Geometric Reasoning Research Articles

Visual-Tactile Perception Based Control Strategy for Complex Robot Peg-in-Hole Process via Topological and Geometric Reasoning

Visual-Tactile Perception Based Control Strategy for Complex Robot Peg-in-Hole Process via Topological and Geometric Reasoning

Mapping the landscape of social behavior.

Social interaction is integral to animal behavior. However, we lack tools to describe it with quantitative rigor, limiting our understanding of its principles and neuropsychiatric disorders, like autism, that perturb it. Here, we present a technique for high-resolution 3D tracking of postural dynamics and social touch in freely interacting animals, solving the challenging subject occlusion and part assignment problems using 3D geometric reasoning, graph neural networks, and semi-supervised learning. We collected over 140 million 3D postures in interacting rodents, featuring new monogenic autism rat lines lacking reports of social behavioral phenotypes. Using a novel multi-scale embedding approach, we identified a rich landscape of stereotyped actions, interactions, synchrony, and body contact. This enhanced phenotyping revealed a spectrum of changes in autism models and in response to amphetamine that were inaccessible to conventional measurements. Our framework and large library of interactions will greatly facilitate studies of social behaviors and their neurobiological underpinnings.

Developing Hidden Talent: An Exploratory Study of Advanced Math Curriculum Implementation and Its Effects on Young Mathematically Promising English Learners

In this qualitative case study, we investigated the role of an advanced math curriculum with language scaffolding on the development of geometric and visual-spatial reasoning skills in young mathematically promising English learners (MPELs). Specifically, we examined the effects of providing MPELs with challenging and supportive math curriculum to facilitate the development of mathematical potential through a geometry unit. Twenty-one MPELs were randomly assigned to either an intervention or a comparison class. The data sources included audio recordings of class sessions, documents of math tasks, field notes, and test scores. Findings revealed that in the intervention class, tasks required more cognitive demands, and students were encouraged to contribute to classroom discourse more than in the comparison class. MPELs in the intervention class demonstrated greater gains in geometric reasoning compared to their peers in the comparison class. These findings underscore the point that challenging and supportive curricula can effectively nurture the mathematical potential of underrepresented gifted students.

Experimental Comparison of Two 6D Pose Estimation Algorithms in Robotic Fruit-Picking Tasks

This paper presents an experimental comparison between two existing methods representative of two categories of 6D pose estimation algorithms nowadays commonly used in the robotics community. The first category includes purely deep learning methods, while the second one includes hybrid approaches combining learning pipelines and geometric reasoning. The hybrid method considered in this paper is a pipeline of an instance-level deep neural network based on RGB data only and a geometric pose refinement algorithm based on the availability of the depth map and the CAD model of the target object. Such a method can handle objects whose dimensions differ from those of the CAD. The pure learning method considered in this comparison is DenseFusion, a consolidated state-of-the-art pose estimation algorithm selected because it uses the same input data, namely, RGB image and depth map. The comparison is carried out by testing the success rate of fresh food pick-and-place operations. The fruit-picking scenario has been selected for the comparison because it is challenging due to the high variability of object instances in appearance and dimensions. The experiments carried out with apples and limes show that the hybrid method outperforms the pure learning one in terms of accuracy, thus allowing the pick-and-place operation of fruits with a higher success rate. An extensive discussion is also presented to help the robotics community select the category of 6D pose estimation algorithms most suitable to the specific application.

The shading isophotes: Model and methods for Lambertian planes and a point light

Structure-from-Motion (SfM) and Shape-from-Shading (SfS) are complementary classical approaches to 3D vision. Broadly speaking, SfM exploits geometric primitives from textured surfaces and SfS exploits pixel intensity from the shading image. We propose an approach that exploits virtual geometric primitives extracted from the shading image, namely the level-sets, which we name shading isophotes. Our approach thus combines the strength of geometric reasoning with the rich shading information. We focus on the case of untextured Lambertian planes of unknown albedo lit by an unknown Point Light Source (PLS) of unknown intensity. We derive a comprehensive geometric model showing that the unknown scene parameters are in general all recoverable from a single image of at least two planes. We propose computational methods to detect the isophotes, to reconstruct the scene parameters in closed-form and to refine the results densely using pixel intensity. Our methods thus estimate light source, plane pose and camera pose parameters for untextured planes, which cannot be achieved by the existing approaches. We evaluate our model and methods on synthetic and real images.

