geometry

A blog post exploring John Coltrane's mysterious tone circle diagram — a chromatic-scale circle of fifths — and how it encodes the chord structures behind 'Giant Steps', rebuilt as an interactive web widget.

Mathigon 介绍一个关于数学折纸的互动学习页面，涵盖柏拉图立体、阿基米德立体等多面体主题，并链接到 Polypad 上的 3D 模型探索工具。

This paper studies how transformer intermediate residual states become specialized to their own final output states, finding that directional alignment and endpoint rank improve even when Euclidean distance barely changes across six pretrained language models. It offers a high-dimensional model separating norm, alignment, and endpoint geometry, proving that straight-path convergence cannot introduce new competitors, and linking residual geometry to output token rankings.

The article explores mathematical billiards, including varieties like inner billiards, and discusses using computer experiments to study trajectories in dynamical systems.

This paper decomposes transformer representation updates into parallel and perpendicular components to study evolution geometry, linking it to editing robustness, compression diagnosis, and training improvements.

This 1832 paper by Necker describes an optical phenomenon where a two-dimensional drawing of a geometrical solid can be perceived as a three-dimensional cube that flips orientation.

The paper introduces the 'direction of ignorance' in LLMs' unembedding geometry, which encodes the training corpus's unigram distribution and acts as a Bayesian prior. It demonstrates how this prior is tempered based on context informativeness, providing insights into model behavior.

The article explains Carl Sagan's analogy of how higher-dimensional beings perceive our 3D world, using geometric logic to illustrate concepts that challenge human perception of reality.

This article improves the lower bound for the smallest square that can pack 17 unit squares, achieving a new value of 4.5058, building on previous work and incorporating AI-assisted methods.

This paper introduces an open-source benchmark for evaluating diagrammatic reasoning in geometry, revealing that foundation models like GPT and Claude struggle to produce faithful diagrams despite their problem-solving abilities.

A detailed write-up on solving an OSINT challenge to geolocate an island from a photo using geometric analysis, CUDA programming, and OpenStreetMap data filtering.

A new mathematical theory is reviving the field of enumerative geometry, allowing mathematicians to solve ancient counting problems in various number systems.

A mathematics animation by Daniel Piker shows dots following paths and returning to their starting points in 3.5 hours, resembling ants walking.

This paper introduces a continuous metric field framework trained by a single causal contrastive loss that unifies geometric structure discovery from robot navigation to black hole emergence, demonstrating zero-shot generalization across dimensions.

Presents ArchEGraph, a large-scale graph dataset for building energy modeling with aligned geometry, topology, weather, and thermal loads, along with benchmark tasks for graph reconstruction and load prediction.

A tweet promoting the book 'The Shape of Data: Geometry-Based Machine Learning and Data Analysis in R' by Colleen M. Farrelly and Yaé Ulrich Gaba, highlighting its practical approach to using geometry and topology in data science.

This paper introduces the Expert Subspace Separation Index (ESSI) to disentangle route coherence, candidate quality, and candidate-by-context interaction in sparse Mixture-of-Experts language models, revealing a pattern of coherent overlap where routing selects token-relevant experts from a shared geometric neighborhood yet multi-expert computation remains useful.

This preregistered study tests whether holonomy (a geometric measure) concentrates on active SAE feature planes in the Gemma 2 2B language model. Contrary to the semantic-concentration prediction, active-feature planes carried less holonomy than matched mixed-feature controls, resulting in a narrow operational reversal with the underlying cause remaining open.

This paper proposes a geometric analysis of transformer residual streams across depth, using relative displacement and orthogonal Procrustes analysis to reveal structured regularities in six instruction-tuned models on code generation and translation tasks.

This article examines the equivalence between the component and geometric definitions of the vector dot product in Euclidean space, providing a geometric proof using the law of cosines and a projection proof using orthonormal basis vectors.