No matter where you put six dots on a ball, you can always pair them up using three circles that never touch each other.
April 2, 2026
Original Paper
Any six points on the Riemann sphere can be split into three pairs by a triple of disjoint discs
arXiv · 2604.00351
The Takeaway
This is a surprisingly simple geometric rule that was previously unproven. It shows that for any six distinguished points on a globe, you can always find three disjoint 'discs' that each contain exactly two points, proving that a specific numerical method used to map complex 3D surfaces is universally valid.
From the abstract
We prove that for any six points on the Riemann sphere there exist three disjoint closed (or open) discs, each of which contains exactly two of the six distinguished points. This statement shows that recently proposed method to numerically evaluate Kleinian hyperelliptic functions of genus 2 is applicable to any complex curve of genus 2.