Problems In Geometry Research Articles

Diffusion dynamics for an infinite system of two-type spheres and the associated depletion effect

We consider a random diffusion dynamics for an infinite system of hard spheres of two different sizes evolving in Rd, its reversible probability measure, and its projection on the subset of the large spheres. The main feature is the occurrence of an attractive short-range dynamical interaction – known in the physics literature as a depletion interaction – between the large spheres, which is induced by the hidden presence of the small ones. By considering the asymptotic limit for such a system when the density of the particles is high, we also obtain a constructive dynamical approach to the famous discrete geometry problem of maximising the contact number of n identical spheres in Rd. As support material, we propose numerical simulations in the form of movies.

108.01 A use of Pythagorean triples in a problem in elementary geometry

108.01 A use of Pythagorean triples in a problem in elementary geometry

Intersection Properties of Finite Disk Collections

In this article, we study the intersection of a finite collection of disks in Euclidean space by examining spheres of various dimensions and their poles (extreme values with respect to canonical projections) contained within the intersection’s boundary. We derive explicit formulae for computing these extreme values and present two applications. The first application involves computing the smallest common rescaling factor for the radii of the disk system, which brings the system to a single point of intersection. This calculation allows us to compute the generalized Čech filtration, a crucial tool for the topological data analysis of weighted point clouds. The second application focuses on determining the minimal Axis-Aligned Bounding Box (AABB) for the intersection of a finite collection of disks in Euclidean space, addressing a significant problem in computational geometry. We consider that this work aims to contribute to the fields of topological data analysis and computational geometry by providing new tools for analyzing complex geometric structures.

Scoring rubric design to measure the ability to prove plane geometry problems not accompanied by image visualization

Proof of the type of problem not accompanied by image visualization will require a longer flow and process than proof of the kind of problem accompanied by image visualization. The ability of students to prove problems, especially problems not accompanied by image visualization, must be adequately expressed and objectively. For that, we need an instrument that can reveal the ability to prove the case of these problems. This research has successfully designed a scoring rubric that can be explicitly used to measure students' proving abilities on problems not accompanied by image visualization. Aspects developed in the scoring rubric include making image visualizations according to the information in the questions. These include sub-aspects of image accuracy and completeness of labels, initial steps of proving, preparation of conjectures, flow of proving, and support for valid arguments for statements made. Based on the validation from the experts, the scoring rubric developed was declared valid and ready to be used to measure the student's proving ability on plane geometry, proving problems not accompanied by image visualization.

Investigating problem-solving behaviours of university students through an eye-tracking system using GeoGebra in geometry: A case study

Problem solving is one of the high-level thinking skills and essential in teaching mathematical concepts and procedures. The eye-tracking method allows educators to see and interpret different problem-solving behaviours of students. The purpose of this study was to investigate the problem-solving behaviours of university students based on eye tracking statistics when faced with a geometry problem, and to explain these behaviours in terms of the students' success and learning styles. The research design was a case study approach that combined quantitative and qualitative data. Eight first-year students from the department of elementary mathematics education in a private University, Turkey, participated in this study. They were asked to solve a specific geometry question using the GeoGebra software. While solving this question, their eye movements were recorded and analysed using an eye-tracking program. Results were analysed to investigate their problem-solving skills and behaviours based on Polya’s mathematical problem-solving stages. Results of the study suggest that the effort devoted to the understanding and planning stages of problem-solving are important factors that contribute to success in this case. Results also revealed that participated students with a converger learning style were more successful in each step of the problem-solving process. Additionally, it could be concluded that the input area had only been used by converger students, indicating that they devoted time and effort to creating and trying formulas.

