Geometry vs Binary: Unsolved Mathematics — unsolved problems in computational geometry 2025 — E8 Intelligence Research

## Binary Failure → Geometry Solution **ANALYSIS:** **1. Moser's Worm Problem** (from "Jedes ungelöste Geometrieproblem, das einfach klingt") - **Binary/statistical failure:** The problem asks for the minimal area shape that can cover any curve of length 1. Conventional approaches try to optimize via discretization — sampling thousands of worm shapes, running gradient descent on polygon vertices. This hits a wall because the space of *all continuous curves* is uncountably infinite; no finite sample captures the worst-case worm. Neural nets overfit to sampled curves and miss pathological spirals or fractal-like worms. - **Geometric lattice success:** The problem is fundamentally about *covering* and *projection*. A root lattice framework treats the worm as a vector in a normed space, and the covering problem becomes a question of lattice packing and Voronoi cells. The minimal cover is tied to the *covering radius* of the lattice — a purely geometric invariant. Instead of sampling Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com