The perpendicular bisectors of the sides of a hexagon, whose opposite sides are parallel, produce a point symmetric hexagon. Michael de Villiers already gave two proofs of this theorem, firstly an elaborate one with the aid of dynamic geometry and secondly a merely verifying one with the help of coordinates and computer algebra. In this note, we…

This thesis focuses on the challenge of characterizing multipartite entanglement. While the study of bipartite entanglement is well-documented in scientific literature, recognizing that entanglement can involve more than two parties--i.e. three or more parties---is crucial, as multipartite entanglement enables the completion of more complicated…

Polyiamonds are the 'and so on' when unit-squares are replaced with unit-equilateral triangles. Polyiamonds are used to play the satisfying Blokus variant, Blokus Trigon, where the playing grid is also hexagonal, and the playing pieces are all the polyiamonds from mono- to hex. This article discusses the use of polyiamonds in puzzles.

In this work, we delve into geometric analysis, particularly examining the interplay between lower bounds on Ricci curvature and specific functionals. Our exploration begins with an investigation into the implications of Yamabe invariants for asymptotically Poincare-Einstein manifolds and their conformal boundaries under conditions of…

This dissertation explores bridge n-sections of knotted surfaces in 4-manifolds by defining invariants that measure the complexity of their topology and by giving geometric constructions that determine Lagrangian surfaces in 4-manifolds under certain conditions. First, we present an elegant geometric construction that generates all triple grid…

Certain terms in mathematics were created according to conventions that are not obvious to students who will use the term. When this is the case, investigating the choice of a name can reveal interesting and unforeseen connections among mathematical topics. In this study, we tasked prospective and practicing teachers to consider: What is geometric…

Prior research suggests that creating art can reduce mental health symptoms, such as anxiety. It has also been proposed that creating art within the geometric shape of a circle may confer additional therapeutic benefits. However, it remains unclear whether creating art within a circle or creating art with no restrictions reduces anxiety in both…

This study provides a framework for explaining the way individuals might construct the final result of infinite iterative processes in a geometrical context. To this end, 30 undergraduate students were interviewed. The analysis of the data collected yielded two different views: first, there were those who believed that the presented infinite…

An exciting recent development in chemistry has been the ability to control nanoparticle crystal morphology or shape. Nanoparticles of different shapes present different crystal surfaces (or facets) with respect to the surrounding environment. Synthetic control over nanoparticle morphology has enabled the study of the influence of surface facets…

The Van Hiele model of geometric reasoning establishes five levels of development, from level 1 (visual) to level 5 (rigor). Despite the fact that this model has been deeply studied, there are few research works concerning the fifth level. However, there are some works that point out the interest of working with activities at this level to promote…

The area under the curve is a fundamental concept for students to build their understanding of the Definite Integral. This research reveals how students comprehend the area under the curve in given contextual problems and how the Hypothetical Learning Trajectory (HLT) can help students find the concept. This research follows the development…

A very common Applied Optimization Problem in Calculus deals with minimizing a distance given certain constraints, using Calculus, the general method for solving these problems is to find a function formula for the distance that we need to minimize, take the derivative of the distance function, set it equal to zero, and solve for the input value,…

There has been little written about the ways in which teachers understand draggable geometric objects in a dynamic geometry environment with respect to variance and invariance--two important mathematical ideas related to the development of spatial perception and geometric reasoning. In this article, we investigated such understanding from two…

We use Action-Process-Object-Schema (APOS) theory to study students' geometric understanding of partial derivatives of functions of two variables. This study contributes to research on the teaching and learning of differential multivariable calculus and its didactics. This is an important area due to its multiple applications in science,…

The reconstruction of two-dimensional (2D) slices to three-dimensional (3D) digital anatomical models requires technical skills and software that are becoming increasingly important to the modern anatomist, but these skills are rarely taught in undergraduate science classrooms. Furthermore, learning opportunities that allow students to…