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In the disciplines of dance and mathematics, binary identities commonly split individuals into either (non)dance and/or (non)math people. However, these shorthand assessments become dangerous because they constrain disciplinary knowledge, and limiting one's identity and capabilities remains antithetical to broad education goals. Capitalizing on dance as a situated, embodied form of knowledge, as well as an engaging, interactive, and social endeavor, we facilitate instruction that uses a dance-based approach to teach geometric transformations in a teacher education program. In this paper, we discuss findings from a case study intervention in a graduate-level geometry education course where we engaged participants in this transformations dance activity. We ask: What disciplinary practices did participants use in a dance-based intervention on geometric transformations? And, how did they use them? We found that the parallel disciplinary practices that the participants engaged in, in dance and mathematics, sheds light on a need in education to resist binary identities and to encourage inquiry-based practices. We aim to inspire future professional development and collaborative teaching opportunities for dance teachers in math and math teachers in dance, along with research that investigates intersections of learning, practice, and identity, particularly in dance + math.

This letter from the editor presents Vol. 25(2) of the Nexus Network Journal. This issue has two connected themes, the first is about the role of regular modules, patterns, or proportions in architecture, and the second, the ways rules or systems underpin architectural form. The papers in this issue address either regularity, rules, or in most cases, both, because spatial and formal patterns are typically derived from rules. Collectively, the eleven research contributions in this issue have topics that span chronologically from the thirteenth century to the present day, and geographically from Asia to Africa and Europe. [ABSTRACT FROM AUTHOR]

In what way does a mathematical proof depend on the notation used in its presentation? This paper examines this question by analysing the computational differences, in the sense of Larkin and Simon's ‘Why a diagram is (sometimes) worth 10,000 words’, between diagrammatic and sentential notations as a means for presenting geometric proofs. Wittgenstein takes up the question of mathematical notation and proof in Section III of Remarks on the Foundations of Mathematics. After discussing his observations on a proof's ‘characteristic visual shape’ in Section III with respect to arithmetical proofs, the paper shows how the notion of a characteristic visual shape illuminates the special effectiveness of diagrammatic notation in geometry. [ABSTRACT FROM AUTHOR]

Nexus Network Journal guest editor Roberta Spallone introduces papers selected from the Nexus Conference 2023 in Turin for detailed review, expansion and development. These papers identify close links between architecture, mathematics, and specific areas of interest related to historical periods, architectural cultures, typological elements, and analysis tools. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new family of the generalized k-Leonardo numbers and study their properties. We investigate the Gaussian Leonardo numbers and associated new families of these Gaussian forms. We obtain combinatorial identities like Binet formula, Cassini’s identity, partial sum, etc. in the closed form. Moreover, we give various generating and exponential generating functions. [ABSTRACT FROM AUTHOR]

A signed graph (G, σ) is called a parity signed graph if there exists a bijective mapping f : V (G) → {1, . . ., |V (G)|} such that for each edge uv in G, f(u) and f(v) have same parity if σ(uv) = +1, and opposite parity if σ(uv) = −1. The rna number σ−(G) of G is the least number of negative edges among all possible parity signed graphs over G. Equivalently, σ−(G) is the least size of an edge-cut of G that has nearly equal sides. In this paper, we show that for the generalized Petersen graph Pn,k, σ−(Pn,k) lies between 3 and n. Moreover, we determine the exact value of σ−(Pn,k) for k ∈ {1, 2}. The rna numbers of some famous generalized Petersen graphs, namely, Petersen graph, Dürer graph, Möbius-Kantor graph, Dodecahedron, Desargues graph and Nauru graph are also computed. Recently, Acharya, Kureethara and Zaslavsky characterized the structure of those graphs whose rna number is 1. We use this characterization to show that the smallest order of a (4n + 1)-regular graph having rna number 1 is 8n + 6. We also prove the smallest order of (4n − 1)-regular graphs having rna number 1 is bounded above by 12n − 2. In particular, we show that the smallest order of a cubic graph having rna number 1 is 10. [ABSTRACT FROM AUTHOR]