On Using GeoGebra and ChatGPT for Geometric Discovery

This paper explores the performance of ChatGPT and GeoGebra Discovery when dealing with automatic geometric reasoning and discovery. The emergence of Large Language Models has attracted considerable attention in mathematics, among other fields where intelligence should be present. We revisit a couple of elementary Euclidean geometry theorems discussed in the birth of Artificial Intelligence and a non-trivial inequality concerning triangles. GeoGebra succeeds in proving all these selected examples, while ChatGPT fails in one case. Our thesis is that both GeoGebra and ChatGPT could be used as complementary systems, where the natural language abilities of ChatGPT and the certified computer algebra methods in GeoGebra Discovery can cooperate in order to obtain sound and—more relevant—interesting results.

Geometric reasoning on the euclidean traveling salesperson problem in answer set programming1

The Traveling Salesperson Problem (TSP) is one of the best-known problems in computer science. Many instances and real world applications fall into the Euclidean TSP special case, in which each node is identified by its coordinates on the plane and the Euclidean distance is used as cost function. It is worth noting that in the Euclidean TSP more information is available than in the general case; in a previous publication, the use of geometric information has been exploited to speedup TSP solving for Constraint Logic Programming (CLP) solvers. In this work, we study the applicability of geometric reasoning to the Euclidean TSP in the context of an ASP computation. We compare experimentally a classical ASP approach to the TSP and the effect of the reasoning based on geometric properties. We also compare the speedup of the additional filtering based on geometric information on an ASP solver and a CLP on Finite Domain (CLP(FD)) solver.

Special Issue: Selected papers from the AIxIA 2023 Workshops

The 2023 edition of the AIxIA Conference, held in Rome, brought together a large number of researchers and practitioners to discuss the most recent and important advancements in Artificial Intelligence (AI). The conference featured 19 workshops, organized by 77 experts, attracting 248 submissions and resulting in 16 proceedings. This special issue presents extended versions of selected papers initially showcased at these workshops. Each paper underwent rigorous review and represents a diverse array of topics, reflecting the multifaceted nature of the Italian AI community. The topics covered include ethical foundations to symbiotic AI, symbolic knowledge extraction from black-box models, creative influence prediction using graph theory, AI approaches to multidimensional poverty prediction, an assessment of AI-based supports for informal caregivers, deep learning-based EEG denoising, AI-assisted board-game-based learning, large language models for assessment and feedback in higher education, geometric reasoning in the Traveling Salesperson Problem, defeasible reasoning in weighted knowledge bases, and conditional computation in neural networks. These contributions demonstrate the innovative and interdisciplinary research within the AI community, offering valuable insights and advancing the field.

Dynamic model of single-DOF spherical mechanisms based on instantaneous pole axes and Eksergian's equation

Instantaneous pole axes (IPAs) fully describe instantaneous kinematics of spherical mechanisms. In single-degree-of-freedom (single-DOF) mechanisms, IPAs’ locations uniquely depend on the mechanism configuration. Such a property allows the deduction of instantaneous-motion characteristics by means of analytic techniques based on geometric features of the mechanism configuration. Moreover, these geometric/analytic approaches are extendable to mechanism's static analyses since the virtual work principle relates mechanism's statics to its instantaneous kinematics. Analytic approaches based on geometric reasoning are of interest in mechanism design and their further extension to dynamic analyses is appealing in that context. This work proposes a possible extension of IPA-based techniques to dynamic analyses of single-DOF spherical mechanisms by using Eksergian's equation. A novel general dynamic model for single-DOF spherical mechanisms is proposed, which is based on IPAs’ locations. Then, the effectiveness of the proposed model is applied to a relevant single-DOF spherical mechanism.

