powerful tools for understanding quantum systems. However, realistic systems are often subject to disorders, dissipation and decoherence. In many circumstances, symmetries are not exact but only on average. This work investigates the constraints on mixed states resulting from non-commuting average symmetries. We will focus on the cases where the commutation relations of the average symmetry generators are violated by nontrivial phases, and call such average symmetry anomalous. We show that anomalous average symmetry implies degeneracy in the density matrix eigenvalues, and present several lattice examples with average symmetries, including XY chain, Heisenberg chain, and deformed toric code models. In certain cases, the results can be further extended to reduced density matrices, leading to a new lower bound on the entanglement entropy. We discuss several applications in the contexts of many body localization, quantum channels, entanglement phase transitions and also derive new constraints on the Lindbladian evolution of open quantum systems.

Optimizing Ni-Zr-Ti metallic glasses: Cluster design and molecular dynamics evaluation of glass forming ability

Optimizing Ni-Zr-Ti metallic glasses: Cluster design and molecular dynamics evaluation of glass forming ability

Na Vacancy-Driven Phase Transformation and Fast Ion Conduction in W-Doped Na3SbS4 from Machine Learning Force Fields.

Solid-state sodium batteries require effective electrolytes that conduct at room temperature. The Na3PnCh4 (Pn = P, Sb; Ch = S, Se) family has been studied for their high Na ion conductivity. The population of Na vacancies, which mediate ion diffusion in these materials, can be enhanced through aliovalent doping on the pnictogen site. To probe the microscopic role of extrinsic doping and its impact on diffusion and phase stability, we trained a machine learning force field for Na3-x W x Sb1-x S4 based on an equivariant graph neural network. Analysis of large-scale molecular dynamics trajectories shows that an increased Na vacancy population stabilizes the global cubic phase at lower temperatures with enhanced Na ion diffusion and that the explicit role of the substitutional W dopants is limited. In the global cubic phase, we observe large and long-lived deviations of atoms from the averaged symmetry, echoing recent experimental suggestions. Evidence of correlated Na ion diffusion is also presented that underpins the suggested superionic nature of these materials.

Tetrahedra Rotational and Displacive Disorder in the Scheelite-Type Oxide CsReO4.

Scheelite-type metal oxides are a notable class of functional materials, with applications including ionic conductivity, photocatalysis, and the safe storage of radioactive waste. To further engineer these materials for specific applications, a detailed understanding of how their properties can change under different conditions is required─not just in the long-range average structure but also in the short-range local structure. This paper outlines a detailed investigation of the metal oxide CsReO4, which exhibits an uncommon orthorhombic Pnma pseudo-scheelite-type structure at room temperature. Using synchrotron X-ray diffraction, the average structure of CsReO4 is found to undergo a transformation from the orthorhombic Pnma pseudo-scheelite-type structure to the tetragonal I41/a scheelite-type structure at ∼440 K. In the X-ray pair distribution function analysis, lattice strain and rotations of the ReO4 tetrahedra are apparent above 440 K despite the increase in long-range average symmetry, revealing a disconnect between the structural models at different length scales. This study demonstrates how the bonding requirements and ionic radii of the A-site cation can induce disorder that is detectable at different length scales, affecting the physical properties of the material.

No Association Between Injury-Related Fear and Isokinetic Quadriceps Strength in Individuals With a History of Anterior Cruciate Ligament Reconstruction.

Injury-related fear and quadriceps strength are independently associated with secondary anterior cruciate ligament (ACL) injury risk. It is not known whether injury-related fear and quadriceps strength are associated, despite their individual predictive capabilities of secondary ACL injury. The purpose of this study was to examine the association between injury-related fear and quadriceps strength in individuals at least 1year after ACL reconstruction (ACLR). Cross-sectional study. Forty participants between the ages of 18 and 35years at least 1year post unilateral primary ACLR. Participants completed the Tampa Scale of Kinesiophobia-11 (TSK-11) and a standard isokinetic quadriceps strength assessment using the Biodex Isokinetic Dynamometer. Pearson Product-Moment correlations were used to examine the linear association between the TSK-11 scores and peak torque (in nanometers per kilogram) for each limb and between the TSK-11 scores and limb symmetry indices for each limb. Pearson Product-Moment correlation coefficients (r) were interpreted as very high (.90-1.00), high (.70-.90), moderate (.50-.70), low (.30-.50), and no correlation (.00-.30). The average TSK-11 score was 18.2 (5.3), average ACLR peak quadriceps torque was 1.9 (0.50)N·m/kg, average contralateral peak quadriceps torque was 2.3 (0.48)N·m/kg, and average limb symmetry index was 85.3% (12.6%). There was no statistically significant correlation between the TSK-11 and peak quadriceps torque on the ACLR limb (r = .12, P = .46), the TSK-11 and contralateral limb (r = .29, P = .07), or the TSK-11 and limb symmetry index (r = -.18, P = .27). There was no association between kinesiophobia and peak isokinetic quadriceps strength in individuals at least 1year post-ACLR. Both factors, independently, have been shown to influence risk of secondary injury in patients after ACLR.