Poincaré's conventionalism has been interpreted in many writings as a philosophical position emerged by reflection on certain scientific problems, such as the applicability of geometry to physical space or the status of certain scientific principles. In this paper I would like to consider conventionalism as a philosophical position that emerged from Poincaré's scientific practice. But not so much from dealing with scientific problems, as from the use of two specific methodologies proper to modern mathematics and the modern natural sciences: methodological structuralism and the hypothetical-deductive method—thus, as a philosophical position which emerged from a way (or rather, two ways) of doing science. With this approach, I try to deepen the analysis of connections between Poincaré's scientific practice and his philosophy. [ABSTRACT FROM AUTHOR]

Modern color science finds its birth in the middle of the nineteenth century. Among the chief architects of the new color theory, the name of the polymath Hermann von Helmholtz stands out. A keen experimenter and profound expert of the latest developments of the fields of physiological optics, psychophysics, and geometry, he exploited his transdisciplinary knowledge to define the first non-Euclidean line element in color space, i.e., a three-dimensional mathematical model used to describe color differences in terms of color distances. Considered as the first step toward a metrically significant model of color space, his work inaugurated researches on higher color metrics, which describes how distance in the color space translates into perceptual difference. This paper focuses on the development of Helmholtz's mathematical derivation of the line element. Starting from the first experimental evidence which opened the door to his reflections about the geometry of color space, it will be highlighted the pivotal role played by the studies conducted by his assistants in Berlin, which provided precious material for the elaboration of the final model proposed by Helmholtz in three papers published between 1891 and 1892. Although fallen into oblivion for about three decades, Helmholtz's masterful work was rediscovered by Schrödinger and, since the 1920s, it has provided the basis for all subsequent studies on the geometry of color spaces up to the present time. [ABSTRACT FROM AUTHOR]

The architectonic is key for situating Kant's understanding of science in the coming century. For Kant the faculty of reason turns to ideas to form a complete system. The coherence of the system rests on these ideas. In contrast to technical unity which can be abstracted a posteriori, architectonic ideas are the source of a priori unity for the system of reason because they connect our reasonable pursuit to essential human ends. Given Kant's focus on mathematics, in the architectonic and his critical philosophy more generally, we must have some sense of the architectonic idea of mathematics. In this paper, I argue for the key principles of the architectonic idea of mathematics: 1) because mathematics is grounded in a priori intuition, it is a peculiarly human activity; 2) the method of mathematics is one of a priori construction, a method only mathematics can employ and: 3) the objects of mathematics are extensive magnitudes. Given these principles, we can use the architectonic idea to have some clarity about how mathematics has dealt with historical development. [ABSTRACT FROM AUTHOR]

Foundry Degassing Units (FDU) are used for refining aluminum alloys. For an ideal refining process using an FDU unit, it is necessary to select several parameters, which are linked to each other. For a rotary impeller, we searched for several parameters, such as its optimal shape, speed in the liquid alloy, and distance from the bottom of the refining ladle, where the aforementioned parameters contribute to the overall wear and life of the rotor and, consequently, of the rotor shaft. The Computational Fluid Dynamics (CFD) method can be used to determine the above-mentioned parameters. This paper describes the particular steps of preparation of rotor geometry for the subsequent setting up of the basic numerical model. [ABSTRACT FROM AUTHOR]

In this paper, a joint study of the behavior of solutions of the Heavy Ball ODE and Heavy Ball type algorithms is given. Since the pioneering work of Polyak (USSR Comput Math Math Phys 4(5):1–17, 1964), it is well known that such a scheme is very efficient for C 2 strongly convex functions with Lipschitz gradient. But much less is known when only growth conditions are considered. Depending on the geometry of the function to minimize, convergence rates for convex functions, with some additional regularity such as quasi-strong convexity, or strong convexity, were recently obtained in Aujol et al. (Convergence rates of the Heavy-Ball method for quasi-strongly convex optimization, 2020). Convergence results with much weaker assumptions are given in the present paper: namely, linear convergence rates when assuming a growth condition (which amounts to a Łojasiewicz property in the convex case). This analysis is firstly performed in continuous time for the ODE, and then transposed for discrete optimization schemes. In particular, a variant of the Heavy Ball algorithm is proposed, which converges geometrically whatever the parameters choice, and which has the best state of the art convergence rate for first order methods to minimize composite non smooth convex functions satisfying a Łojasiewicz property. [ABSTRACT FROM AUTHOR]