Students' Thinking Process in Solving Geometry Problems based on Harel Theory on Malay Traditional House Design

Students' Thinking Process in Solving Geometry Problems based on Harel Theory on Malay Traditional House Design

Exploring junior high school students' geometry self-efficacy in solving 3D geometry problems through 5E instructional model intervention: A grounded theory study

Geometry self-efficacy is an essential affective aspect that will influence students in solving mathematics problems, especially geometry material. Therefore, teachers must be able to develop learning instructions that not only affect students' mathematical abilities but also strengthen students' affective aspects. This research explores students' geometry self-efficacy when learning to solve three-dimensional geometry problems through the 5E Instructional Model intervention. A grounded theory design was used to reveal the aims of this research. Participants in this research were one mathematics teacher and 22 students (12 girls and 10 boys) in class VIII at a state Junior High School in Indramayu Regency, Indonesia. The research involved the qualitative analysis of gathered data obtained through observation, questionnaires, interviews, and documentation, employing grounded theory analysis techniques, including open coding, axial coding, and selective coding. The findings revealed that students with high self-efficacy in geometry display confidence in describing and calculating the surface area and volume of three-dimensional geometric objects. Those with moderate self-efficacy in geometry are self-assured in addressing straightforward assignments but may need more confidence in tackling more complex tasks. Conversely, students with low self-efficacy in geometry tend to need more confidence and are prone to giving up easily. Therefore, this research emphasizes that the geometry self-efficacy level can influence how students act and complete 3D geometry tasks given by teachers in learning, especially 3D geometry learning.

Optimal modeling of line paths based on multibeam bathymetric system

Multibeam ocean bathymetry is one of the major issues in exploring the ocean. For problems such as multibeam line ocean bathymetry, this paper starts from the related theory, analyzes the inherent geometric features of the problem, comprehensively establishes a multibeam line model with generalized significance within a certain error range, and according to the method of least squares, gives a model of the line with a fixed opening angle of the multibeam transducer in a certain range of the sea area. Firstly, from the geometrical point of view, the interpreted diagram and the schematic diagram given in the title are combined and simplified into a problem of solving a triangle. The trigonometric equations are listed through the sine theorem and solved algebraically. A mathematical model is developed about the coverage width and adjacent strips. Then, from the question, we know that we need to solve the modeling differences caused by the different angles of the measuring lines in different directions, i.e., to elevate the two-dimensional problem of solving triangles to a three-dimensional problem of three-dimensional geometry. By finding the angle between the coverage width and the horizontal plane , we know the relationship between it and the angle between the direction of the survey line and the slope angle , so as to establish the model of the coverage width of multibeam bathymetry. Summarized by the verification of the previous conclusions, this paper obtains: when controlling the width of two adjacent survey lines exactly meets the overlap rate of 10% with the previous survey line, and the survey line direction is perpendicular to the seafloor slope in the projection, the measurement length is the shortest at this time. Using the above summary as a constraint, the shortest path is calculated to be 66 nautical miles using the trigonometric method.

Solving Some Geometry Problems of the Náboj 2023 Contest with Automated Deduction in GeoGebra Discovery

Solving Some Geometry Problems of the Náboj 2023 Contest with Automated Deduction in GeoGebra Discovery

Deskripsi Kemampuan Pemahaman Konsep Dalam Menyelesaikan Soal Geometri Berdasarkan Perbedaan Gender

This research to find out to describe the ability to understand concepts of male and female students in class VIII of SMP Negeri 3 Sungguminasa in solving geometry problems. The type of research used is describtive research with a qualitative approach. The subjects in this research was students of class VIII.D SMP Negeri 3 Sungguminasa as many as 2 students, 1 male students and 1 female students who were selected based on the data of the highest report card scores. The instrument used is a concept understanding test and interview guidelines. The results showed that there were differences in the ability to understand concepts between female students and male students with the following descriptions: (1) the ability to understand concepts of female students in the triangle material was able to meet 5 indicators of understanding the concept of interpretting, exemplifying, summarizing, inferring, comparing and not fulfilling 2 indicators of understanding the concept of classifying and explaining. (2) the concept understanding ability of male students in the triangle material is able to meet the 4 indicators of understanding the concept of interpretting, exemplifying, classifying, inferring and not meeting the 3 indicators of understanding the concept of summarizing, comparing, and explaining. Thus it can be said that female students have a better understanding of concepts than male students.

