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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]

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]

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]

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]

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]

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 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]

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]

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 note we introduce a new technique to answer an issue posed in Fávaro et al. (Bull Braz Math Soc 51:27–46, 2020) concerning geometric properties of the set of non-surjective linear operators. We also extend and improve a related result from the same paper. [ABSTRACT FROM AUTHOR]

It is well established that spatial reasoning skills (i) support mathematics achievement, (ii) are malleable, and (iii) can be improved through training. More recently, there has been interest in using spatial training to causally support corresponding improvements in mathematics achievement; however, findings so far appear to be mixed. The current study explores the effect of a spatial reasoning intervention on Year 11 students' spatial reasoning skills and mathematics achievement and considers the role of a pedagogical framework and the multidimensional nature of mathematics and spatial reasoning in the design of the intervention. The Experience-Language-Pictorial-Symbolic-Application (ELPSA) pedagogical framework was used to modify an existing spatial intervention program for delivery by high-school educators to Year 11 students (an important but understudied population). The spatial intervention involved training a range of spatial skills over an extended timeframe. Students were randomly assigned to the intervention condition or to a business-as-usual control (n = 73). Using a pre-/post-test design, we found the intervention was successful in improving participants' spatial reasoning skills and performance on measurement and geometry items compared to the control condition but not on number and algebra items. These findings demonstrate that spatial training can support mathematics achievement in certain contexts, highlighting the importance of identifying how individual spatial skills support specific mathematics tasks. Consideration was given for how to use strong pedagogical techniques to scaffold transfer, finding utility in the ELPSA framework. Implications for how to embed spatial training within real mathematics classrooms, as done in the current study, are discussed. [ABSTRACT FROM AUTHOR]

Brief information on sessions of the workshop in 2017 is presented. [ABSTRACT FROM AUTHOR]

This research reflects on using mathematics for processing and interpreting data obtained from digital surveying. The experiments outlined in this study demonstrate how mathematical algorithms, which can be written in computational environments like Wolfram Mathematica, can reveal the geometric structure of an architectural artifact. [ABSTRACT FROM AUTHOR]

We develop a combinatorial rule to compute the real geometry of type B Schubert curves S (λ ∙) in the orthogonal Grassmannian O G (n , C 2 n + 1) , which are one-dimensional Schubert problems defined with respect to orthogonal flags osculating the rational normal curve. Our results are natural analogs of results previously known only in type A [J. Algebraic Combin. 45(1), 191–243 (2017)]. First, using the type B Wroński map studied in [Adv. Math. 224(3), 827–862 (2010)], we show that the real locus of the Schubert curve has a natural covering map to R P 1 , with monodromy operator ω defined as the commutator of jeu de taquin rectification and promotion on skew shifted semistandard tableaux. We then introduce two different algorithms to compute ω without rectifying the skew tableau. The first uses the crystal operators introduced in [Algebr. Comb. 3(3), 693–725 (2020)], while the second uses local switches much like jeu de taquin. The switching algorithm further computes the K-theory coefficient of the Schubert curve: its nonadjacent switches precisely enumerate Pechenik and Yong's shifted genomic tableaux. The connection to K-theory also gives rise to a partial understanding of the complex geometry of these curves. [ABSTRACT FROM AUTHOR]

We explore extensions of tropical methods to arithmetic enumerative problems such as A 1 -enumeration with values in the Grothendieck–Witt ring and rationality over Henselian valued fields, using bitangents to plane quartics as a test case. We consider quartic curves over valued fields whose tropicalizations are smooth and satisfy a mild genericity condition. We then express obstructions to rationality of bitangents and their points of tangency in terms of twisting of edges of the tropicalization; the latter depends only on the tropicalization and the initial coefficients of the defining equation modulo squares. We also show that the GW -multiplicity of a tropical bitangent, i.e. the multiplicity with which its lifts contribute to the A 1 -enumeration of bitangents as defined by Larson and Vogt (Res Math Sci 8(26):1–21, 2021), can be computed from the tropicalization of the quartic together with the initial coefficients of the defining equation. As an application, we show that the four lifts of most tropical bitangent classes contribute 2 H , twice the class of the hyperbolic plane, to the A 1 -enumeration. These results rely on a degeneration theorem relating the Grothendieck–Witt ring of a Henselian valued field to the Grothendieck–Witt ring of its residue field, in residue characteristic not equal to two. [ABSTRACT FROM AUTHOR]

