Geometric Intuition Research Articles

Music-driven geometric and topologic intuition: a case study with the Klein bottle

The comprehension of mathematical notions can profit from the personal background of a population of students, drawing upon their former knowledge in another field. In particular, the embodiment of mathematics provides us with a bridge to transfer competencies from music to intuitive understanding of non-trivial geometries and topologies. In this study, we analyse an experiment in which music composition students are introduced to the Klein bottle for the first time. Through the Ontological and Semiotic Approach (OSA), a conceptual framework in mathematics education, we evaluate the evolution from an initial intuition of the Klein bottle to a formal understanding of the object via musical renditions and verbal presentations to both neophyte and specialist audiences in mathematics.

Collective Variable-Based Enhanced Sampling: From Human Learning to Machine Learning.

Enhanced-sampling algorithms relying on collective variables (CVs) are extensively employed to study complex (bio)chemical processes that are not amenable to brute-force molecular simulations. The selection of appropriate CVs characterizing the slow movement modes is of paramount importance for reliable and efficient enhanced-sampling simulations. In this Perspective, we first review the application and limitations of CVs obtained from chemical and geometrical intuition. We also introduce path-sampling algorithms, which can identify path-like CVs in a high-dimensional free-energy space. Machine-learning algorithms offer a viable approach to finding suitable CVs by analyzing trajectories from preliminary simulations. We discuss both the performance of machine-learning-derived CVs in enhanced-sampling simulations of experimental models and the challenges involved in applying these CVs to realistic, complex molecular assemblies. Moreover, we provide a prospective view of the potential advancements of machine-learning algorithms for the development of CVs in the field of enhanced-sampling simulations.

A biomechanical model of a vehicle passenger in the sagittal plane

Musculoskeletal biomechanical models have wide applications in ergonomics, rehabilitation, and injury estimation. Their use can be extended to enable quantitatively explaining and estimating ride comfort for a vehicle's passenger. A biomechanical model of the upper body in the sagittal plane is constructed, which allows for curved motion to simulate the propagation of disturbance energy within a seated passenger aboard a moving vehicle. The dynamic predictions of the model are validated against experimental results within the literature. Frequency responses show that within the vehicular frequency range, the L4L5 and the L5S1 discs in the lower lumbar region are susceptible to the highest vibration transmission. It was also found that vibration transmission is maximized at around 4.5 Hz. The model provides analytical and geometric intuition into the motion of the various segments of the upper body using a few simple geometric assumptions and can be employed to develop a quantitative ride-comfort metric, such that the most comfortable ride would be that which would induce the least internal motion within the passenger model.

Research on Implementing the Core Literacy of Intuitive Imagination in Senior High School Mathematics Teaching

In the process of senior high school mathematics learning, improving students' intuitive imagination core literacy can enhance their ability to combine numbers and shapes. It also plays an important role in developing geometric intuition, spatial imagination, and understanding the essence of things. We first summarize the current strategies for cultivating the core literacy of intuitive imagination, then provide an overview of the intuitive imagination in core literacy. The author interviewed 12 senior high school mathematics teachers who were teaching locally in various cities in Shandong province with a large population and balanced education levels. Through interviews, it is learned that, students always lack numerical evidence when doing exercises, and most student s only look at questions without sketching. During teaching, teachers closely follow the textbook but lack explanation of flexible problem-solving methods using geometry, and they may experience improper use of teaching aids or incoherent explanations. Finally, corresponding strategies are provided.