Anderson critical metal phase in trivial states protected by average magnetic crystalline symmetry

Transitions between distinct obstructed atomic insulators (OAIs) protected by crystalline symmetries, where electrons form molecular orbitals centering away from the atom positions, must go through an intermediate metallic phase. In this work, we find that the intermediate metals will become a scale-invariant critical metal phase (CMP) under certain types of quenched disorder that respect the magnetic crystalline symmetries on average. We explicitly construct models respecting average C2zT, m, and C4zT and show their scale-invariance under chemical potential disorder by the finite-size scaling method. Conventional theories, such as weak anti-localization and topological phase transition, cannot explain the underlying mechanism. A quantitative mapping between lattice and network models shows that the CMP can be understood through a semi-classical percolation problem. Ultimately, we systematically classify all the OAI transitions protected by (magnetic) groups Pm,P2′,P4′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Pm,P{2}^{{\\prime} },P{4}^{{\\prime} }$$\\end{document}, and P6′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P{6}^{{\\prime} }$$\\end{document} with and without spin-orbit coupling, most of which can support CMP.

The Average Symmetry Index of Minerals Co-Varies with Their Hydrogen Content, Rarity, and Paragenetic Mode

Variations in the Dolivo-Dobrovol’sky symmetry index for minerals through time reveal several factors that influence the emergence of crystalline symmetry in natural processes. Of special interest in this regard are the numerous paragenetic modes—different processes of mineral genesis that reflect changes in physical, chemical, and ultimately biological environments that foster the emergence of new mineral species. Here, we consider the roles of hydrogen content, rarity, formation temperature and pressure, and age on the average symmetry of minerals from 57 different modes of formation (i.e., paragenetic modes). We find four significant trends in the average mineral symmetry index for all minerals in each paragenetic mode: specifically, this average index is (1) lower for minerals with greater hydrogen content; (2) greater for minerals formed at higher pressure; (3) lower for minerals of greater rarity; and (4) greater for older paragenetic modes. These findings elucidate some of the intricate relationships among paragenetic modes, average mineral attributes, and the Dolivo-Dobrovol’sky symmetry index, providing insights into the geological processes governing mineral formation.

Site-directed cation ordering in chabazite-type AlxGa1-xPO4-34 frameworks revealed by NMR crystallography.

We report the first synthesis of the mixed-metal chabazite-type AlxGa1-xPO4-34(mim) solid solution, containing 1-methylimidazolium, mim, as structure directing agent (SDA), from the parent mixed-metal oxide solid solution, γ-(AlxGa1-x)2O3. This hitherto unreported family of materials exhibits complex disorder, arising from the possible distributions of cations over available sites, the orientation of the SDA and the presence of variable amounts of water, which provides a prototype for understanding structural subtleties in nanoporous materials. In the as-made forms of the phosphate frameworks, there are three crystallographically distinct metal sites: two tetrahedral MO4 and one octahedral MO4F2 (M = Al, Ga). A combination of solid-state NMR spectroscopy and periodic DFT calculations reveals that the octahedral site is preferentially occupied by Al and the tetrahedral sites by Ga, leading to a non-random distribution of cations within the framework. Upon calcination to the AlxGa1-xPO4-34 framework, all metal sites are tetrahedral and crystallographically equivalent in the average R3̄ symmetry. The cation distribution was explored by 31P solid-state NMR spectroscopy, and it is shown that the non-random distribution demonstrated to exist in the as-made materials would be expected to give remarkably similar patterns of peak intensities to a random distribution owing to the change in average symmetry in the calcined materials.