Impact of cooperative learning on teaching mensuration to senior secondary students

Scientific and technological developments rely strongly on mathematical skills. However, mensuration is a challenging topic in school mathematics, leading to poor performance by many students. Various factors are responsible for the unsatisfactory performance, including inappropriate teaching methods where students passively absorb knowledge transmitted through reading and explanations of textbook contents. Thus, this study examined how a cooperative learning method affected students' performance in mensuration. A quasi-experimental research design was used in the study. The sample comprised two intact classes of 80 students assigned to experimental and control groups selected from one senior secondary school in Ogun State. Data were collected using an adopted standardised mensuration achievement test (MAT), which takes an analytical approach that includes problem-solving, geometric reasoning, formula application, and conceptual comprehension. The results showed a statistically significant difference between the mean score (M=11.18, SD=3.09) of students taught mensuration using the cooperative learning method and the mean score (M=7.66, SD=3.32) of students taught mensuration without using the learning method of teaching (t (78)=9.50, p=0.00). Also, the mean score varied statistically significantly (M=7.66, SD=2.63) of male students taught mensuration using the cooperative learning method and the mean score (M=8.73, SD=3.08) of female students taught mensuration without using the learning method of teaching (t (78) = 6.58, p=0.00). Therefore, cooperative learning in mathematics classrooms fosters students' ability to create and achieve learning objectives.

Product Design Trends within the Footwear Industry: A Review

Computer technology influences the capability to enhance the functionality of manufacturing and product design technologies. Innovations in computational design and digital manufacturing empower designers and manufacturers to create novel representations and algorithms for designing, analyzing, and planning the production of highly complicated products achievable through state-of-the-art technology. Various principles, including computational physics, geometric reasoning, and automated spatial planning, enable engineers to generate entirely new categories of products in the footwear industry. This study aims to review the methods and tools that have been published in the literature for the last twenty years, and provide a better understanding of the parameters, tools, and controls that contribute to the design and manufacturing processes of shoes. The main focus is on highlighting the product design-related trends within the footwear industry. A structured framework becomes apparent in the literature through the grouping and presentation of information. This framework facilitates drawing conclusions about the trends and existing needs derived from in-depth research in the field of footwear. Additionally, it reveals the upcoming methods and tools that will contribute to the enhancement and development of this emerging and promising industry sector. In conclusion, the categorization limitation within the footwear industry could serve as the foundation for exploring key areas to be analyzed further in other industries, for instance, in furniture, clothing, and packaging.

Parametric Analysis Of Wave-Induced Forces And Overtopping On Composite Vertical Breakwaters With Retreated Crown Wall

Composite vertical breakwaters are monolithic structures often used to protect port basins, especially in deep-water conditions. At present, the design of these structures is mainly based on Goda's formulae (Goda, 2010) or on the probabilistic design tools (PROVERBS) proposed by Oumeraci et al. (2001), while their hydraulic performance in terms of overtopping can be predicted by using the tools described in the EurOtop Manual (EurOtop Manual, 2018). Due to the size of these structures, the optimization and the improvement of the hydraulic performances (e.g. reduction of wave loads, wave overtopping, etc.) of these breakwaters can lead to significant economic saving. Typically, designers try to make small geometric changes without modifying the main geometrical dimensions of these structures. One of these technical solutions consists in placing the cast-in-situ concrete crown wall at a retreated position with respect to the front caisson face. It is assumed that the retreat of the crown wall, for geometric reasons, induces a time lag between the loads acting on the lower front external seawall face and on the crown wall of the caisson; furthermore, this could introduce a change in the pulsating nature of the loads, introducing also turbulent dissipations, consequently reducing the reflection coefficient and modifying the wave overtopping. To the best authors’ knowledge, in the literature there is a lack of guidelines to consider the effects of crown wall retreat in terms of wave actions and hydraulic performance of the structure. Recently, Romano & Bellotti (2023), based on physical model tests, provided a first experimental insight on the increase/reduction of the wave loads acting on deep water vertical breakwaters with retreated crown wall.

