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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


Three different algorithms, as implemented in three different computer programs, were put to the task of extracting direct space lattice parameters from four sets of synthetic images that were per design more or less periodic in two dimensions (2D). One of the test images in each set was per design free of noise and, therefore, genuinely 2D periodic so that it adhered perfectly to the constraints of a Bravais lattice type, Laue class, and plane symmetry group. Gaussian noise with a mean of zero and standard deviations of 10 and 50% of the maximal pixel intensity was added to the individual pixels of the noise-free images individually to create two more images and thereby complete the sets. The added noise broke the strict translation and site/point symmetries of the noise-free images of the four test sets so that all symmetries that existed per design turned into pseudo-symmetries of the second kind. Moreover, motif and translation-based pseudo-symmetries of the first kind, a.k.a. genuine pseudo-symmetries, and a metric specialization were present per design in the majority of the noise-free test images already. With the extraction of the lattice parameters from the images of the synthetic test sets, we assessed the robustness of the algorithms' performances in the presence of both Gaussian noise and pre-designed pseudo-symmetries. By applying three different computer programs to the same image sets, we also tested the reliability of the programs with respect to subsequent geometric inferences such as Bravais lattice type assignments. Partly due to per design existing pseudo-symmetries of the first kind, the lattice parameters that the utilized computer programs extracted in their default settings disagreed for some of the test images even in the absence of noise, i.e., in the absence of pseudo-symmetries of the second kind, for any reasonable error estimates. For the noisy images, the disagreement of the lattice parameter extraction results from the algorithms was typically more pronounced. Non-default settings and re-interpretations/re-calculations on the basis of program outputs allowed for a reduction (but not a complete elimination) of the differences in the geometric feature extraction results of the three tested algorithms. Our lattice parameter extraction results are, thus, an illustration of Kenichi Kanatani’s dictum that no extraction algorithm for geometric features from images leads todefinitiveresults because they are all aiming at an intrinsically impossible task in all real-world applications (Kanatani in Syst Comput Jpn 35:1-9, 2004). Since 2D-Bravais lattice type assignments are the natural end result of lattice parameter extractions from more or less 2D-periodic images, there is also a section in this paper that describes the intertwined metric relations/holohedral plane and point group symmetry hierarchy of the five translation symmetry types of the Euclidean plane. Because there is no definitive lattice parameter extraction algorithm, the outputs of computer programs that implemented such algorithms are also not definitive. Definitive assignments of higher symmetric Bravais lattice types to real-world images should, therefore, not be made on the basis of the numerical values of extracted lattice parameters and their error bars. Such assignments require (at the current state of affairs) arbitrarily set thresholds and are, therefore, alwayssubjectiveso that they cannot claim objective definitiveness. This is the essence of Kenichi Kanatani’s comments on the vast majority of computerized attempts to extract symmetries and other hierarchical geometric features from noisy images (Kanatani in IEEE Trans Pattern Anal Mach Intell 19:246-247, 1997). All there should be instead for noisy and/or genuinely pseudo-symmetric images are rankings of the relative likelihoods of classifications into higher symmetric Bravais lattice types, Laue classes, and plane symmetry groups.

Concepts: Crystallography, Symmetry, Geometry, Group, Lattice, Symmetry group, Point groups in three dimensions, Wallpaper group


It is well known that freeform surfaces are used to improve the resolution in systems without rotational symmetry. For Scheimpflug systems, the tilted object plane leads to variant magnification in the system imaging. Thus, the system suffers from non-rotationally symmetric aberrations, non-uniform resolution, and non-uniform intensity distribution. In this paper, the paraxial imaging condition of Scheimpflug systems is discussed. From the classical viewpoint, the aberration theory is used to understand, balance, and improve the system performance for variant object distance. For large object distance shift, it is necessary to apply freeform surfaces. With the initial system design method based on Gaussian brackets, the starting configuration of a Scheimpflug system with large object distance shift is obtained. Based on the extension of the Nodal aberration theory concerning the aberrations of freeform surfaces, the rules of selecting the freeform surface position in the system are introduced. By adding two freeform surfaces far away from the pupil, the aberrations are effectively corrected in the Scheimpflug system. The aberrations along the field are decomposed and represented using Zernike fringe polynomials to show the improvement of uniformity and resolution. This work provides insight into Scheimpflug system design with freeform surfaces.

Concepts: Optics, Symmetry, Group, Classical mechanics, Analytic geometry, Surfaces, Rotational symmetry, Point groups in three dimensions


Electrophysiological studies of symmetry have found a difference wave termed the Sustained Posterior Negativity (SPN) related to the presence of symmetry. Yet the extent to which the SPN is modulated by luminance-polarity and colour content is unknown. Here we examine how luminance-polarity distribution across the symmetry axis, grouping by luminance polarity, and the number of colours in the stimuli, modulate the SPN. Stimuli were dot patterns arranged either symmetrically or quasi-randomly. There were several arrangements: ‘segregated’-symmetric dots were of one polarity and random dots of the other; ‘unsegregated’-symmetric dots were of both polarities in equal proportions; ‘anti-symmetric’-dots were of opposite polarity across the symmetry axis; ‘polarity-grouped anti-symmetric’-as anti-symmetric but with half the pattern of one polarity and the other opposite; multi-colour symmetric patterns made of either two, three and four colours. We found that the SPN is: reduced by the amount of position-symmetry, sensitive to luminance-polarity mismatch across the symmetry axis and not modulated by the number of colours in the stimuli. Our results show that the sustained nature of the SPN coincides with the late onset of a topographic microstate sensitive to symmetry. These findings emphasise the importance of not only position symmetry, but also luminance polarity matching across the symmetry axis.

Concepts: Symmetry, Color, Dots, Symmetry group, Rotational symmetry, Point groups in three dimensions, Reflection symmetry, Symmetric relation


Diabetic retinopathy is the leading cause of blindness among working-age adults in most developed countries. It affects eyes bilaterally and is generally believed to be symmetrical, yet there are few studies evaluating the symmetry of diabetic retinopathy. The purpose of the present study was to evaluate the symmetry of the amount of peripheral retinal ischemia in patients with diabetic retinopathy.

Concepts: Stroke, Developed country, Retina, Diabetic retinopathy, Cultural studies, Point groups in three dimensions