Who needs a supercomputer when you can calculate pi with a box of sewing needles? Happy Pi Day! March 14 is the date that otherwise rational people celebrate this irrational number, because 3/14 contains the first three digits of pi. And hey, pi deserves a day. By definition, it's the ratio of the circumference and diameter of a circle, but it shows up in all kinds of places that seem to have nothing to do with circles, from music to quantum mechanics. Pi is an infinitely long decimal number that never repeats.

Technological advancements in soccer have surged over the past decade, transforming aspects of the sport. Unlike binary rules, many soccer regulations, such as the "Offside Rule," rely on subjective interpretation rather than straightforward True or False criteria. The on-field referee holds ultimate authority in adjudicating these nuanced decisions. A significant breakthrough in soccer officiating is the Video Assistant Referee (V AR) system, leveraging a network of 20-30 cameras within stadiums to minimize human errors. V AR's operational scope typically encompasses 10-30 cameras, ensuring high decision accuracy but at a substantial cost. This report proposes an innovative approach to offside detection using a single camera, such as the broadcasting camera, to mitigate expenses associated with sophisticated technological setups.

Geometry Problem Solving (GPS), which is a classic and challenging math problem, has attracted much attention in recent years. It requires a solver to comprehensively understand both text and diagram, master essential geometry knowledge, and appropriately apply it in reasoning. However, existing works follow a paradigm of neural machine translation and only focus on enhancing the capability of encoders, which neglects the essential characteristics of human geometry reasoning. In this paper, inspired by dual-process theory, we propose a Dual-Reasoning Geometry Solver (DualGeoSolver) to simulate the dual-reasoning process of humans for GPS. Specifically, we construct two systems in DualGeoSolver, namely Knowledge System and Inference System. Knowledge System controls an implicit reasoning process, which is responsible for providing diagram information and geometry knowledge according to a step-wise reasoning goal generated by Inference System. Inference System conducts an explicit reasoning process, which specifies the goal in each reasoning step and applies the knowledge to generate program tokens for resolving it. The two systems carry out the above process iteratively, which behaves more in line with human cognition. We conduct extensive experiments on two benchmark datasets, GeoQA and GeoQA+. The results demonstrate the superiority of DualGeoSolver in both solving accuracy and robustness from explicitly modeling human reasoning process and knowledge application.

While static word embedding models are known to represent linguistic analogies as parallel lines in high-dimensional space, the underlying mechanism as to why they result in such geometric structures remains obscure. We find that an elementary contrastive-style method employed over distributional information performs competitively with popular word embedding models on analogy recovery tasks, while achieving dramatic speedups in training time. Further, we demonstrate that a contrastive loss is sufficient to create these parallel structures in word embeddings, and establish a precise relationship between the co-occurrence statistics and the geometric structure of the resulting word embeddings.

This paper introduces a novel approach to evaluating deep learning models' capacity for in-diagram logic interpretation. Leveraging the intriguing realm of visual illusions, we establish a unique dataset, InDL, designed to rigorously test and benchmark these models. Deep learning has witnessed remarkable progress in domains such as computer vision and natural language processing. However, models often stumble in tasks requiring logical reasoning due to their inherent 'black box' characteristics, which obscure the decision-making process. Our work presents a new lens to understand these models better by focusing on their handling of visual illusions -- a complex interplay of perception and logic. We utilize six classic geometric optical illusions to create a comparative framework between human and machine visual perception. This methodology offers a quantifiable measure to rank models, elucidating potential weaknesses and providing actionable insights for model improvements. Our experimental results affirm the efficacy of our benchmarking strategy, demonstrating its ability to effectively rank models based on their logic interpretation ability. As part of our commitment to reproducible research, the source code and datasets will be made publicly available at https://github.com/rabbit-magic-wh/InDL

It is our human nature to tinker with rules, as anyone who has argued over dubious interpretations of the game Monopoly will attest. Rule-breaking may hold the key to giving us an edge over machines that, for all their emerging capabilities, are tied to fixed directives. A close look at how mathematics comes to be demonstrates the power of departing from accepted convention, mathematician and author Junaid Mubeen writes in Mathematical Intelligence: A Story of Human Superiority Over Machines.How Humans have long been dreaming up concepts and creatures that exist outside our physical reality. The oldest known figurative art object, discovered in a cave in the Lone Valley of south-western Germany, is the Lion Man of Hohlenstein-Stadel, a chimeric figurine that is half-human, half-lion. Sculpted around 40,000 years ago for purposes unknown, the Lion Man is a product of pure human imagination.

We describe a prototype of a new experimental GeoGebra command and tool Discover that analyzes geometric figures for salient patterns, properties, and theorems. This tool is a basic implementation of automated discovery in elementary planar geometry. The paper focuses on the mathematical background of the implementation, as well as methods to avoid combinatorial explosion when storing the interesting properties of a geometric figure.

In Statistical Learning, the Vapnik-Chervonenkis (VC) dimension is an important combinatorial property of classifiers. To our knowledge, no theoretical results yet exist for the VC dimension of edited nearest-neighbour (1NN) classifiers with reference set of fixed size. Related theoretical results are scattered in the literature and their implications have not been made explicit. We collect some relevant results and use them to provide explicit lower and upper bounds for the VC dimension of 1NN classifiers with a prototype set of fixed size. We discuss the implications of these bounds for the size of training set needed to learn such a classifier to a given accuracy. Further, we provide a new lower bound for the two-dimensional case, based on a new geometrical argument.