A historical analysis of the independent development of calculus by Newton and Leibniz

This paper undertakes a historical investigation of the separate and independent development of calculus by Isaac Newton and Gottfried Leibniz in the late 17th century. Through analysis of primary sources and historiographical perspectives, it explores the differences in notation, methods, and applications used by each mathematician to formulate foundational concepts of calculus. The research demonstrates that Newton relied more on geometric intuition, developing calculus concepts like fluxions and fluents rooted in kinematic problems. His 1687 Philosophiae Naturalis Principia Mathematica synthesized many calculus innovations. Meanwhile, Leibniz approached calculus from an algebraic mindset, utilizing infinitesimal differentials and comprehensively explaining integral and differential calculus in publications like Nova Methodus pro Maximis et Minimis. Evaluation of letters and documents from the 1670s and 1680s shows no direct collaboration or communication about calculus between Newton and Leibniz. This lack of transmission, coupled with the disparities in their notation and calculus techniques, provides evidence for independent creation. However, Newton and Leibniz shared key insights regarding rates of change, derivatives and integrals, hinting at a broader zeitgeist in early modern mathematics and science. Thus, this dual achievement illustrates how the Scientific Revolution facilitated conceptual convergence despite geographic separation between great thinkers. Investigating this case study offers perspective on the interplay between individual genius and wider social contexts in driving scientific progress. This paper concludes by assessing the legacy of the Newton-Leibniz debate over priority and analyzing work that paved the way for modern unified calculus notation and applications.

Self-adaptive subspace representation from a geometric intuition

In this paper, we address the unsupervised subspace learning task from a geometric intuition, which offers a direct understanding of the data representation. First, we consider all subspaces of the Euclidean space as a series of graded Grassmannian manifolds, and represent them by orbits of rotation group action. Then, we reformulate the unsupervised subspace learning by a least square problem with respect to rotation and projection operator. Second, we introduce a low-rank regularization to obtain a low-dimensional and robust subspace representation. Then, the model is translated into a minimization problem on the Grassmannian manifold. By dividing the model into two subproblems of solving the optimal rotation and subspace dimension, we design an alternating iteration strategy, where the locally geodesic structure of the rotation group and the unconstrained quadratic 0-1 programming are utilized. Finally, we apply the proposed method to the image classification problem on five benchmark datasets and compare it with nine state-of-the-art methods. Numerical results show that our proposed method has better feature representation ability and almost achieves the best performance in terms of classification accuracy and robustness.

Reflection on Implementing Geometric Intuition in the Junior High School Mathematics Classroom

As one of the core literacies emphasized in the Mathematics Curriculum Standards for Compulsory Education (2022 Edition), geometric intuition has been widely valued in the current junior high school mathematics teaching. Based on the method of theoretical analysis, this paper analyzes the connotation of geometric intuition literacy combines it with the practice of current mathematics classroom teaching, and proposes four implementation strategies for geometric intuition literacy in junior high school mathematics classrooms, which are as follows: 1. Add operational practice to classroom teaching; 2. Provide thematic courses related to ruler drawing; 3. Reasonable use of modern educational technology to assist classroom teaching; 4. Organize special questions about charts to practice with students. These strategies can be used for reference by junior high school mathematics teachers in practical teaching.

Praxis, Diagramm, Körper. Die epistemologischen turns und die Rehabilitation von Kants Euklidizitätsthese

Abstract This paper aims to show how Kant’s concept of mathematical intuition and his thesis of the Euclidicity of space have been rehabilitated by recent interpretations that refer to the concepts of practice, diagrams, and the body. This involves criticising the idea of mathematics as a purely logical and axiomatic science, reading Kant’s theory of geometric intuition as a theory of diagrammatic visualizations, and showing how Kant’s idea of the Euclidicity of space can be justified with reference to embodied experience. These readings contradict the critique of Kant voiced by logicism and logical empiricism. They show that Kant’s notion of non-empirical intuition refers to the embeddedness of human cognition in embodied, media-based practices that function as a third epistemological realm alongside the conceptual and the empirical.