Acknowledgement- "This paper is part of a research project work titled- 'The Ancient Indian Mathematics Education in the Context of Sulba-Sutra: A Qualitative Study' funded by the Indian Council of Historical Research (ICHR), Ministry of Education, Government of India. We gratefully acknowledge ICHR, New Delhi for the financial support, which enabled us to deeply study the ancient Indian text the Sulba-Sutra." Abstract: This study explores the pedagogical insights inherent in the Sulba Sutras, ancient Indian mathematical texts detailing the construction of Vedic ritualistic altars. Findings reveal that the Sutras employ the equation method written in Sanskrit, emphasizing the importance of ratios, proportions, and measurements following a constructivist learning approach. Real-world objects and tools were utilized, fostering hands-on engagement and problem-solving skills. The study concludes that the Sulba Sutras offer a holistic and joyful learning experience, showcasing the immediate relevance of these pedagogical strategies of mathematics in deep geometrical understanding among students. [ABSTRACT FROM AUTHOR]

Consider a point particle moving through a Poisson distributed array of cubes all oriented along the axes—the random wind-tree model introduced in Ehrenfest–Ehrenfest (1912) as reported by Ehrenfest, Ehrenfest (Begriffliche Grundlagen der statistischen Auffassung in der Mechanik Encykl. d. Math. Wissensch. IV 2 II, Heft 6, 90 S (1912) (Translated:) The conceptual foundations of the statistical approach in mechanics. Dover Books on Physics, 1912). We show that in the joint Boltzmann–Grad and diffusive limit this process satisfies an invariance principle. That is, the process converges in distribution to Brownian motion in a particular scaling limit. In a previous paper (2020) (Lutsko, Tóth in Commun. Math. Phys. 379:589–632, 2020) the authors used a novel coupling method to prove the same statement for the random Lorentz gas with spherical scatterers. In this paper we show that, despite the change in dynamics, a similar strategy with some modification can be used to prove the invariance principle for the random wind-tree model. The key differences from our previous work are that the individual path segments of the underlying Markov process are no longer fully independent and the geometry of recollisions is simpler. [ABSTRACT FROM AUTHOR]

This survey introduces 101 new publications on applications of Clifford's geometric algebras (GAs) newly published during 2022 (until mid‐January 2023). The selection of papers is based on a comprehensive search with Dimensions.ai, followed by detailed screening and clustering. Readers will learn about the use of GA for mathematics, computation, surface representations, geometry, image, and signal processing, computing and software, quantum computing, data processing, neural networks, medical science, physics, electric engineering, control and robotics. [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a simple connected graph. A perfect matching (or Kekulé structure in chemical literature) of G is a set of disjoint edges which covers all vertices of G. The anti-forcing number of G is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching and is denoted by af(G). For every m ∈ N, the mth power of G, denoted by Gm, is a graph with the same vertex set as G such that two vertices are adjacent in Gm if and only if their distance is at most m in G. In this paper, we study the anti-forcing number of the powers of some graphs. [ABSTRACT FROM AUTHOR]

A Roman dominating function (RDF) on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of an RDF f is the sum of the weights of the vertices under f. The Roman domination number, γR(G) of G is the minimum weight of an RDF in G. The Roman domination polynomial of a graph G of order n is the polynomial RD(G, x) = Σ2ni=γR(G) dR(G, i)xi, where dR(G, i) is the number of RDFs of G with weight i. In this paper we prove properties of Roman domination polynomials and determine RD(G, x) in several classes of graphs G by new approaches. We also present bounds on the number of all Roman domination polynomials in a graph. [ABSTRACT FROM AUTHOR]