Connecting the threads: the role of multiplicative thinking in algebraic, geometrical, and statistical reasoning

ABSTRACT 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.

FGeo-DRL: Deductive Reasoning for Geometric Problems through Deep Reinforcement Learning

Human-like automatic deductive reasoning has always been one of the most challenging open problems in the interdisciplinary field of mathematics and artificial intelligence. This paper is the third in a series of our works. We built a neural-symbolic system, named FGeo-DRL, to automatically perform human-like geometric deductive reasoning. The neural part is an AI agent based on deep reinforcement learning, capable of autonomously learning problem-solving methods from the feedback of a formalized environment, without the need for human supervision. It leverages a pre-trained natural language model to establish a policy network for theorem selection and employ Monte Carlo Tree Search for heuristic exploration. The symbolic part is a reinforcement learning environment based on geometry formalization theory and FormalGeo, which models geometric problem solving (GPS) as a Markov Decision Process (MDP). In the formal symbolic system, the symmetry of plane geometric transformations ensures the uniqueness of geometric problems when converted into states. Finally, the known conditions and objectives of the problem form the state space, while the set of theorems forms the action space. Leveraging FGeo-DRL, we have achieved readable and verifiable automated solutions to geometric problems. Experiments conducted on the formalgeo7k dataset have achieved a problem-solving success rate of 86.40%.

An analysis of the essential understandings in elementary geometry and a comparison to the common core standards with teaching implications

Geometry and spatial reasoning form the foundations of learning in mathematics. However, geometry is a subject often ignored by curriculum writers and teachers until high school, leading to students lacking in critical skills in geometric reasoning. As the United States moves into a new curriculum epoch, heralding the commencement of the national common core standards (CCS), one could question if CCS in geometry align with the essential understandings children need to be successful geometric thinkers. This paper begins with an examination of the essential understandings of geometric reasoning leading to an interpretation and critique of the elementary geometry CCS. Finally, the instructional implications are discussed, considering the common core progression through what we know about how children learn geometry.

FGeo-SSS: A Search-Based Symbolic Solver for Human-like Automated Geometric Reasoning

Geometric problem solving (GPS) has always been a long-standing challenge in the fields of automated reasoning. Its problem representation and solution process embody rich symmetry. This paper is the second in a series of our works. Based on the Geometry Formalization Theory and the FormalGeo geometric formal system, we have developed the Formal Geometric Problem Solver (FGPS) in Python 3.10, which can serve as an interactive assistant or as an automated problem solver. FGPS is capable of executing geometric predicate logic and performing relational reasoning and algebraic computation, ultimately achieving readable, traceable, and verifiable automated solutions for geometric problems. We observed that symmetry phenomena exist at various levels within FGPS and utilized these symmetries to further refine the system’s design. FGPS employs symbols to represent geometric shapes and transforms various geometric patterns into a set of symbolic operation rules. This maintains symmetry in basic transformations, shape constructions, and the application of theorems. Moreover, we also have annotated the formalgeo7k dataset, which contains 6981 geometry problems with detailed formal language descriptions and solutions. Experiments on formalgeo7k validate the correctness and utility of the FGPS. The forward search method with random strategy achieved a 39.71% problem-solving success rate.