CAT-Seg: cascaded medical assistive tool integrating residual attention mechanisms and Squeeze-Net for 3D MRI biventricular segmentation

Cardiac image segmentation is a critical step in the early detection of cardiovascular disease. The segmentation of the biventricular is a prerequisite for evaluating cardiac function in cardiac magnetic resonance imaging (CMRI). In this paper, a cascaded model CAT-Seg is proposed for segmentation of 3D-CMRI volumes. CAT-Seg addresses the problem of biventricular confusion with other regions and localized the region of interest (ROI) to reduce the scope of processing. A modified DeepLabv3+ variant integrating SqueezeNet (SqueezeDeepLabv3+) is proposed as a part of CAT-Seg. SqueezeDeepLabv3+ handles the different shapes of the biventricular through the different cardiac phases, as the biventricular only accounts for small portion of the volume slices. Also, CAT-Seg presents a segmentation approach that integrates attention mechanisms into 3D Residual UNet architecture (3D-ResUNet) called 3D-ARU to improve the segmentation results of the three major structures (left ventricle (LV), Myocardium (Myo), and right ventricle (RV)). The integration of the spatial attention mechanism into ResUNet handles the fuzzy edges of the three structures. The proposed model achieves promising results in training and testing with the Automatic Cardiac Diagnosis Challenge (ACDC 2017) dataset and the external validation using MyoPs. CAT-Seg demonstrates competitive performance with state-of-the-art models. On ACDC 2017, CAT-Seg is able to segment LV, Myo, and RV with an average minimum dice symmetry coefficient (DSC) performance gap of 1.165%, 4.36%, and 3.115% respectively. The average maximum improvement in terms of DSC in segmenting LV, Myo and RV is 4.395%, 6.84% and 7.315% respectively. On MyoPs external validation, CAT-Seg outperformed the state-of-the-art in segmenting LV, Myo, and RV with an average minimum performance gap of 6.13%, 5.44%, and 2.912% respectively.

Average symmetry protected higher-order topological amorphous insulators

While topological phases have been extensively studied in amorphous systems in recent years, it remains unclear whether the random nature of amorphous materials can give rise to higher-order topological phases that have no crystalline counterparts. Here we theoretically demonstrate the existence of higher-order topological insulators in two-dimensional amorphous systems that can host more than six corner modes, such as eight or twelve corner modes. Although individual sample configuration lacks crystalline symmetry, we find that an ensemble of all configurations exhibits an average crystalline symmetry that provides protection for the new topological phases. To characterize the topological phases, we construct two topological invariants. Even though the bulk energy gap in the topological phase vanishes in the thermodynamic limit, we show that the bulk states near zero energy are localized, as supported by the level-spacing statistics and inverse participation ratio. Our findings open an avenue for exploring average symmetry protected higher-order topological phases in amorphous systems without crystalline counterparts.

Automatic orbital segmentation using deep learning-based 2D U-net and accuracy evaluation: A retrospective study

The purpose of this study was to verify whether the accuracy of automatic segmentation (AS) of computed tomography (CT) images of fractured orbits using deep learning (DL) is sufficient for clinical application. In the surgery of orbital fractures, many methods have been reported to create a 3D anatomical model for use as a reference. However, because the orbit bone is thin and complex, creating a segmentation model for 3D printing is complicated and time-consuming. Here, the training of DL was performed using U-Net as the DL model, and the AS output was validated with Dice coefficients and average symmetry surface distance (ASSD). In addition, the AS output was 3D printed and evaluated for accuracy by four surgeons, each with over 15 years of clinical experience. One hundred twenty-five CT images were prepared, and manual orbital segmentation was performed in all cases. Ten orbital fracture cases were randomly selected as validation data, and the remaining 115 were set as training data. AS was successful in all cases, with good accuracy: Dice, 0.860±0.033 (mean±SD); ASSD, 0.713±0.212mm. In evaluating AS accuracy, the expert surgeons generally considered that it could be used for surgical support without further modification. The orbital AS algorithm developed using DL in this study is extremely accurate and can create 3D models rapidly at low cost, potentially enabling safer and more accurate surgeries.

Temperature dependence of incommensurate modulation in Ca0.28Ba0.72Nb2O6

We present an electron microscopy and diffraction study of a CaxBa1−xNb2O6 ceramic with x=0.28 (CBN28), a ferroelectric material with a partially filled tetragonal tungsten-bronze structure. The microstructure has strong similarities to that of SrxBa1−xNb2O6, with an average orthorhombic symmetry and an intimate intermixture of merohedral twin variants at a length scale of tens of nanometers. Superstructure spots in diffraction patterns are displaced by a one-dimensional incommensurate modulation, characterized by a propagation vector δδ0. Heating experiments show that δ is strongly coupled to ferroelectric polarization, decreasing as 180° ferroelectric domains become more finely spaced and needle-like as the Curie temperature TC is approached during heating and increasing once more above TC. No change in symmetry is observed at TC, consistent with a transition from ferroelectric to antiferroelectric (or ferrielectric) relaxor properties. The superstructure spots and incommensurate modulation disappear ∼250°C above TC, consistent with polar regions in the material becoming fully transient.