This paper introduces certain relations between 1-edge contraction and the total vertex stress and the confluence number of a graph. A main result states that if a graph G with ζ(G) = k ≥ 2 has an edge vivj and a ζ-set CG such that vi, vj ∈ CG then, ζ(G/vivj ) = k − 1. In general, either S(G/ei) ≤ S(G/ej ) or S(G/ej ) ≤ S(G/ei) is true. This observation leads to an investigation into the question: for which edge(s) ei will S(G/ei) = max{S(G/ej ) : ej ∈ E(G)} and for which edge(s) will S(G/ej ) = min{S(G/el) : el∈ E(G)}? [ABSTRACT FROM AUTHOR]

For a graph G, we define a total k-labeling ϕ as a combination of an edge labeling ϕe : E(G) → {1, 2, . . ., ke} and a vertex labeling ϕv : V (G) → {0, 2, . . ., 2kv}, where k = max {ke, 2kv}. The total k-labeling ϕ is called a vertex irregular reflexive k-labeling of G if any pair of vertices u, u' have distinct vertex weights wtϕ(u) ≠ wtϕ(u'), where wtϕ(u) = ϕ(u) + Σuu'∈E(G) ϕ(uu' ) for any vertex u ∈ V (G). The smallest value of k for which such a labeling exists is called the reflexive vertex strength of G, denoted by rvs(G). In this paper, we present a new lower bound for the reflexive vertex strength of any graph. We investigate the exact values of the reflexive vertex strength of generalized friendship graphs, corona product of two paths, and corona product of a cycle with isolated vertices by referring to the lower bound. This study discovers some interesting open problems that are worth further exploration. [ABSTRACT FROM AUTHOR]

The PI index of a graph G is given by PI(G) = Σe∈E(G) (|V (G)|−NG(e)), where NG(e) is the number of equidistant vertices for the edge e. Various topological indices of bicyclic graphs have already been calculated. In this paper, we obtained the exact value of the PI index of bicyclic graphs. We also explore the extremal graphs among all bicyclic graphs with respect to the PI index. Furthermore, we calculate the PI index of a cactus graph and determine the extremal values of the PI index among cactus graphs. [ABSTRACT FROM AUTHOR]

The coalition in a graph G consists of two disjoint sets of vertices V1 and V2, neither of which is a dominating set but whose union V1 ∪ V2, is a dominating set. A coalition partition in a graph G is a vertex partition π = {V1, V2, . . ., Vk} such that every set Vi ∈ π is not a dominating set but forms a coalition with another set Vj ∈ π which is not a dominating set. The coalition number C(G) equals the maximum k of a coalition partition of G. In this paper, we compute the coalition numbers of all cubic graphs of order at most 10. [ABSTRACT FROM AUTHOR]

A graphoidal cover of a graph G (not necessarily finite) is a collection ψ of paths (not necessarily finite, not necessarily open) satisfying the following axioms: (GC-1) Every vertex of G is an internal vertex of at most one path in ψ, and (GC-2) every edge of G is in exactly one path in ψ. The pair (G, ψ) is called a graphoidally covered graph and the paths in ψ are called the ψ-edges of G. In a graphoidally covered graph (G, ψ), two distinct vertices u and v are ψ-adjacent if they are the ends of an open ψ-edge. A graphoidally covered graph (G, ψ) in which no two distinct vertices are ψ-adjacent is called ψ-independent and the graphoidal cover ψ is called a totally disconnecting graphoidal cover of G. Further, a graph possessing a totally disconnecting graphoidal cover is called a graphoidally independent graph. The aim of this paper is to establish complete characterization of graphoidally independent infinite cactus. [ABSTRACT FROM AUTHOR]