Solving olympiad geometry without human demonstrations

Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning1–4, owing to their reputed difficulty among the world’s best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges1,5, resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004.

A digital geometric approach for discrete Voronoi diagram construction using GPU

SummaryVoronoi diagram is a versatile geometric structure used in solving many computational geometry problems in the field of image analysis, computer vision, engineering, medicine, molecular biology and various other disciplines. As a result it is a popular research domain. The literature boasts of many elegant and efficient methods for the construction of the Voronoi diagram in the real space. But these algorithms often fail to perform well when applied to the discrete domain. We present in this paper a novel algorithm for the construction of discrete Voronoi diagram using digital geometric concepts that takes care of the geometric perspective of the Voronoi diagram in the discrete domain. Although our technique is more suited for the CPU but by careful use of some cut‐off heuristics we are able to design a parallel programming approach for the construction of the Voronoi diagram making use of the GPU computational power. The speedup of our algorithm is compatible with the popular GPU technique for Voronoi diagram construction, namely, Jump flood algorithm. But since our algorithm uses the geometric perspective for the construction of Voronoi diagram it gives more accurate output and hence requires no correction step as is the case for the Jump flood algorithm. Furthermore, our approach focuses on extensibility, allowing for easy amenability for construction of higher dimensional Voronoi diagram. This ensures that the Voronoi diagram generation process can be adapted and expanded upon to meet specific requirements or incorporate future enhancements. We present an implementation of the proposed algorithm on a GPU using TAICHI language and describe its performance benefits on multiple input sites.

Students' Deductive Reasoning in Evaluating Solutions to Geometry Problems

Deductive reasoning is very important in learning mathematics and is one of the formal goals that emphasizes the arrangement of reasoning. This is a qualitative research that aims to describe the deductive reasoning abilities of junior high school students in evaluating problem solutions in geometry material. The subjects of this study were two junior high school students in Gresik who had high mathematical abilities and were of different genders. The instrument used was task-based interviews. The task in question is a deductive reasoning ability test in which there are problems and their solutions. Data analysis was carried out using three indicators of deductive reasoning according to Sumarmo: carrying out calculations based on certain formulas, conducting proofs, and drawing conclusions. The results showed that the deductive reasoning abilities of female and male students were relatively the same. In solving the problem, Both male and female students had similar initial idea by making an example of the length of the side of the given object. However, male students are more capable of proving and considering more effective strategies for solving problems. Therefore, students and teachers need to improve their deductive reasoning abilities and consider all effective strategies for solving problems.

Ricci Flow Under Kato-Type Curvature Lower Bound

In this work, we extend the existence theory of non-collapsed Ricci flows from point-wise curvature lower bound to Kato-type curvature lower bound. As an application, we prove that any compact three-dimensional non-collapsed strong Kato limit space is homeomorphic to a smooth manifold. Moreover, similar result also holds in higher dimension under stronger curvature condition. We also use the Ricci flow smoothing to study stability problem in scalar curvature geometry.

Students’ Inconsistency when Solving a Geometry Problem in Three-Dimensional Context

There is an issue regarding students' consistency in completing geometry assignments, as indicated by several research findings and assessments. Diagnosing how students integrate concepts in the conceptual design and understanding the reasons behind their inconsistency in completing assignments are the main focuses of this case study. This research was conducted with 58 high school students in Tanjungpandan, Indonesia. The data were obtained from students' answers to problems of the three initial levels of geometry thinking and retrospective reports about their answers. The data were analyzed based on three phases: the concept-eliciting and integrating phase, the relationship-eliciting phase, and the relationship-integrating phase. The study revealed that students' performance in geometry analysis aligned with the epistemological concept issue. Visual objects garnered the most attention from students, leading to their analysis techniques being primarily object-oriented. Some stages of property analysis were skipped, causing students to make claims about objects of thought when they should have been establishing relationships between properties to classify shapes through rigorous geometry analysis. Numerical computation remains an essential aspect of geometry analysis. The conceptual design has not yet reached the abstraction stage, resulting in experiments to solve problems not always yielding the correct solutions. In education, this highlights the need for a deep understanding of concept epistemology, efficient concept integration, and the cultivation of abstract thinking skills.