Algorithmic Elementary-Age Nasal Reconstruction for Binder Syndrome

Background: Nasomaxillary hypoplasia, or Binder syndrome, is conventionally treated at skeletal maturity (SKM). However, our institution has employed a stratified treatment algorithm, treating severe patients with serial nasal expansion beginning at elementary school age (5-11 years) to expand the soft tissues and improve appearance prior to definitive rhinoplasty at SKM. This study evaluates the treatment patterns and outcomes of Binder syndrome employing this pathway. Methods: Eight cases of nasomaxillary hypoplasia were identified from a database of rhinoplasties performed between 2006 and 2020 at a tertiary children’s hospital. Photomorphometric analyses were conducted on basal, frontal, and lateral photographs. Goode’s ratio was used to assess change in projection, and pronasale to ala distance was measured bilaterally as a proxy of symmetry. Changes in symmetry were calculated using bilateral nasal width ratio. Results: In our sample, 6 patients (75.0%) had initial rhinoplasty prior to SKM, with 3 undergoing serial silastic implant placement, one with complicated implant placement followed by autologous reconstruction, and 2 undergoing serial expansion with allograft rib cartilage grafts. Average age at initial rhinoplasty was 6.3 (range 5.2-8.8) years for the pre-SKM group and 15.6 (range 14.1-17.1) years for the post-SKM group. Patients in the pre-SKM rhinoplasty group underwent more procedures than those at SKM (2.3 vs 1.5, respectively). The average change in nasal parameters was −0.029 for Goode’s ratio and −9.5° for nasolabial angle. The average post-operative nasal symmetry index was 0.98 representing near perfect symmetry. There was no significant difference in the degree of improvement in nasolabial angle, Goode’s ratio, or nasal symmetry between SKM and pre-SKM cohorts (all P > .05). Conclusion: Among patients with Binder syndrome, those treated initially at elementary-age were able to achieve similar improvements in nasal indices to those treated at SKM. This approach allows patients to live their key developmental years with less stigmata of their facial difference but requires more surgical intervention.

Lattice instability and magnetic phase transitions in strongly correlated MnAs

Using variable temperature x-ray total scattering in magnetic field, we study the interaction between lattice and magnetic degrees of freedom in MnAs, which loses its ferromagnetic order and hexagonal (‘H’) lattice symmetry at 318 K to recover the latter and become a true paramagnet when the temperature is increased to 400 K. Our results show that the 318 K transition is accompanied by highly anisotropic displacements of Mn atoms that appear as a lattice degree of freedom bridging the ‘H’ and orthorhombic phases of MnAs. This is a rare example of a lowering of an average crystal symmetry due to an increased displacive disorder emerging on heating. Our results also show that magnetic and lattice degrees of freedom appear coupled but not necessarily equivalent control variables for triggering phase transitions in strongly correlated systems in general and in particular in MnAs.

Can a powered knee-ankle prosthesis improve weight-bearing symmetry during stand-to-sit transitions in individuals with above-knee amputations?

BackgroundAfter above-knee amputation, the missing biological knee and ankle are replaced with passive prosthetic devices. Passive prostheses are able to dissipate limited amounts of energy using resistive damper systems during “negative energy” tasks like sit-down. However, passive prosthetic knees are not able to provide high levels of resistance at the end of the sit-down movement when the knee is flexed, and users need the most support. Consequently, users are forced to over-compensate with their upper body, residual hip, and intact leg, and/or sit down with a ballistic and uncontrolled movement. Powered prostheses have the potential to solve this problem. Powered prosthetic joints are controlled by motors, which can produce higher levels of resistance at a larger range of joint positions than passive damper systems. Therefore, powered prostheses have the potential to make sitting down more controlled and less difficult for above-knee amputees, improving their functional mobility.MethodsTen individuals with above-knee amputations sat down using their prescribed passive prosthesis and a research powered knee-ankle prosthesis. Subjects performed three sit-downs with each prosthesis while we recorded joint angles, forces, and muscle activity from the intact quadricep muscle. Our main outcome measures were weight-bearing symmetry and muscle effort of the intact quadricep muscle. We performed paired t-tests on these outcome measures to test for significant differences between passive and powered prostheses.ResultsWe found that the average weight-bearing symmetry improved by 42.1% when subjects sat down with the powered prosthesis compared to their passive prostheses. This difference was significant (p = 0.0012), and every subject’s weight-bearing symmetry improved when using the powered prosthesis. Although the intact quadricep muscle contraction differed in shape, neither the integral nor the peak of the signal was significantly different between conditions (integral p > 0.01, peak p > 0.01).ConclusionsIn this study, we found that a powered knee-ankle prosthesis significantly improved weight-bearing symmetry during sit-down compared to passive prostheses. However, we did not observe a corresponding decrease in intact-limb muscle effort. These results indicate that powered prosthetic devices have the potential to improve balance during sit-down for individuals with above-knee amputation and provide insight for future development of powered prosthetics.