The Application of Geometric Intuition in High School Mathematics Teaching—Take the Su Education Edition Textbook Ellipse as an Example

The Application of Geometric Intuition in High School Mathematics Teaching—Take the Su Education Edition Textbook <i>Ellipse</i> as an Example

Diagrams forMethod12 in the Archimedes Palimpsest

This paper discusses four diagrams in the Archimedes Palimpsest, a manuscript that provides among other texts the only extant witness to Archimedes’ Method. My study of the two diagrams for Method 12 aims to open up discussions about the following two questions. First, I want to question the assumed relationship between diagram and geometric configuration. Rather than a representation-represented relation, I argue that the two diagrams for Method 12 have a stronger independence from the geometric configuration they are related to. Their connection with the theorem rather lies in their role in shaping the way in which the proof is written. Secondly, I want to examine the connection between geometric intuition and figure. Geometric intuition is the recognition of certain properties or relations in a geometric configuration not through deduction and is held to be parallel to, if not dependent on, the empirical observation of a physical figure. Method 12 provides a curious case where an intuitive fact is not only unrepresented, but even hindered by the diagrams, and the proof assumes that the reader has come to realise that fact, which is most easily achieved when the diagrams are ignored.

BRST cohomology is Lie algebroid cohomology

In this paper we demonstrate that the exterior algebra of an Atiyah Lie algebroid generalizes the familiar notions of the physicist's BRST complex. To reach this conclusion, we develop a general picture of Lie algebroid isomorphisms as commutative diagrams between algebroids preserving the geometric structure encoded in their brackets. We illustrate that a necessary and sufficient condition for such a diagram to define a morphism of Lie algebroid brackets is that the two algebroids possess gauge-equivalent connections. This observation indicates that the aforementioned set of Lie algebroid isomorphisms should be regarded as equivalent to the set of local diffeomorphisms and gauge transformations. Moreover, a Lie algebroid isomorphism being a chain map in the exterior algebra sense ensures that isomorphic algebroids are cohomologically equivalent. The Atiyah Lie algebroids derived from principal bundles with common base manifolds and structure groups may therefore be divided into equivalence classes of isomorphic algebroids. Each equivalence class possesses a local representative which we refer to as the trivialized Lie algebroid, and we show that the exterior algebra of the trivialized algebroid gives rise to the BRST complex. We conclude by illustrating the usefulness of Lie algebroid cohomology in computing quantum anomalies, including applications to the chiral and Lorentz-Weyl (LW) anomalies. In particular, we pay close attention to the fact that the geometric intuition afforded by the Lie algebroid (which was absent in the naive BRST complex) provides hints of a deeper picture that simultaneously geometrizes the consistent and covariant forms of the anomaly. In the algebroid construction, the difference between the consistent and covariant anomalies is simply a different choice of basis.

Research on the Cognitive Degree of Pre-service Junior High School Mathematics Teachers for Geometric Intuition Literacy in China

Currently, geometric intuition literacy has attracted extensive attention from all walks of life. Many relevant problems about it have been studied except the cognitive degree of geometric intuition literacy of pre-service junior high school mathematics teachers. In this study, 40 pre-service junior high school mathematics teachers in China are surveyed using an open-ended interview method. The analysis of the data reveals that: (1) The accuracy of the cognition about geometric intuition literacy is not high. (2) The cognitive scope of pre-service high school mathematics teachers about geometric intuition literacy is not extensive and pre-service teachers' knowledge of geometric intuition literacy is focused on the more generalized elements. (3) The cognition of many pre-service teachers is not deep or comprehensive and concentrates at a lower level. (4) The cognition of many pre-service teachers for many aspects of geometric intuition literacy is not very clear. Therefore, it is suggested that: Teacher training schools should reform their curricula and pay attention to teacher trainees' learning and education on geometric intuition literacy. Special training and education on core literacy should be strengthened in pre-service teacher training so that geometric intuition literacy can be valued. Pre-service junior high school mathematics teachers should actively learn and study core literacy to improve their understanding of geometric intuition literacy.

