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Concept: Point groups in three dimensions


Symmetry is a biologically relevant, mathematically involving, and aesthetically compelling visual phenomenon. Mirror symmetry detection is considered particularly rapid and efficient, based on experiments with random noise. Symmetry detection in natural settings, however, is often accomplished against structured backgrounds. To measure salience of symmetry in diverse contexts, we assembled mirror symmetric patterns from 101 natural textures. Temporal thresholds for detecting the symmetry axis ranged from 28 to 568 ms indicating a wide range of salience (1/Threshold). We built a model for estimating symmetry-energy by connecting pairs of mirror-symmetric filters that simulated cortical receptive fields. The model easily identified the axis of symmetry for all patterns. However, symmetry-energy quantified at this axis correlated weakly with salience. To examine context effects on symmetry detection, we used the same model to estimate approximate symmetry resulting from the underlying texture throughout the image. Magnitudes of approximate symmetry at flanking and orthogonal axes showed strong negative correlations with salience, revealing context interference with symmetry detection. A regression model that included the context-based measures explained the salience results, and revealed why perceptual symmetry can differ from mathematical characterizations. Using natural patterns thus produces new insights into symmetry perception and its possible neural circuits.

Concepts: Statistics, Mathematics, Symmetry, Estimation, Symmetry group, Rotational symmetry, Point groups in three dimensions, Reflection symmetry


We report the first occurrence of an icosahedral quasicrystal with composition Al62.0(8)Cu31.2(8)Fe6.8(4), outside the measured equilibrium stability field at standard pressure of the previously reported Al-Cu-Fe quasicrystal (AlxCuyFez, with x between 61 and 64, y between 24 and 26, z between 12 and 13%). The new icosahedral mineral formed naturally and was discovered in the Khatyrka meteorite, a recently described CV3 carbonaceous chondrite that experienced shock metamorphism, local melting (with conditions exceeding 5 GPa and 1,200 °C in some locations), and rapid cooling, all of which likely resulted from impact-induced shock in space. This is the first example of a quasicrystal composition discovered in nature prior to being synthesized in the laboratory. The new composition was found in a grain that has a separate metal assemblage containing icosahedrite (Al63Cu24Fe13), currently the only other known naturally occurring mineral with icosahedral symmetry (though the latter composition had already been observed in the laboratory prior to its discovery in nature). The chemistry of both the icosahedral phases was characterized by electron microprobe, and the rotational symmetry was confirmed by means of electron backscatter diffraction.

Concepts: Chondrite, Observation, Knowledge, Atmospheric pressure, Vacuum, Meteorite, Rotational symmetry, Point groups in three dimensions


Restricting our scope to the dynamical motion of the leaflets, we present a computational model for a symmetric, tri-leaflet, bioprosthetic heart valve (BHV) at the end of five complete cardiac pressure cycles, reaching the steady state of deformation during both closing and opening phases. To this end, we utilized a highly anisotropic material model for the large deformation behavior of the tissue material, for which an experimental validation was provided. The important findings are: (1) material anisotropy has significant effect on the valve opening/closing; (2) the asymmetric deformations, especially in the fully closed configuration, justify the use of cyclic symmetry; (3) adopting the fully-open position as an initial/reference configuration has the advantage of completely bypassing any complications arising from the need to assume the size and shape of the contact area in the coaptation regions of the leaflets that is necessary when the alternative, commonly-used, approach of selecting the fully-closed position is used as a reference; and (4) with proper treatments for both material anisotropy and tissue-to-tissue contact, the overall BHV model provide realistic results in conformity with the ex vivo/in vitro experiments.

Concepts: Thermodynamics, Symmetry, Geometry, Young's modulus, Attractor, Artificial heart valve, Asymmetry, Point groups in three dimensions


We describe a general computational method for designing proteins that self-assemble to a desired symmetric architecture. Protein building blocks are docked together symmetrically to identify complementary packing arrangements, and low-energy protein-protein interfaces are then designed between the building blocks in order to drive self-assembly. We used trimeric protein building blocks to design a 24-subunit, 13-nm diameter complex with octahedral symmetry and a 12-subunit, 11-nm diameter complex with tetrahedral symmetry. The designed proteins assembled to the desired oligomeric states in solution, and the crystal structures of the complexes revealed that the resulting materials closely match the design models. The method can be used to design a wide variety of self-assembling protein nanomaterials.

Concepts: Protein, Crystallography, Materials science, Design, Design management, Octahedron, Rotational symmetry, Point groups in three dimensions


The formation of quasi-spherical cages from protein building blocks is a remarkable self-assembly process in many natural systems, where a small number of elementary building blocks are assembled to build a highly symmetric icosahedral cage. In turn, this has inspired synthetic biologists to design de novo protein cages. We use simple models, on multiple scales, to investigate the self-assembly of a spherical cage, focusing on the regularity of the packing of protein-like objects on the surface. Using building blocks, which are able to pack with icosahedral symmetry, we examine how stable these highly symmetric structures are to perturbations that may arise from the interplay between flexibility of the interacting blocks and entropic effects. We find that, in the presence of those perturbations, icosahedral packing is not the most stable arrangement for a wide range of parameters; rather disordered structures are found to be the most stable. Our results suggest that (i) many designed, or even natural, protein cages may not be regular in the presence of those perturbations and (ii) optimizing those flexibilities can be a possible design strategy to obtain regular synthetic cages with full control over their surface properties.

