Particles sitting on a Penrose tiling graph refuse to follow standard statistical laws because the crystal's weird geometry forces them into a specific density.
April 24, 2026
Original Paper
A Nearest-Neighbor Hard-Core Model on a Penrose Graph
arXiv · 2604.21086
The Takeaway
Quasicrystals with a Penrose P3 tiling structure prevent particles from organizing into the expected phases of matter. Standard bipartite graphs usually allow for a specific type of phase coexistence, but this geometric layout caps particle density at precisely 0.549. The geometric arrangement of the atoms essentially overrides the fundamental statistical behavior of the particles themselves. This result proves that the packing of particles changes entirely when the underlying grid is a quasicrystal. Engineers could use these mathematical limits to design new materials that are physically incapable of reaching certain unstable states.
From the abstract
We prove that the maximal graph-density of an independent set in a Penrose P3 tiling considered as a planar non-directed graph is equal to $(57 - 25 \sqrt{5})/2 \approx 0.54915$ despite the fact that the graph is bipartite. Accordingly, the extreme Gibbs measure of the nearest-neighbor hard core particle model on this graph is unique for sufficiently large values of the particle activity. This invalidates a natural expectation to observe the coexistence of even and odd phases.