Three-dimensional contingency tables are analyzed, with one variable (e.g., sex) as a factor, and with a natural relation between the other variables (e.g. left and right eye vision). Models of special interest, like symmetry and proportional symmetry between the related variables, and homogeneity across the factor levels, are investigated. Maximum likelihood estimators of parameters and partitions of chi-square goodness-of-fit statistics are explicitly presented; the independence of certain models is noted, and an example is discussed.

Symmetry is a salient visual feature in the natural world, yet the perception of symmetry may be influenced by how natural lighting conditions (e.g., shading) fall on the object relative to its symmetry axis. Here, we investigate how symmetry detection may interact with luminance polarity grouping, and whether this modulates neural responses to symmetry, as evidenced by the Sustained Posterior Negativity (SPN) component of Event-Related Potentials (ERPs). Stimuli were dot patterns arranged either symmetrically (reflection, rotation, translation) or quasi-randomly, and by luminance polarity about a grouping axis (i.e., black dots on one side and white dots on the other). We varied the relative angular separation between the symmetry and polarity-grouping axes: 0, 30, 60, 90 deg. Participants performed a two interval-forced-choice (2IFC) task indicating which interval contained the symmetrical pattern. We found that accuracy for the 0 deg polarity-grouped condition was higher compared to the single-polarity condition for rotation and translation (but not reflection symmetry), and higher than all other angular difference (30, 60, 90) conditions for all symmetry types. The SPN was found to be separated topographically into an early and late component, with the early SPN being sensitive to luminance polarity grouping at parietal-occipital electrodes, and the late SPN sensitive to symmetry over central electrodes. The increase in relative angular differences between luminance polarity and symmetry axes highlighted changes between cardinal (0, 90 deg) and other (30, 60 deg) angles. Critically, we found a polarity-grouping effect in the SPN time window for noise only patterns, which was related to symmetry type, suggesting a task/ symmetry pattern influence on SPN processes. We conclude that luminance polarity grouping can facilitate symmetry perception when symmetry is not readily salient, as evidenced by polarity sensitivity of early SPN, yet it can also inhibit neural and behavioral responses when luminance polarity and symmetry axes are not aligned.

Helices are one of the most frequently encountered symmetries in biological assemblies. Helical symmetry has been exploited in electron microscopic studies as a limited number of filament images, in principle, can provide all the information needed to do a three-dimensional reconstruction of a polymer. Over the past 25 years, three-dimensional reconstructions of helical polymers from cryo-EM images have shifted completely from Fourier–Bessel methods to single-particle approaches. The single-particle approaches have allowed people to surmount the problem that very few biological polymers are crystalline in order, and despite the flexibility and heterogeneity present in most of these polymers, reaching a resolution where accurate atomic models can be built has now become the standard. While determining the correct helical symmetry may be very simple for something like F-actin, for many other polymers, particularly those formed from small peptides, it can be much more challenging. This review discusses why symmetry determination can be problematic, and why trial-and-error methods are still the best approach. Studies of many macromolecular assemblies, such as icosahedral capsids, have usually found that not imposing symmetry leads to a great reduction in resolution while at the same time revealing possibly interesting asymmetric features. We show that for certain helical assemblies asymmetric reconstructions can sometimes lead to greatly improved resolution. Further, in the case of supercoiled flagellar filaments from bacteria and archaea, we show that the imposition of helical symmetry can not only be wrong, but is not necessary, and obscures the mechanisms whereby these filaments supercoil.

We consider a class of nonhomogeneous elliptic equations in the half-space with critical singular boundary potentials and nonlinear fractional derivative terms. The forcing terms are considered on the boundary and can be taken as singular measure. Employing a functional setting and approach based on localization-in-frequency and Littlewood–Paley decomposition, we obtain results on solvability, regularity, and symmetry of solutions.

From the symmetry point of view, micas may be classified as follows: those with all three octahedrally coordinated sites occupied by the same cation (homo-octahedral micas), those with only two of these sites occupied by the same cation (meso-octahedral micas), and those with the three sites occupied by different cations or by two different cations and a void, in an ordered manner (hetero-octahedral micas). For any of these three classes, mica polytypes, idealized in accordance with the generalized Pauling model, can be interpreted as OD structures consisting of octahedral OD layering and tetrahedral OD layering in which an interlayer cation plane is sandwiched between tetrahedral sheets. A mica layer built up by an octahedral sheet and two halves of tetrahedral sheets on either side consists of two OD packets linked by a two-fold rotation.

The orientation of any OD packet may be given by a number from 0 to 5 (related to a hexagonal coordinate system). A dot behind or before these numbers is used to denote the position of the octahedral layer (number + dot = orientational character). The displacement of a packet against its predecessor is characterized by a vector from the origin of a packet pn (or qn-1) to the origin of the adjacent packet pn+1 (or p2n). These displacements may also be symbolized by numbers from 0 to 5 (displacement characters); a zero displacement is symbolized by *. Any mica polytype (ordered or disordered) can thus be described by a two-line symbol. The orientational characters are located on the first line, and the displacement characters on the second. Any symbol, therefore denotes unequivocally the stacking layers in a polytype. The space-group symmetry of ordered polytypes follows directly from the symbol.

Single crystals of sudoite from Ottré, Belgium, allow confirmation of a dioctahedral 2:1 layer and a trioctahedral interlayer in a IIb arrangement. A regular 2-layer s structure is formed in which the octahedral stagger within both 2:1 layers is directed along X1 and adjacent layers are alternately displaced by a2/3 and a3/3. Poor quality crystals and twinning prevented three-dimensional refinement. One-dimensional refinement suggests that the smaller d(001) value of dioctahedral chlorites relative to trioctahedral species is due primarily to the thinner dioctahedral sheet.

Morphological symmetry abnormalities in cheliped appendages of alpheid shrimps are extremely rare and poorly recorded in the literature. A symmetric minor cheliped were, for example, observed in queen females belonging to Synalpheus eusocial species. Symmetric major chelipeds were now described in Synalpheus fritzmuelleri individuals living in shallow Brazilian waters. These individuals were found in symbiotic association with the bryozoan Schizoporella sp. (biogenic substrate) adhering to the pilings of Ubatuba Bay docks, São Paulo State. Only one of 20 sampled S. fritzmuelleri individuals presented anomalous symmetric chelipeds. Based on carapace length, size, and morphological features, the analyzed specimens seemed to be juvenile; thus, the hypothesis of anomalous condition can be directly linked to genetic inhibition of the mechanism accounting for major cheliped development in this ontogeny phase. Studies like the present one often provide remarkable information on animal morphology and can be used as reference in evolutionary assessments to be conducted in the future.

Hypertrophic ‘giant’ handaxes are a rare component of Acheulean assemblages, yet have been central to debates relating to the social, cognitive and cultural ‘meaning’ of these enigmatic tools. The authors examine giant handaxes from the perspective of the British record and suggest that they are chronologically patterned, with the great majority originating from contexts broadly associated with Marine Isotope Stage 9. Giant handaxes tend to have higher symmetry than non-giants, and extravagant forms, such as ficrons, are better represented; they may therefore be linked to incipient aesthetic sensibilities and, potentially, to changing cognition at the transition between the Lower and Middle Palaeolithic.