Making connections within and between different aspects of mathematics is recognised as fundamental to learning mathematics with understanding. However, exactly what these connections are and how they serve the goal of learning mathematics is rarely made explicit in curriculum documents with the result that mathematics tends to be presented as a set of discrete, disconnected topics. Interest in establishing a more coherent approach to the teaching and learning of school mathematics has led to a focus on big ideas. That is, networks of related concepts, skills and ways of thinking that facilitate learning mathematics with understanding. Research on learning progressions has helped identify what these big ideas are and how they serve to build connections within and between different aspects of mathematics. This paper draws on research that provides an evidenced-based learning progression for multiplicative reasoning to illustrate the connective role of multiplicative thinking in the development of algebraic, geometrical, and statistical reasoning. [ABSTRACT FROM AUTHOR]

In this paper, we present a relaxed CQ algorithm involving the Armijo-line search and the alternated inertial technique to solve the multiple-sets split feasibility problem in infinite dimensional real Hilbert spaces. We prove the convergence of the sequence generated by our method under some mild assumptions. Finally, numerical simulations show the efficiency of our algorithm. [ABSTRACT FROM AUTHOR]

Discussions about teaching area measurement in primary school have been ongoing over some decades. However, investigations that thoroughly examine the current research on conceptual understanding in area measuring in elementary schools are still lacking. The objective of this paper is to review whether conceptual knowledge in area measurement may support students to obtain better results in primary schools. This study is to gain insight into how conceptual knowledge in area measurement has been portrayed for primary school students, and reveal possible omissions and gaps in the synthesized literature on the subject. To gather information, two databases were used: Scopus and Web of Science. Primary searches pulled up many studies on the subject of investigation. After analyzing abstracts and eliminating duplicates, our systematic review indicates that there seems a direct link between conceptual understanding and area measurement in primary school mathematics. Hence, teaching children the principle of area measurement rather than a procedure for solving problems seems to be the most effective way of improving problem-solving skills and conceptual understanding for primary students. [ABSTRACT FROM AUTHOR]

In this study, we redesigned paper-and-pencil activities and adapted them to an introductory multivariate calculus course supported by 3D dynamic geometry software. We used semi-structured interviews and students’ written productions during the semester to analyse the use of technology in supporting students’ learning. Considering the work of two students, we find that the software has transformative potential but that it did not promote students’ learning as expected. We discuss possible reasons for this. [ABSTRACT FROM AUTHOR]

Between 1873 and 1876, Felix Klein published a series of papers that he later placed under the rubric anschauliche Geometrie in the second volume of his collected works (1922). The present study attempts not only to follow the course of this work, but also to place it in a larger historical context. Methodologically, Klein's approach had roots in Poncelet's principle of continuity, though the more immediate influences on him came from his teachers, Plücker and Clebsch. In the 1860s, Clebsch reworked some of the central ideas in Riemann's theory of Abelian functions to obtain complicated results for systems of algebraic curves, most published earlier by Hesse and Steiner. These findings played a major role in enumerative geometry, whereas Plücker's work had a strongly qualitative character that imbued Klein's early studies. A leitmotif in these works can be seen in the interplay between real curves and surfaces as reflected by their transformational properties. During the early 1870s, Klein and Zeuthen began to explore the possibility of deriving all possible forms for real cubic surfaces as well as quartic curves. They did so using continuity methods reminiscent of Poncelet's earlier approach. Both authors also relied on visual arguments, which Klein would later advance under the banner of intuitive geometry (anschauliche Geometrie). [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a new subclass of normal- ized analytic and univalent functions in the open unit disk associated with Sigmoid function. Coeficient estimates, convolution conditions, convex- ity and some other geometric properties for functions in this class are investigated. Also, subordination and inclusion results are obtained. [ABSTRACT FROM AUTHOR]

In view of a better understanding of the geometry of scalar flat Kähler metrics, this paper studies two families of scalar flat Kähler metrics constructed by Hwang and Singer (Trans Am Math Soc 354(6):2285–2325, 2002) on C n + 1 and on O (- k) . For the metrics in both the families, we prove the existence of an asymptotic expansion for their ϵ -functions and we show that they can be approximated by a sequence of projectively induced Kähler metrics. Further, we show that the metrics on C n + 1 are not projectively induced, and that the Burns–Simanca metric is characterized among the scalar flat metrics on O (- k) to be the only projectively induced one as well as the only one whose second coefficient in the asymptotic expansion of the ϵ -function vanishes. [ABSTRACT FROM AUTHOR]