Influence of the Hydride Content on the Local Structure of a Perovskite Oxyhydride BaTiO3–xHx

The local structure of BaTiO3 perovskite has been investigated for studying phase transitions by several groups. While it has been shown that H– doping into BaTiO3 changes its long-range average crystal symmetry from tetragonal to cubic, factors contributing to the structural change have not been fully understood. In this work, we investigated the dependence of the local structure of BaTiO3–xHx on the doped H– amount by means of Raman and X-ray absorption fine-structure (XAFS) spectroscopy measurements. The x value of BaTiO3–xHx could vary in the range of 0–0.5 by changing the amount of the CaH2 precursor in the topochemical synthesis of BaTiO3–xHx. Raman spectroscopy indicated that the cubic phase is induced in the initial tetragonal phase when the doped H– amount was relatively large (x ≥ 0.2). On the other hand, XAFS measurements revealed that the symmetry of the TiO6 octahedral in BaTiO3–xHx is strongly affected by the doped H– amount. The local distortion of the Ti [111] displacement is relaxed with an increase of H–, causing the tetragonal-to-cubic structural change.

Consequences of average time-reversal symmetry in disordered antiferromagnetic topological insulators

Average symmetry protects the topological surface states of topological (crystalline) insulators with time-reversal symmetry from disorder-induced localization. However, the nature of such average symmetry for magnetic topological insulators and, in particular, its connection to surface transport await inspection. Here, we investigate the impact of imperfect magnetic order on an antiferromagnetic topological insulator, which is a solid with a bulk axion field $\ensuremath{\theta}=\ensuremath{\pi}$. We find that the disordered topological surfaces of an antiferromagnetic topological insulator can be generally gapped and localized. The behavior of topological surfaces is now controlled by a mesoscopic average time-reversal symmetry that requires a magnetically imperfect system to be divisible into finite and magnetically neutral slabs. In the presence of this mesoscopic average time-reversal symmetry, the topological surface states will be gapless in the thermodynamic limit and tend to delocalize at a single energy similar to the delocalization transition in the chiral universality class. The notion of average-symmetry-induced delocalization is thus extended to account for magnetic topological insulators, and the spectroscopic and transport signatures clarified herein are relevant to future experimental investigations.

Evolution of symmetry index in minerals

AbstractCrystal structures of minerals are defined by a specific atomic arrangement within the unit‐cell, which follows the laws of symmetry specific to each crystal system. The causes for a mineral to crystallize in a given crystal system have been the subject of many studies showing their dependency on different formation conditions, such as the presence of aqueous fluids, biotic activity and many others. Different attempts have been made to quantify and interpret the information that we can gather from studying crystal symmetry and its distribution in the mineral kingdom. However, these methods are mostly outdated or at least not compatible for use on large datasets available today. Therefore, a revision of symmetry index calculation has been made in accordance with the growing understanding of mineral species and their characteristics. In the gathered data, we observe a gradual but significant decrease in crystal symmetry through the stages of mineral evolution, from the formation of the solar system to modern day. However, this decrease is neither uniform nor linear, which provides further implications for mineral evolution from the viewpoint of crystal symmetry. The temporal distribution of minerals based on the number of essential elements in their chemical formulae and their symmetry index has been calculated and compared to explore their behaviour. Minerals with four to eight essential elements have the lowest average symmetry index, while being the most abundant throughout all stages of mineral evolution. There are many open questions, including those pertaining to whether or not biological activity on Earth has influenced the observed decrease in mineral symmetry through time and whether or not the trajectory of planetary evolution of a geologically active body is one of decreasing mineral symmetry/increasing complexity.