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Colloidal particles, which consist of clusters of hundreds or thousands of atoms, can still resemble atomic systems. In particular, colloids have been used to study the packing of spheres and the influence of short-range interactions on crystallization and melting. Manoharan reviews these similarities, as well as the cases in which colloidal particles show behavior not seen in atomic systems. For example, the packing of nonspherical objects, where geometry or topology may matter, can give insights into the role of entropy in packing. Science , this issue 10.1126/science.1253751 Colloidal particles with well-controlled shapes and interactions are an ideal experimental system for exploring how matter organizes itself. Like atoms and molecules, these particles form bulk phases such as liquids and crystals. But they are more than just crude analogs of atoms; they are a form of matter in their own right, with complex and interesting collective behavior not seen at the atomic scale. Their behavior is affected by geometrical or topological constraints, such as curved surfaces or the shapes of the particles. Because the interactions between the particles are often short-ranged, we can understand the effects of these constraints using geometrical concepts such as packing. The geometrical viewpoint gives us a window into how entropy affects not only the structure of matter, but also the dynamics of how it forms.

Topology and geometry are essential to our understanding of modern physics, underlying many foundational concepts from high-energy theories, quantum information, and condensed-matter physics. In condensed-matter systems, a wide range of phenomena stem from the geometry of the band eigenstates, which is encoded in the matrix-valued Wilson line for general multiband systems. Using an ultracold gas of rubidium atoms loaded in a honeycomb optical lattice, we realize strong-force dynamics in Bloch bands that are described by Wilson lines and observe an evolution in the band populations that directly reveals the band geometry. Our technique enables a full determination of band eigenstates, Berry curvature, and topological invariants, including single- and multiband Chern and Z₂ numbers.

This study investigated the effects of students’ constructions of geometrical concepts related to the circle on their construction and validation of proofs. The participants were 110 high school students. A questionnaire was administered; both qualitative and quantitative methods were used to analyze the results. Afterwards, in-depth interviews were conducted with some of the participants. The findings from the interviews enriched and strengthened the findings from the questionnaire. Together, the findings highlight the impact of two factors on the ability to construct or evaluate proofs: (1) the use of the self-attributes of a single presented drawing instead of the critical attributes of the concept; and (2) the use of prototypical or non-prototypical examples. In this study, the position of the drawing attached to the assignment affected students’ construction of proofs.

This study aims to reveal how the embodied cognition of certain geometrical concepts of secondary-school students arises via gestures and what kinds of gestures they produce while engaging with different concepts. The study participants comprised four eleventh-grade students studying at a state high school in Turkey. The study focused on the gestures of students related to angle, a measure of an angle, congruence-similarity, and translation. Data were gathered via video-recorded focus group discussions and individual interviews, and the cognition of the students for each concept was coded using content analysis. According to the research findings, it was found that the deictic gestures of the participants reflect the grounding of cognition in the physical environment; representational gestures manifest mental simulations of action and perception, and some metaphoric gestures reflect body-based conceptual metaphors.

In the Cayley–Klein model, we review some basic results concerning the geometry of hyperbolic triangles. We introduce a new definition of the circumcircle of a hyperbolic triangle, guaranteed to exist in every case, and describe its main properties. Our central theorem establishes, by means of purely elementary projective geometric arguments, that a hyperbolic triangle has a nine-point conic if and only if it is a right triangle. Subject Classification: 51M09

Shape knowledge, a key aspect of school readiness, is part of early mathematical learning. Variations in how children are exposed to shapes may affect the pace of their learning and the nature of their shape knowledge. Building on evidence suggesting that child-centered, playful learning programs facilitate learning more than other methods, 4- to 5-year-old children (N = 70) were taught the properties of four geometric shapes using guided play, free play, or didactic instruction. Results revealed that children taught shapes in the guided play condition showed improved shape knowledge compared to the other groups, an effect that was still evident after 1 week. Findings suggest that scaffolding techniques that heighten engagement, direct exploration, and facilitate \"sense-making,\" such as guided play, undergird shape learning.

Primary school students often find it difficult to understand the differences between two dimensional and three-dimensional geometric shapes. Taking advantage of the ability of virtual and augmented reality to visualize 3D objects, we investigate the potential of using virtual and augmented reality technologies for teaching the lesson of geometric solids to primary school children. As part of the study 30 fourth, fifth and sixth class primary school students were divided into three groups that include a control group and two experimental groups. The first and second experimental groups used dedicated virtual and augmented reality applications to learn about geometric solids, while students from the control group used traditional printed material as part of the learning process. The results indicate that the implementation of new technologies in education of virtual and augmented reality improve interactivity and student interest in mathematics education, contributing to more efficient learning and understanding of mathematical concepts when compared to traditional teaching methods. No significant difference was found between virtual and augmented reality technologies with regard to the efficiency of the methods that contribute to the learning of mathematics, suggesting that both virtual and augmented reality display similar potential for educational activities in Mathematics.

This paper examines mathematics teachers’ level of acceptance and intention to use the Augmented Reality Geometry Tutorial System (ARGTS), a mobile Augmented Reality (AR) application developed to enhance students’ 3D geometric thinking skills. ARGTS was shared with mathematics teachers, who were then surveyed using the Technology Acceptance Model (TAM) to understand their acceptance of the technology. We also examined the external variables of Anxiety, Social Norms and Satisfaction. The effect of the teacher’s gender, degree of graduate status and number of years of teaching experience on the subscales of the TAM model were examined. We found that the Perceived Ease of Use (PEU) had a direct effect on the Perceived Usefulness (PU) in accordance with the Technology Acceptance Model (TAM). Both variables together affect Satisfaction (SF), however PEU had no direct effect on Attitude (AT). In addition, while Social Norms (SN) had a direct effect on PU and PEU, there was no direct effect on Behavioural Intention (BI). Anxiety (ANX) had a direct effect on PEU, but no effect on PU and SF. While there was a direct effect of SF on PEU, no direct effect was found on BI. We explain how the results of this study could help improve the understanding of AR acceptance by teachers and provide important guidelines for AR researchers, developers and practitioners.

This study describes the implementation and effects of a 32-week teacher-led spatial reasoning intervention in K-2 classrooms. The intervention targeted spatial visualization skills as an integrated feature of regular mathematics instruction. Compared to an active control group, children in the spatial intervention demonstrated gains in spatial language, visual-spatial reasoning, 2D mental rotation, and symbolic number comparison. Overall, the findings highlight the potential significance of attending to and developing young children's spatial thinking as part of early mathematics instruction.