There is convincing evidence that training spatial abilities leads to improved mathematics performance in typically developing (TD) children. However, a lack of information on mathematical development and spatial-mathematical associations in people with Down syndrome (DS) hinders the translation of these interventions. Here, we established developmental trajectories of mathematics and explored whether spatial ability predicts attainment on different mathematics measures in individuals with DS. Participants with DS (n = 36; ages 9-35 years) and TD children (n = 132; ages 4-11 years) completed three groups of tasks: spatial tasks assessing different subdomains of spatial thinking; mathematics tasks assessing early mathematics skills, mathematical reasoning, arithmetic, and geometry; and IQ tasks. The developmental trajectories of mathematics performance against mental age revealed similar starting points of the trajectories and similar rates of development for DS and TD groups. Furthermore, after controlling for verbal skills, spatial skills explained 5.8% to 18.1% of the variation in mathematical performance across different mathematics tasks, and the pattern of spatial-mathematical relations was similar for DS and mental age-matched TD groups. This shows that mathematical development in DS groups appears to mirror that in TD children, indicative of delay only. Strong spatial-mathematical relations were observed for individuals with DS, like those seen for TD participants. This is the vital preliminary knowledge needed to support the design and use of spatial intervention for improving mathematics in individuals with DS.

Current mechanical models of the bladder largely idealize the bladder as spherical with uniform thickness. This present study aims to investigate this idealization using micro-CT to generate 3D reconstructed models of rat bladders at 10-20 micrometer resolution in both voided and filled states. Applied to three rat bladders, this approach identifies shape, volume, and thickness variations under different pressures. These results demonstrate the filling/voiding process is far from the idealized spherical inflation/contraction. However, the geometry idealizations may be reasonable in cases where the filled bladder geometry is of importance, such as in studies of growth and remodeling.

Of Euclid\'s lost manuscripts, few have elicited as much scholarly attention as the Porisms, of which a couple of brief summaries by late-Antiquity commentators are extant. Despite the lack of textual sources, attempts at restoring the content of this absent volume became numerous in early-modern Europe, following the diffusion of ancient mathematical manuscripts preserved in the Arabic world. Later, one similar attempt was that of French geometer Michel Chasles (1793-1880). This paper investigates the historiographical tenets and practices involved in Chasles\' restoration of the porisms, as well as the philosophical and mathematical claims tentatively buttressed therewith. Echoes of the Quarrel of the Ancients and the Moderns, and of a long-standing debate on the authority and usefulness of the past, are shown to have decisively shaped Chasles\' enterprise-and, with it, his integration of mathematical and historical research.

BACKGROUND: The female locust is equipped with unique digging tools, namely two pairs of valves-a dorsal and a ventral-utilized for excavating an underground hole in which she lays her eggs. This apparatus ensures that the eggs are protected from potential predators and provides optimal conditions for successful hatching. The dorsal and the ventral valves are assigned distinct roles in the digging process. Specifically, the ventral valves primarily function as anchors during propagation, while the dorsal valves displace soil and shape the underground tunnel.
RESULTS: In this study, we investigated the noticeable asymmetry and distinct shapes of the valves, using a geometrical model and a finite element method. Our analysis revealed that although the two pairs of valves share morphological similarities, they exhibit different 3D characteristics in terms of absolute size and structure. We introduced a structural characteristic, the skew of the valve cross-section, to quantify the differences between the two pairs of valves. Our findings indicate that these structural variations do not significantly contribute to the valves\' load-bearing capabilities under external forces.
CONCLUSIONS: The evolutionary development of the form of the female locust digging valves is more aligned with fitting their respective functions rather than solely responding to biomechanical support needs. By understanding the intricate features of these locust valves, and using our geometrical model, valuable insights can be obtained for creating more efficient and specialized tools for various digging applications.

The shape of a flat-faceted octahedral crystal can be uniquely defined by the measured distances between pairs of its parallel facets and the length of one of its false edges. In total, only five numerical values are involved in this approach. Some interdependencies of parameters that allow one to control the correctness of measurements were derived. The proposed method is suitable for describing the shape as full-faceted, or as incomplete octahedral crystals (e.g. diamond) with unequally developed facets. This so-called `real crystal form\' can be considered as one of the typomorphic features of minerals, connecting the dissymmetry to the anisotropy of the host rock. The measurement results can be used in crystallo-morphological analysis, restoration of the lost crystal shape in the case of man-made damage and in the practice of diamond prospecting.

