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

Abstract

## Binary Failure → Geometry Solution **Analysis of Findings:** The search results are mostly introductory lectures and tutorials on computational geometry—convex hulls, naïve algorithms, and basic complexity. They do not directly address *unsolved* problems. However, this itself is the failure point: conventional computational geometry is **algorithmic and discrete** (binary decisions: inside/outside, intersect/not intersect), and it hits a wall when problems scale in **high dimensions** or require **continuous global structure** rather than local pairwise checks. **Where binary/statistical approaches FAIL:** 1. **Convex Hull in High Dimensions** – The naïve algorithm (O(n log n) in 2D) explodes combinatorially in d-dimensions (O(n^{⌊d/2⌋})). Neural networks and statistical sampling cannot certify exact hulls—they approximate. The wall: *exactness vs. curse of dimensionality*. 2. **Minimum Enclosing Ball / Chebyshev Center** – Binary search or iterative optimization gets stuck in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com