There are a number of (co-)homology theories on coarse spaces. Controlled operator K-theory is by far the most popular one of them. Our approach is geometric. We study when does the Roe-algebra of a space restrict to a subspace. Then we show the Roe-algebra is a cosheaf on the coarse topology. A result is a Mayer–Vietoris exact sequence in the presence of a coarse cover. We compute examples as an application. [ABSTRACT FROM AUTHOR]

Leonardo da Vinci's extensive drawings and notes devoted to anatomy do not arise in a medical context. He does not engage with surgery or "physic." Rather, his aim is to reveal what he understood to be the divine engineering of God's greatest creation. His earliest anatomical drawings map the conduits for the "spirits" at a deep level not practiced by other artists interested in the human body. The first set of drawings he produced in 1489 describes skulls with brilliant draftsmanship. His notes for these drawings are predominantly concerned with the location of the "common sense" (sensus communis) at the geometrical center of the cranium. The next great set of drawings completed in ca. 1510 expounds the form and functions of the bones and muscles in meticulous mechanical detail, probably in collaboration with the young Paduan doctor, Marcantonio della Torre. Leonardo's greatest realization of mathematics in anatomy comes when he looks at the valves of the heart; in studying the heart he also conceived a casting technique that he adapted to determine the form of the ventricles in the brain, radically revising the traditional arrangement of these structures. In a range of portrayals from diagrammatic to pictorial and from static to dynamic, Leonardo's anatomical research is unrivalled in revealing the mechanics of the human body. His work in this context is an underappreciated precursor of the modern science of biomechanics. (Larger-sized versions of the Leonardo drawings presented here are available as supplementary material in the online version of this article.) [ABSTRACT FROM AUTHOR]

We study the existence of positive solutions for the following class of scalar field problem on the hyperbolic space: - Δ B N u - λ u = a (x) | u | p - 1 u in B N , u ∈ H 1 (B N) , where B N denotes the hyperbolic space, 1 < p < 2 ∗ - 1 : = N + 2 N - 2 , if N ⩾ 3 ; 1 < p < + ∞ , if N = 2 , λ < (N - 1) 2 4 , and 0 < a ∈ L ∞ (B N). We prove the existence of a positive solution by introducing the min–max procedure in the spirit of Bahri–Li in the hyperbolic space and using a series of new estimates involving interacting hyperbolic bubbles. [ABSTRACT FROM AUTHOR]

We establish a pointwise estimate of | A | along the mean curvature flow in terms of the initial geometry and the | H A | bound. As corollaries we obtain the blowup rate estimate of | H A | and an extension theorem with respect to | H A | . [ABSTRACT FROM AUTHOR]

The treatises of Mohammad Abu'l-Wafa al-Buzjani (tenth c.) have influenced many architects, who utilized his innovative geometric principles and techniques in their works. The studies carried out thus far have focused on forms and have largely ignored the undergirding mathematical aspects. The present study aims to investigate the geometric principles used in the architectural works of Qavamuddin Shirazi, one of the architects of the Timurid period. Using selected problems from Abu'l-Wafa's treatise, the plans of the Shirazi's buildings are investigated by means of grid-based plan drawings, substituting the obtained numerical values in mathematical formulae, and adapting them to geometric problems. The analysis shows that Shirazi was aware of geometric problems presented by Abu'l-Wafa and the underlying mathematical principles, and that he tried to make use of specific geometries in his works. [ABSTRACT FROM AUTHOR]

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