Concepts: System, Group, Design, Aesthetics, Symmetry group, Rotational symmetry, Point groups in three dimensions, Platonic solid


Self-assembly of rigid building blocks with explicit shape and symmetry is substantially influenced by the geometric factors and remains largely unexplored. We report the selective assembly behaviors of a class of precisely defined, nanosized giant tetrahedra constructed by placing different polyhedral oligomeric silsesquioxane (POSS) molecular nanoparticles at the vertices of a rigid tetrahedral framework. Designed symmetry breaking of these giant tetrahedra introduces precise positional interactions and results in diverse selectively assembled, highly ordered supramolecular lattices including a Frank-Kasper A15 phase, which resembles the essential structural features of certain metal alloys but at a larger length scale. These results demonstrate the power of persistent molecular geometry with balanced enthalpy and entropy in creating thermodynamically stable supramolecular lattices with properties distinct from those of other self-assembling soft materials.

Concepts: Chemistry, Nanotechnology, Symmetry, Geometry, Polyhedron, Point groups in three dimensions, Polyhedral compound, Platonic solid


On 10 August 2016 China launched the GF-3, its first C-band polarimetric synthetic aperture radar (SAR) satellite, which was put into operation at the end of January, 2017. GF-3 polarimetric SAR has many advantages such as high resolution and multi-polarization imaging capabilities. Polarimetric SAR can fully characterize the backscatter property of targets, and thus it is of great interest to explore the physical scattering mechanisms of terrain types, which is very important in interpreting polarimetric SAR imagery and for its further usages in Earth observations. In this paper, focusing on target scattering characterization and feature extraction, we generalize the Δ α B / α B method, which was proposed under the reflection symmetric assumption, for the general backscatter process to account for both the reflection symmetry and asymmetry cases. Then, we evaluate the performances of physical scattering mechanism analysis methods for GF-3 polarimetric SAR imagery. Radarsat-2 data acquired over the same area is used for cross validation. Results show that GF-3 polarimetric SAR data has great potential for target characterization, especially for ocean area observation.

Concepts: Symmetry, Group, Radar, Symmetry group, Asymmetry, Point groups in three dimensions, Reflection symmetry


We study the symmetric properties of waveguide modes in presence of gain/losses, anisotropy/bianisotropy, or continuous/discrete rotational symmetry. We provide a comprehensive approach to identity the modal symmetry by constructing a 4 × 4 waveguide Hamiltonian and searching the symmetric operation in association with the corresponding waveguides. We classify the chiral/time reversal/parity/parity time/rotational symmetry for different waveguides, and provide the criterion for the aforementioned symmetry operations. Lastly, we provide examples to illustrate how the symmetry operations can be used to classify the modal properties from the symmetric relation between modal profiles of several different waveguides.

Concepts: Angular momentum, Symmetry, Group, Rotational symmetry, Point groups in three dimensions, Symmetric relation, Symmetry combinations, Antisymmetric relation


Covering: 2012 to June 2017This review aims to show that complex natural product synthesis can be streamlined by taking advantage of molecular symmetry. Various strategies to construct molecules with either evident or hidden symmetry are illustrated. Insights regarding the origins and adjustments of these strategies as well as inspiring new methodological developments are deliberated. When a symmetric strategy fails, the corresponding reason is analysed and an alternative approach is briefly provided. Finally, the importance of exploiting molecular symmetry and future research directions are discussed.

Concepts: Chemical substance, Symmetry, Group, Group theory, Aesthetics, Point groups in three dimensions, Point group, M. C. Escher


The isomerizations of 3-aza-3-ium-dihydrobenzvalene, 3,4-diaza-3-ium-dihydrobenzvalene, and 3,4-diaza-diium-dihydrobenzvalene to their respective cyclic-diene products have been studied using electronic structure methods with a multiconfigurational wavefunction and several single reference methods. Transition states for both the allowed (conrotatory) and forbidden (disrotatory) pathways were located. The conrotatory pathways of each structure all proceed through a cyclic intermediate with a trans double bond in the ring: this trans double bond destroys the aromatic stabilization of the π electrons due to poor orbital overlap between the cis and trans π bonds. The 3, 4-diaza-3-ium-dihydrobenzvalene structure has C1 symmetry, and there are four separate allowed and forbidden pathways for this structure. The 3-aza-3-ium-dihydrobenzvalene structure is Cs symmetric, and there are two separate allowed and forbidden pathways for this structure. For 3, 4-diaza-3, 4-diium-benzvalene, there was a single allowed and single forbidden pathway due to the C2v symmetry. The separation of the barrier heights for all three molecules was studied, and we found the difference in activation barriers for the lowest allowed and lowest forbidden pathways in 3, 4-diaza-3-ium-dihydrobenzvalene, 3-aza-3-ium-dihydrobenzvalene, and 3, 4-diaza-diium-benzvalene to be 9.1, 7.4, and 3.7 kcal/mol, respectively. The allowed and forbidden barriers of 3, 4-diaza-diium-dihydrobenzvalene were separated by 3.7 kcal/mol, which is considerably less than the 12-15 kcal/mol expected based on the orbital symmetry rules. The addition of the secondary ammonium group tends to shift the conrotatory and disrotatory barriers up in energy (approximately 12-14 kcal/mol (conrotatory) and 5-10 kcal/mol (disrotatory) per secondary NH2 group) relative to the barriers of dihydrobenzvalene, but there is negligible effect on E, Z to Z, Z isomerization barriers, which remain in the expected range of > 4 kcal/mol.

Concepts: Addition, Symmetry, Group, Group theory, Physical organic chemistry, Point groups in three dimensions, Conrotatory and disrotatory, Conrotatory