This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. As part of model development, we evaluate our models on open research problems. We expanded these evaluations after performance on our existing mathematical evaluations saturated. … Read more
Platonic Solids Platonic Solids are the most regular polyhedra: all faces are the same regular polygon, and they look the same at every vertex. The Greek philosopher Plato discovered that there are only five solids with these properties. He believed … Read more
Contents Transformer language models are an emerging technology that is gaining increasingly broad real-world use, for example in systems like GPT-3 , LaMDA , Codex , Meena , Gopher , and similar models. However, as these models scale, their open-endedness … Read more
Raph Levien, August 8, 2026 I have for many decades been fascinated by the prospect of a curve family better suited for interactive design than cubic Béziers. In that search, I have come to a new-found respect for those Béziers. … Read more
The Maya temple Tikal in Guatemala is about one day’s walk from Xultun, where researchers discovered mathematical formula scribbled on the walls.Credit: Kryssia Campos/Getty A mathematical formula inscribed on a wall at the Maya site of Xultun in Guatemala has … Read more
The mathematical secrets of Barcelona’s Sagrada Familia Mapping IgnoranceMapping Ignorance BlueskyTelegramMastodonWebsiteE-mailNextPreviousDateAuthorShareReplyPrint
I came across my 10 year old repo for project Euler solutions. (N.B! Euler spoilers onwards). Naturally it’s full of python files. One file stood out. It was just called problem15.txt. I pulled up the problem. I imagined getting this … Read more
Living on light is a dangerous game. Not only do the sun’s rays carry ultraviolet waves that can snap DNA strands and degrade molecules, but they also vary wildly in intensity. Plants must endure and thrive through soft morning light … Read more
Abstract:Artificial intelligence (AI) is the name popularly given to a broad spectrum of computer tools designed to perform increasingly complex cognitive tasks, including many that used to solely be the province of humans. As these tools become exponentially sophisticated and … Read more
The simulation hypothesis — the idea that our universe might be an artificial construct running on some advanced alien computer — has long captured the public imagination. Yet most arguments about it rest on intuition rather than clear definitions, and … Read more