Geometrical intuition components for designing mathematical tasks

Geometrical intuition is the ability to visualize, construct, and manage geometrical shapes in the mind when solving geometry problems. Geometrical intuition requires four skills: the ability to construct and manage geometrical figures in mind, perceive geometrical properties, connect pictures to concepts and theories in geometry, and determine where and how to begin when solving geometry problems. This geometric intuition ability is important for developing problem-solving. Therefore, we need a task that can be used to identify and develop students' geometric intuition abilities. This research aims to design a geometric intuition task. We employ design research methods to design geometrical intuition tasks by conducting a literature review on geometric intuition and geometry tasks, creating geometrical intuition tasks, and estimating and noting the possible student responses. This study produced three types of tasks based on the four components of geometric intuition. We provide a list of possible responses that junior high school students may provide, as well as practical suggestions for teachers. We recommend research using our developed task to evaluate students' geometrical intuition.

Solutions of the Ortvay Rudolf International Competition in Physics: Mass of a Force Carrier Particle (2014/24 Problem)

This paper is the fourth in a series of published solutions discussing problems of the Ortvay Rudolf international competition. The problem treated below is an exercise in special relativity, applied to classical description of particle-exchange during scattering. As it is known, introducing rapidity can radically simplify the calculations in special relativity, as well as offer a simple geometric intuition. However, students often prefer doing calculations directly through formulas involving velocities, resulting in lengthy calculations, often increasing the chance of errors. It is our belief that giving the students simple, but not trivial problems before demonstrating the short and elegant solution presented below can help students to appreciate and understand the geometrical formulation of special relativity. Our calculation does not use mathematical techniques beyond those expected of high-school students, and the first half of this work may be used in special high-school classes.

Research on Geometric Intuition of Junior High School Students in China

Recently, geometric intuition, as one of the core literacies of mathematics in Junior high schools, has received widespread attention. Many scholars have researched this area in China, but it lacks literature that provides a complete overview and collation of geometric intuition. We searched with the keyword geometric intuition through CNKI, and carefully read and categorised the opinions and research approaches of existing research, and draw the following conclusions: (1) Current research in China mainly focuses on four aspects: the connotation and expression of geometric intuition, the value of geometric intuition, the measurement and current situation of geometric intuition, and the cultivation strategies of geometric intuition. (2) There are some shortcomings in previous studies. For example, the methodology is monotonous and the content of geometric intuition require further expanding. (3) There are some gaps in the research on this topic. For example, the connotation of geometric intuition in the new era needs to be explored in-depth. Additionally, there is a lack of operational strategies, which needs to be verified through practical teaching. Therefore, it is necessary to improve the research methodology and expand the scope of research on geometric intuition in order to make this research more comprehensive, systematic, and in-depth.

Performance evaluation and optimization of fixture adapter for oil drilling top drives

Abstract A top drive is an essential mechanical device in oil field drilling since it provides the necessary torque for the drilling operations. Manufacturers in the oil and gas industry typically perform in-housing testing and classify the Safe Working Load of top drives. Testing a top drive requires a unique test stand, thus making testing top drives from other manufacturers a difficult challenge. A fixture adapter can be designed using geometric constraints and intuition to make testing apparatus semi-universal, yet they are often bulky and heavy, posing more significant safety concerns. This study aims to first numerically assess the existing fixture adapter and then structurally optimize it for enhancing its structural integrity and efficiency under various severe working environments. Therefore, finite element analysis (FEA) was performed on the existing fixture adapter, and compliance minimization topology optimization was employed. Four load and boundary conditions were used from the three most frequent operation scenarios for the fixture adapters: (i) drilling standby, (ii) staging area, (iii) drilling make-up, and (iv) break-up. The FEA results indicated that no safety factor was compromised with a 50% and 60% mass retention constraint via topology optimization compared to the original fixture adapter. The optimized fixture adapter was also tested under compression using printed 3D prototypes to validate the finite analysis and topology optimization processes.

