Mathematicians proved that a shape can fit together perfectly for a trillion layers and yet still be mathematically impossible to tile a floor with.
March 31, 2026
Original Paper
Unboundedness of the Heesch Number for Hyperbolic Convex Monotiles
arXiv · 2603.27827
The Takeaway
This resolves the 'Heesch problem' by proving that in curved space, there is no limit to how many layers a non-tiling shape can 'fake' before hitting a mathematical dead end. It means a tile could appear to be a perfect fit at any scale a human could ever check, only to be eventually proven impossible to complete.
From the abstract
We provide a resolution of the Heesch problem for homogeneous (also known as semi-regular) tilings, and as a corollary, for tilings by convex monotiles in the hyperbolic plane. We also provide the first known example of weakly aperiodic convex monotiles arising from the dual of homogeneous tilings.