This is a personal account of an intersection of the schools of research in foundations of geometry founded by Alfred Tarski and Friedrich Bachmann. Their academic lineages and the origins of the schools are also described, as well as the mathematics that resulted from this intersection. [ABSTRACT FROM AUTHOR]

This paper investigated the effect of using LEGO EV3 robots to teach Newton's second law with conceptual understanding to a group of 14 to 18-year-olds in an after-school STEM education program. 74 teenagers participated in this research study. A quantitative methods approach involving descriptive analyses, paired-sample t-test, and repeated measures ANOVA were used to answer the research questions. The results showed that the LEGO EV3 robots positively affected participants' understanding of Newton's second law of motion and their interest in pursuing STEM education and careers. In addition, the descriptive analyses from the pre- and post-interest questionnaire revealed that participants were more confident and willing to learn with robotic devices after the activity than before. Repeated measures ANOVA analysis indicated that scaffolded programming tasks affected participants' computational thinking skills. Implications include the purposeful use of LEGO EV3 robotics and potentially other educationally focused programmable devices (e.g., micro:bit, Spheros, Arduinos, etc.). [ABSTRACT FROM AUTHOR]

The purpose of this systematic literature review (SLR) is to identify: (a) the topic of the study, (b) the research methods used, and (c) the results of research on Mathematics education in Malaysia. This study discusses the use of teaching aid (TA) in the field of syllabus and geometry for Form 2 students. The use of TA is considered highly successful and relevant for educators to improve the quality of the teacher's instructions and students' understanding. Therefore, using the rules of optional reporting items for systematic review and meta-analysis (PRISMA) by Moher et al. (2015), a review system was carried out to determine the appropriate strategies and variables for the field. Four stages constitute the PRISMA paradigm used in this study: identification, screening, qualification, and admission. Using criteria opted by researchers from multiple searches, including Google Scholar, Researchgate, Scopus, and Emerald, over 20 papers were identified for additional investigation. The data were then analysed quantitatively to describe the research's findings. From the results, two main research themes were found, namely (a) learning to use TA; and (b) the field of measurement and geometry of Mathematics. The results of the article analysis indicate that Mathematics education in Malaysia is currently at a moderate level and is ineffective at fostering students' understanding and interest. These results are anticipated to serve as the foundation for teachers, students, schools, and the Ministry of Education to undertake more engaging and interactive learning, particularly in the subject areas of mathematics and geometry. [ABSTRACT FROM AUTHOR]

This paper briefly describes a heuristic investigation using dynamic geometry of some concurrency, collinearity and other properties of a particular hexagon that would be of interest to mathematics olympiad enthusiasts. After managing to prove the initial result, further reflection on the proof led to an immediate generalisation, illustrating the so-called ‘discovery’ function of proof. [ABSTRACT FROM AUTHOR]

This paper describes a model of mathematics bianshi teaching using one problem with several solutions, changes, and applications in an inquiry-based classroom. Based on the model of mathematics bianshi teaching, the author discusses a class-room experience where one Chinese mathematics teacher takes a simple plane geometry problem to guide students to generate different solutions and pose several bianshi problems. Students then apply the conclusions obtained to solve new problems, and develop their own creativity. [ABSTRACT FROM AUTHOR]

A key question of the 'maverick' tradition of the philosophy of mathematical practice is addressed, namely what is mathematical progress. The investigation is based on an article by Penelope Maddy devoted to this topic in which she considers only contributions 'of some mathematical importance' as progress. With the help of a case study from contemporary mathematics, more precisely from tropical geometry, a few issues with her proposal are identified. Taking these issues into consideration, an alternative account of 'mathematical importance', broadly within the framework of progress Maddy offers, is developed with a special focus on mathematicians' peer-review practice. [ABSTRACT FROM AUTHOR]