Random Melnikov Method and Induced Chaos in Bistable Vibro-Impact Oscillators with Bounded Noise

In this paper, the Melnikov method for an abstract stochastic nonsmooth hybrid system is derived in detail and employed to study the homoclinic bifurcations and induced chaos in a bistable vibro-impact SD oscillator with bounded noise first proposed by Li et al. 2021, in which the geometric nonlinearity is used to produce the irrational restoring force and an impact map is introduced to describe energy loss during collisions. The initial Poincaré section is selected on the right rigid constraint so that rigorous perturbation analysis can be carried out to avoid the extension of trajectories near the switching manifold. The stochastic Melnikov function with geometrical intuition is obtained to detect the threshold of parameters for homocinic tangency of the perturbed stochastic stable and unstable manifolds by calculating the energy difference of their initial points on the Poincaré section. Finally, the analytical Melnikov analysis combined with numerical simulations is carried out to validate the developed Melnikov method for analyzing the global dynamics of the bistable vibro-impact SD oscillator subject to bounded noise.

Nucleon-nucleon potential from instanton holonomies

We derive the nucleon-nucleon interaction from the Skyrme model using the instanton and product approximations to skyrmion dynamics. In doing so, we also calculate the classical potential and metric for skyrmion dynamics in each of the approximations. This is the first time they have been compared in detail and the results show major disagreements between the approximations. We derive the eight low energy nucleon-nucleon interaction potentials and compare them with the Paris model. For the instanton approximation we find strong negative isoscalar and isovector spin-orbit potentials, matching phenomenological models and our geometric intuition. Results for the other potentials are mixed, in part due to the zero pion mass limit used in this approximation.

Ubiquitous bias and false discovery due to model misspecification in analysis of statistical interactions: The role of the outcome's distribution and metric properties.

Studies of interaction effects are of great interest because they identify crucial interplay between predictors in explaining outcomes. Previous work has considered several potential sources of statistical bias and substantive misinterpretation in the study of interactions, but less attention has been devoted to the role of the outcome variable in such research. Here, we consider bias and false discovery associated with estimates of interaction parameters as a function of the distributional and metric properties of the outcome variable. We begin by illustrating that, for a variety of noncontinuously distributed outcomes (i.e., binary and count outcomes), attempts to use the linear model for recovery leads to catastrophic levels of bias and false discovery. Next, focusing on transformations of normally distributed variables (i.e., censoring and noninterval scaling), we show that linear models again produce spurious interaction effects. We provide explanations offering geometric and algebraic intuition as to why interactions are a challenge for these incorrectly specified models. In light of these findings, we make two specific recommendations. First, a careful consideration of the outcome's distributional properties should be a standard component of interaction studies. Second, researchers should approach research focusing on interactions with heightened levels of scrutiny. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Hermann Cohen’s logic of the pure knowledge as a philosophy of science

The connection of Hermann Сohen’s “The Logic of Pure Knowledge” with the revolutionary transformations in physics and mathematics at the end of the 19th century is shown. Сohen criticised Kant’s answer to the question “How is mathematics possible”? If Kant refers to a priori forms of pure intuition, Сohen sees in it a restriction of freedom of mathematical thinking by limits of intuition. It has been shown that Cohen's position is in accordance with the main development of mathematics in the last decades of the 19th century, in particular, with K. Weierstrass’ striving to get rid of geometrical or mechanical images and intuitions in the mathematical analysis. Cohen was also well informed about the latest ideas in physics of his time. They are also discussed in “The Logic of Pure Knowledge.” The revolutionary spirit of physics and mathematics was appealing to Cohen, and he felt a corresponding enthusiasm for it. The ongoing scientific revolution is consonant with Cohen’s assertion that the foundations of science are hypotheses. The purity of pure thinking does not guarantee the correctness of any of its constructions. Each step in the development of science requires a critique of existing notions. The development of knowledge from the naive to the critical one shows a movement from a picture of the world as a set of stable things to a picture of continuous movement and change, where change is more important than what changes. Cohen sees such a development in an evolution of the concept of substance in modern physics and he welcomes the replacement of material substance by energy, seeing this movement as a confirmation of critical idealism. Finally, it is discussed whether we can speak of the actuality of Cohen’s logic of pure knowledge.