geometry

A critique of the AI tutor Koji, highlighting flaws in its math teaching approach, such as allowing students to fumble without guidance and missing key conceptual explanations.

A pictorial introduction to differential geometry, showing how Maxwell's equations emerge from geometric concepts, based on a 2017 arXiv paper.

Geomatic is a command-driven geometry studio that leverages automatic differentiation for interactive geometric operations.

This paper introduces LOES (Layer-wise Optimal Embedding Selection) and GeoReg (Geometric Regularization Loss), methods that select and fuse task-relevant intermediate layers from deep models to improve transfer learning performance, demonstrating consistent gains across architectures and modalities.

A webpage presenting known optimal packings of unit squares into a larger square, with interactive SVG diagrams for various numbers of squares.

A digital reproduction of Oliver Byrne's 1847 colorful edition of Euclid's Elements, featuring interactive diagrams, posters, and puzzles.

This paper introduces the Representation Gap, a metric for neural network generalization error with better asymptotic dynamics. Using a geometric perspective and optimal quantization theory, the authors show it is governed by the intrinsic dimension of the task, and verify this empirically on synthetic and realistic datasets.

OpenAI claims its unreleased reasoning model has solved the 80-year-old planar unit distance problem in mathematics, producing an original proof that outperforms traditional grid-based arrangements.

StampFormer is a physics-guided deep learning framework that fuses geometry and material properties to predict FEA outcomes for sheet metal stamping in under a second, achieving high fidelity with less than 8.5% relative error.

This paper investigates whether fMRI representations from different subjects' visual cortices can be aligned using unsupervised geometric methods, finding evidence for approximately isometric structure across individuals, extending the Platonic Representation Hypothesis to human brains.

This paper uses shape analysis tools to characterize how different data augmentation strategies reshape the geometry of neural network representations, finding that augmentation strength and type lead to distinct, well-behaved trajectories in shape space.

This paper introduces a unified geometric framework showing that weighted InfoNCE objectives can be interpreted as Distance Geometry Problems, providing exact characterizations of optimal embeddings for supervised and weakly supervised contrastive learning methods and revealing when such embeddings are geometrically realizable, degenerate, or inconsistent.

The article presents an elegant geometric proof for Buffon's Needle problem by extending the concept to curved 'noodles' and using a circle to determine the probability constant, avoiding complex integrals.

Researchers introduce GeoRepEval, a framework to evaluate LLM robustness across equivalent geometric problem representations (Euclidean, coordinate, vector). Testing 11 LLMs on 158 geometry problems, they find accuracy gaps up to 14 percentage points based solely on representation choice, with vector formulations being a consistent failure point.

RDP-LoRA uses geometric trajectory analysis and the Ramer-Douglas-Peucker algorithm to automatically select the most impactful layers for parameter-efficient fine-tuning, outperforming full-layer and random LoRA baselines.

Researcher analyzes LLM internal representations across 8 languages and multiple models, finding that concept thinking occurs in geometric space in middle transformer layers independent of input language, supporting a universal deep structure hypothesis similar to Chomsky's theory rather than Sapir-Whorf linguistic relativism.