Understanding the three-dimensional anatomy of the talar neck is essential in assessing the accuracy of reduction in talar neck fractures as well as for planning surgical correction for talar malunions. However, the geometrical parameters that describe this anatomy are sparsely reported in the orthopedics literature. We aimed to identify from the existing literature, geometrical parameters that describe the anatomy of the talar neck, determine how these are measured, and their normative values. A scoping literature review was conducted in accordance with the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) extension for scoping reviews (PRISMA-ScR) guidelines. The primary searches were conducted on the PubMed, Embase, and Scopus databases. Any original research study looking at the human talus neck geometry was included. Parameters that described the anatomy of the talar neck were identified, and pooled estimates were determined by the random-effects meta-analysis model. Heterogeneity was assessed by the I2 test and leave-one-out meta-analysis. Subgroup analysis was done to compare the values of parameters between the Asian and Non-Asian populations. The risk of bias was assessed by the National Institutes of Health (NIH) Case Series Tool. The combined searches yielded 6326 results, of which 21 studies were included in the review and 15 in six different sets of metanalysis. The majority of the studies (n=19, 90.5%) evaluated adult tali, and only two (9.5%) evaluated pediatric tali. In most of the studies (n=13, 61.9%), talus neck geometry was evaluated on dry bones or anatomical specimens; evaluation by imaging techniques (radiographs, CT, MRI, and radiostereometric analysis) was used in eight studies, (39.1%). A total of eight different geometrical parameters (neck length, height, width, declination angle, inclination angle, torsion angle, circumference, and cross-sectional area) were identified. Except for talar torsion, variability was noted in methods of measurement of all other parameters. Subgroup analysis revealed that Asians had a higher neck height as compared to non-Asians; other parameters were not significantly different. Although the literature reports geometrical parameters to assess the talar geometry, the methods of measurement of these parameters are variable. Most of the available literature describes measurement techniques on cadaveric tali, and there is no literature on how these parameters should be measured on conventional CT or MRI slices. Further research needs to focus on the standardization of measurement techniques for these parameters on conventional CT and/or MRI scans.

Geometry has been identified as a cognitive domain where deaf individuals exhibit relative strength, yet the neural mechanisms underlying geometry processing in this population remain poorly understood. This fMRI study aimed to investigate the neural correlates of geometry processing in deaf and hearing individuals. Twenty-two adult deaf signers and 25 hearing non-signers completed a geometry decision task. We found no group differences in performance, while there were some differences in parietal activation. As expected, the posterior superior parietal lobule (SPL) was recruited for both groups. The anterior SPL was significantly more activated in the deaf group, and the inferior parietal lobule was significantly more deactivated in the hearing group. In conclusion, despite similar performance across groups, there were differences in the recruitment of parietal regions. These differences may reflect inherent differences in brain organization due to different early sensory and linguistic experiences.

Geometric criteria can be used to assess whether cell intercalation is active or passive during the convergent extension of tissue.

Experimental structural data for bis(terpyridine)iron(II) and a series of related iron(II) complexes, featuring either substituted terpyridine or tris-azinyl analogues of terpyridine, are presented and analyzed in terms of the Mean Absolute Deviation (MAD) from the average experimental data for each specific complex. The experimental structural data are then juxtaposed with density functional theory (DFT) calculated data obtained using various combinations of DFT functionals and basis sets, with and without the inclusion of Grimme D3 empirical dispersion correction. These diverse computational approaches yield optimized geometries that are subsequently compared against the available experimental structural data to assess their accuracy. The aim is to identify a reliable DFT method for determining the geometries of bis(terpyridine)iron(II) and its related complexes.

Visualisations facilitate the interpretation of geometrically structured data and results. However, heterogeneous geometries-such as volumes, surfaces, and networks-have traditionally mandated different software approaches. We introduce hyve, a Python library that uses a compositional functional framework to enable parametric implementation of custom visualisations for different brain geometries. Under this framework, users compose a reusable visualisation protocol from geometric primitives for representing data geometries, input primitives for common data formats and research objectives, and output primitives for producing interactive displays or configurable snapshots. hyve also writes documentation for user-constructed protocols, automates serial production of multiple visualisations, and includes an API for semantically organising an editable multi-panel figure. Through the seamless composition of input, output, and geometric primitives, hyve supports creating visualisations for a range of neuroimaging research objectives.