How to Lay Bricks: Local-Global Alignment Predicts Preference for Shifted Tile Patterns

We examine the aesthetic characteristics of row tile patterns defined by repeating strips of polygons. In experiment 1 participants rated the perceived beauty of equilateral triangle, square and rectangular tilings presented at vertical and horizontal orientations. The tiles were shifted by one-fourth increments of a complete row cycle. Shifts that preserved global symmetry were liked the most. Local symmetry by itself did not predict ratings but tilings with a greater number of emergent features did. In a second experiment we presented row tiles using all types of three- and four-sided geometric figures: acute, obtuse, isosceles and right triangles, kites, parallelograms, a rhombus, trapezoid, and trapezium. Once again, local polygon symmetry did not predict responding but measures of correspondence between local and global levels did. In particular, number of aligned polygon symmetry axes and number of aligned polygon sides were significantly and positively correlated with beauty ratings. Preference was greater for more integrated tilings, possibly because they encourage the formation of gestalts and exploration within and across levels of spatial scale.

Angular momentum and parity projected multidimensionally constrained relativistic Hartree-Bogoliubov model

The nuclear deformations are of fundamental importance in nuclear physics. Recently we developed a multi-dimensionally constrained relativistic Hartree-Bogoliubov (MDCRHB) model, in which all multipole deformations respecting the $V_4$ symmetry can be considered self-consistently. In this work we extend this model by incorporating the angular momentum projection (AMP) and parity projection (PP) to restore the rotational and parity symmetries broken in the mean-field level. This projected-MDCRHB (p-MDCRHB) model enables us to connect certain nuclear spectra to exotic intrinsic shapes such as triangle or tetrahedron. We present the details of the method and an exemplary calculation for $^{12}$C. We develop a triangular moment constraint to generate the triangular configurations consisting of three $\alpha$ clusters arranged as an equilateral triangle. The resulting $^{12}$C spectra are consistent with that from a triangular rigid rotor for large separations between the $\alpha$ clusters. We also calculate the $B(E2)$ and $B(E3)$ values for low-lying states and find good agreement with the experiments.

Examination of 4th Grade Students' Definitions for Square, Rectangle and Triangle Geometric Shapes

The aim of this study is to obtain information about students' definitions, mistakes, misconceptions, and van Hiele geometry thinking levels by using the definitions of 4th grade students for geometric shapes of rectangle, square, isosceles triangle, equilateral triangle, and scalene triangle. The study was carried out with 156 primary school 4th grade students. In the study, the case design, one of the qualitative methods, was used. Students were asked to describe geometric shapes. It was observed that most of the 4th grade students participating in the study were in the visualization stage of van Hiele. There are very few definitions of the hierarchical structure in the study. Most correct definitions are in the partitional form. In the study, misconceptions were detected in some of the students.

Structure of 3He

Using electron scattering data, the diffraction pattern off $$^{3}$$ 3 He shows it to be an equilateral triangle possessing dihedral D$$_{3}$$ 3 point group symmetry (PGS). Previous work showed that $$^{4}$$ 4 He is a 3-base pyramid with C$$_{3v}$$ 3 v PGS. $$^{6}$$ 6 Li is predicted to have C$$_{2v}$$ 2 v PGS. As nuclear $$A \rightarrow $$ A → large, atomic nuclei enter into the ‘protein folding problem’ with many possible groundstate PGS competing for lowest energy.

On Packing Thirteen Points in an Equilateral Triangle

The conversation of how to maximize the minimum distance between points - or, equivalently, pack congruent circles- in an equilateral triangle began by Oler in the 1960s. In a 1993 paper, Melissen proved the optimal placements of 4 through 12 points in an equilateral triangle using only partitions and direct applications of Dirichlet’s pigeon-hole principle. In the same paper, he proposed his conjectured optimal arrangements for 13, 14, 17, and 19 points in an equilateral triangle. In 1997, Payan proved Melissen’s conjecture for the arrangement of fourteen points; and, in September 2020, Joos proved Melissen’s conjecture for the optimal arrangement of thirteen points. These proofs completed the optimal arrangements of up to and including fifteen points in an equilateral triangle. Unlike Melissen’s proofs, however, Joos’s proof for the optimal arrangement of thirteen points in an equilateral triangle requires continuous functions and calculus. I propose that it is possible to continue Melissen’s line of reasoning, and complete an entirely discrete proof of Joos’s Theorem for the optimal arrangement of thirteen points in an equilateral triangle. In this paper, we make progress towards such a proof. We prove discretely that if either of two points is fixed, Joos’s Theorem optimally places the remaining twelve. KEYWORDS: optimization; packing; equilateral triangle; distance; circles; points; thirteen; maximize

Examination of 4th Grade Students' Definitions for Square, Rectangle and Triangle Geometric Shapes

The aim of this study is to obtain information about students' definitions, mistakes, misconceptions, and van Hiele geometry thinking levels by using the definitions of 4th grade students for geometric shapes of rectangle, square, isosceles triangle, equilateral triangle, and scalene triangle. The study was carried out with 156 primary school 4th grade students. In the study, the case design, one of the qualitative methods, was used. Students were asked to describe geometric shapes. It was observed that most of the 4th grade students participating in the study were in the visualization stage of van Hiele. There are very few definitions of the hierarchical structure in the study. Most correct definitions are in the partitional form. In the study, misconceptions were detected in some of the students.

FRACTURAS “FÓSILES” EN ESQUELETOS DE EQUINOIDEOS DEL CRETÁCICO SUPERIOR - “FOSSIL” FRACTURES IN UPPER-CRETACEOUS EQUINOIDS TESTS

Many calcitic skeletons of the fossil echinoid Micraster coranguinum show small triangular holes on their surface originated by the loss of a pyramidal fragment. The geometry of the holes, at first sight surprising, is simply explained by the intersection of fractures of the single crystalline skeleton plates along the three easy cleavage planes of calcite. The base of the pyramidal fragments is often close to an equilateral triangle; the morphology of such fractures is explained by the preferred orientation of the Micraster plates, whose surface is parallel to the basal calcite plane (0001). The fracture process leading to such fractures is most probably of thermo-elastic origin.