NG-DPSM: A neural green-distributed point source method for modelling ultrasonic field emission near fluid-solid interface using physics informed neural network

Distributed point source method (DPSM) is a collocation point-based semi-analytical method to model the scattered ultrasonic fields in isotropic and anisotropic materials. It requires the evaluation of Green's function solutions and its rigorous differentiation for complex geometry problems having a large set of distributed point sources. DPSM is much faster than the finite element method (FEM) however, it is analytically demanding as it requires differentiation of the Green's function. The intricacy associated with differentiation operation hinders the potential application of DPSM in large-scale automation and real-time structural health monitoring. The current paper introduces machine-learning-based strategies within DPSM and yields the proposed Neural Green-DPSM (NG-DPSM) framework. A physics-informed neural network (PINN) is incorporated in the DPSM framework for evaluating the Green's function and its gradients. To train the PINN, displacement Green's function solutions of wave propagation in isotropic solid is used as the governing physics. Once the PINN is trained for displacement Green's function solutions, the NG-DPSM framework leverages the trained network and its automatic differentiation capability to predict displacement and stress fields annihilating the rigorous differentiation of Green's functions. The accuracy and efficacy of NG-DPSM are demonstrated using two numerical experiments. First, the ultrasonic fields are evaluated for the problem geometry of a plate immersed in fluid excited at different angles of incidence. Further, the efficacy of the proposed approach is demonstrated for wave scattering through a circular hole in the plate. The results show a strong agreement between the ultrasonic fields computed using both NG-DPSM and conventional DPSM.

The comparison of general tips for mathematical problem solving generated by generative AI with those generated by human teachers

ABSTRACT In designing an intelligent tutoring system, a core area of the application of AI in education, tips from the system or virtual tutors are crucial in helping students solve difficult questions in disciplines like mathematics. Traditionally, the manual design of general tips by teachers is time-consuming and error-prone. Generative AI, like ChatGPT, presents a new channel for designing general tips. This study utilized prompt engineering and Chain of Thought to summarize general tips for given mathematical problems (one geometry problem and one algebra problem) and their solutions. A Turing test was conducted to compare ChatGPT-generated general tips with human-designed ones. Results from 121 human evaluators, each assessing 6 ChatGPT-generated and 6 human-designed general tips for each of two mathematical problems, showed that the average score for ChatGPT-generated tips is less than that of human-designed tips at a statistically significant level (p < 0.05), and Zero-Shot CoT achieved the best score. However, no evaluator could distinguish the tip types exactly. The average precision, recall and F-value of all ChatGPT-generated tips are less than 40%. AI-generated general tips can serve as a valuable reference for teachers to enhance efficiency and students’ mathematical learning.

Inversion formula for an integral geometry problem over surfaces of revolution

AbstractAn integral geometry problem is considered on a family of ‐dimensional surfaces of revolution whose vertices lie on a hyperplane and directions of symmetry axes are fixed and orthogonal to this plane, in . More precisely, the reconstruction of a function , , , from the integrals of the form extended over a chosen side of all surfaces of revolution of a given family is investigated. Unlike the usual Radon transform, the integrals considered here are not taken with respect to the surface area element. A Fourier slice identity and a backprojection‐type inversion formula are obtained with a method based on the Fourier and Hankel transforms. The reconstruction procedure and some analytical and numerical implementations of the obtained inversion formulas in the cases of and are provided.

Use the Auxiliary Projection Method to Solve Problems in Geometry

Geometry is a science with very wide applications in life. Everything around us is geometric shapes. Solving geometry problems helps us solve many real-life problems. In geometry there are many complex problems. But if a solution is found, many difficult problems will be solved quickly and easily. This article presents the auriliary projection method to solve many problems in geometry. The following data can be determined.

Open Problems in Geometry

Open Problems in Geometry