Faithfully flat descent of pseudo-coherent complexes in rigid geometry was proved by Mathew. In this paper, we generalize the result of Mathew to solid quasi-coherent complexes on rigid analytic varieties, which have been introduced by Clausen and Scholze by means of condensed mathematics. [ABSTRACT FROM AUTHOR]

This study was an investigation of the use of manipulative materials in the mathematics classroom. The researcher guided students to create paper triangles, which were then used as a manipulative to calculate the sum of the interior angles of a triangle. On pre- and post-measures of quantitative data, although the data indicated a positive correlation on attitudes toward math, there was no statistical difference on an 11-item assessment composed of public-released questions taken from the National Assessment of Educational Progress (NAEP) exam. Mixed methods data collection and analysis found that students enjoyed the lesson and wanted more opportunities to engage with manipulative materials. Implications include the need for mathematics teachers to deploy novel approaches as instructional strategies. [ABSTRACT FROM AUTHOR]

In recent decades, with the continuous changes in the mathematics curriculum, many geometry-related topics were neglected or removed from our classroom activities. Thus, some teachers pursue these topics as independent students' projects. There is also a growing interest in various courses and workshops dealing with practical geometric pattern design. The author has taught courses and conducted workshops on geometry and pattern design at the Istanbul Design Center for many years. Participants in these events were high school and university students, artists, and architects. In this paper, we will discuss selected concepts of geometry used by Seljuk architects in designing the famous Seljuk geometric mosaics and stone carvings. We will show how these geometric concepts were used in selected examples from mosques, tombs, hans, and madrasas. We will also demonstrate using geometry software, e.g., Geometer's Sketchpad or GeoGebra, for constructing geometric patterns. [ABSTRACT FROM AUTHOR]

In this paper, we describe an educational experience in the context of the Master's degree that is compulsory in Spain to become a secondary education mathematics teacher. Master's students from two universities in Madrid (Spain) attended lectures that addressed—emphasizing the concourse of a dynamic geometry software package—some historical, didactic and mathematical issues related to linkage mechanisms, such as those arising in the 18th and 19th centuries during the development of the steam engine. Afterwards, participants were asked to provide three different kinds of feedback: (i) working on an assigned group task, (ii) individually answering a questionnaire, and (iii) proposing some classroom activity, imagining it would be addressed to their prospective pupils. All three issues focused on the specific topic of the attended lectures. In the framework of Mason's reflective discourse analysis, the information supplied by the participants has been analyzed. The objective was to explore what they have learned from the experience and what their perception is of the potential interest in linkages as a methodological instrument for their future professional activity as teachers. This analysis is then the basis upon which to reflect on the opportunities (and problems) that this particular bar-joint linkages methodological approach could bring towards providing future mathematics teachers with attractive tools that would contribute to enhancing a STEAM-oriented education. Finally, the students' answers allow us to conclude that the experience was beneficial for these pre-service teachers, both in improving their knowledge on linkages history, mathematics, industrial, technological and artistic applications, and in enhancing the use in the classroom of this very suitable STEAM context. [ABSTRACT FROM AUTHOR]

We consider some families of one relator groups arising as fundamental groups of 3-dimensional manifolds, and calculate their character varieties in SL(2, C). Then we give simple geometrical descriptions of such varieties, and determine the number of their irreducible components. Our paper relates to the work of Baker-Petersen, Qazaqzeh and Morales-Marc'en on the character variety of certain classes of one relator groups, but we use different methods based on the concept of palindrome presentations of given groups. [ABSTRACT FROM AUTHOR]