Dynamical order and many-body correlations in zebrafish show that three is a crowd

Zebrafish constitute a convenient laboratory–based biological system for studying collective behavior. It is possible to interpret a group of zebrafish as a system of interacting agents and to apply methods developed for the analysis of systems of active and even passive particles. Here, we consider the effect of group size. We focus on two– and many–body spatial correlations and dynamical order parameters to investigate the multistate behavior. For geometric reasons, the smallest group of fish which can exhibit this multistate behavior consisting of schooling, milling and swarming is three. We find that states exhibited by groups of three fish are similar to those of much larger groups, indicating that there is nothing more than a gradual change in weighting between the different states as the system size changes. Remarkably, when we consider small groups of fish sampled from a larger group, we find very little difference in the occupancy of the state with respect to isolated groups, nor is there much change in the spatial correlations between the fish. This indicates that fish interact predominantly with their nearest neighbors, perceiving the rest of the group as a fluctuating background. Therefore, the behavior of a crowd of fish is already apparent in groups of three fish.

Mathematical Symbols in Academic Writing: The Case of Incorporating Mathematical Ideals in Academic Writing for Education Researchers

Mathematical symbols, such as those embodying quantum concepts, are indispensable for conveying complex ideas and relationships in academic writing. However, some education researchers and students keep a distance from anything mathematical: algebraic equations, geometrical reasoning, or statistical symbols. How to lower the access threshold for this type of mathematical narrative and reveal the meanings of a range of quantum conceptions to modern educators thus becomes a real problem. Using the pendulum motion equation as a reference point, I argue in this article for the advantages of academic English or French writing genres that fuse a range of mathematical symbols of quantum concepts and conceptual change. Such writings help demonstrate how incorporating the idea of probability (a) refines the debate among conceptual, verbal, and mathematical academic writing; (b) allows new conceptions that draw on the insights from quantum cognition-supported theories; (c) helps explain students’ understanding of mathematical symbols; and (d) offers a new taxonomy for categorizing academic writings.

Neurocognitive Assessment of Mathematics-Related Capacities in Neurosurgical Patients.

A precise neuropsychological assessment is of the utmost importance for neurosurgical patients undergoing the surgical excision of cerebral lesions. The assessment of mathematical abilities is usually limited to arithmetical operations while other fundamental visuo-spatial aspects closely linked to mathematics proficiency, such as the perception of numerical quantities and geometrical reasoning, are completely neglected. We evaluated these abilities with two objective and reproducible psychophysical tests, measuring numerosity perception and non-symbolic geometry, respectively. We tested sixteen neuro-oncological patients before the operation and six after the operation with classical neuropsychological tests and with two psychophysical tests. The scores of the classical neuropsychological tests were very heterogeneous, possibly due to the distinct location and histology of the tumors that might have spared (or not) brain areas subserving these abilities or allowed for plastic reorganization. Performance in the two non-symbolic tests reflected, on average, the presumed functional role of the lesioned areas, with participants with parietal and frontal lesions performing worse on these tests than patients with occipital and temporal lesions. Single-case analyses not only revealed some interesting exceptions to the group-level results (e.g., patients with parietal lesions performing well in the numerosity test), but also indicated that performance in the two tests was independent of non-verbal reasoning and visuo-spatial working memory. Our results highlight the importance of assessing non-symbolic numerical and geometrical abilities to complement typical neuropsychological batteries. However, they also suggest an avoidance of reliance on an excessively rigid localizationist approach when evaluating the neuropsychological profile of oncological patients.

Hierarchies and Panoramic Aspects of Anne Tyng’s Urban Projects and the Contemporary Vision of the City

This article focuses on Anne Tyng’s Urban Hierarchy proposal considering the geometric reasoning, namely the helical structure and intrinsic progressions, as well as some coeval experiences and assumptions, especially regarding the developed visions revising urban standards and images. Starting from ​​these two complementary paths, it is possible to fully comprehend the image of the formulated project, which is essentially based on the configuration of the helical megastructure and the infinite possibilities provided by the application of modularity and aggregation principles. Through the virtual reconstruction of this structure, combining clusters of houses and multiple hierarchical levels of pedestrian and vehicular traffic, it is possible to create different images. This process starts from the traditional terraced house to the multilevel panoramic vision (bilateral unit and its helicoidal aggregation), up to the view of the aggregate of the various units (spiral) in a potentially infinite arrangement based on the Divine Proportion.