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The teaching of Mathematics – in particular, Geometry, through conventional methods has been a challenging task for tutors. Augmented Reality (AR) based applications available in commercial space, have not followed any structured pedagogical approach in the designing process, and also do not ensure that the learning time of students is spent prolifically. In this paper, we explore the use of AR in mathematics for geometry education, to aid visualization of multidimensional objects and long-term retention of concepts by the learners. For designing an appropriate AR application it is necessary to identify some principles which support better memory retention of the students. The application has been specifically designed on the basis of identified principles affecting memory retention. We further explain the development of an AR-based Geometry Learning Assistant (AR-GLA), using a structural approach to pedagogical-design for teaching 3-dimensional geometry to higher school students through improved visualization and enhance their memory retention for related concepts. A sample of 54 K-12 students and 2 teachers with expertise in mathematics were part of the experiment. The students were divided into two different groups; one of the groups was taught with AR-based content whereas the other group was given Interactive Simulation (IS) based learning. The results illustrated that AR-based learning provides better retention of memory as compared to IS-based learning were tested over a period of two months. [ABSTRACT FROM AUTHOR]

There has been increasing interest in best–worst discrete choice experiments (BWDCEs) in health economics, transportation research, and other fields over the last few years. BWDCEs have distinct advantages compared to other measurement approaches in discrete choice experiments (DCEs). A systematic study of best–worst (BW) choice pairs can be traced back to the 1990s. Recently, new ideas have been introduced to the subject. Calculating utility helps measure the attractiveness of BW choices. The goal of this paper is twofold. First, we extend the idea of the BW choice pair to include dynamic, time-dependent transition probability and capture utility at each time and for each choice pair. Second, we used the geometry of BW choice pairs to capture the correlations among them and to characterize and clarify the BW choice pairs in the network, where properties can be derived within each class. This paper discusses BWDCEs, the probability transition matrix of choices over time, and the utility function. The proposed network classification for BW choice pairs is laid out. A detailed simulated example is presented, and the results are compared with the classical K-means classification.

With the rapid change in the society, education must evolve to adapt to the digital native students of the 21st century. A change in the current educational paradigm is thus necessary. Information and Communication Technology (ICT) is a vehicle to achieve this objective of paradigm shift in education [2]. In this paper, we propose gamification approach [3] via a virtual reality (VR) through a report of a didactic experience. A mixed method research design was adopted for this study in which students' classification of polyhedral will be reported for the two different learning experiences using either wooden polyhedral or a VR tool within an immersive learning environment. The VR environment allows the learning of geometry in three-dimensional with a friendly interface that offers a multitude of options. It is worth highlighting the benefits of this VR tool in the teaching-learning process of geometry, making explicit the fundamental role played by ICT in meaningful learning of mathematics. [ABSTRACT FROM AUTHOR]

This paper addresses an article by Felix Klein of 1886, in which he generalized his theory of polynomial equations of degree 5—comprehensively discussed in his Lectures on the Icosahedron two years earlier—to equations of degree 6 and 7. To do so, Klein used results previously established in line geometry. I review Klein's 1886 article, its diverse mathematical background, and its place within the broader history of mathematics. I argue that the program advanced by this article, although historically overlooked due to its eventual failure, offers a valuable insight into a time of crucial evolution of the subject. [ABSTRACT FROM AUTHOR]

In this paper a class of regular surfaces, generated by rotation and twist of a rigid line segment, is considered. Firstly a complete description of those surfaces by means of local charts is obtained and then its orientability is characterized in terms of the number of half-twists, taking into account their direction (clockwise or counterclockwise) since changes in the sense of twist are allowed. Indeed a procedure to construct nonorientable surfaces is provided. Illustrative examples and computer-generated plots using the free software Maxima are included. [ABSTRACT FROM AUTHOR]

A progress report on the (still unresolved) Landsberg-Berwald problem of Finsler geometry: whether there can be non-Berwaldian regular Landsberg spaces. [ABSTRACT FROM AUTHOR]

This paper classifies the Dirac series for complex E 7 . As applications, we verify a few conjectures raised in 2011, 2019 and 2020 for this exceptional Lie group. In particular, according to Conjecture 1.1 of Barbasch and Pandžić [Dirac cohomology and unipotent representations of complex groups, Noncommutative Geometry and Global Analysis, Contemp. Math. 546, American Mathematical Society, Providence (2011), 1–22], our classification should be helpful for understanding the unitary dual of complex E 7 . [ABSTRACT FROM AUTHOR]