The geometry of empty space dictates every property of matter.
Two-fold symmetry: the highway. Energy flows freely across flat planes. Copper, silver, gold, the best conductors on Earth, all share this geometry. Their atoms arrange in face-centered cubic lattices with 12 nearest neighbors, all products of 2 and 3.
Three-fold symmetry: the wall. Energy gets trapped. Diamond, the hardest natural material, has four-fold coordination, a pure power of 2, with deep tetrahedral voids that lock everything in place. Insulators live here.
And the crucial test: we ran this against every element on the periodic table with a known crystal structure. 133 elements. Every single coordination number is a product of 2 and 3. No fives. No sevens. No primes above three. Zero exceptions.
Whether a material conducts electricity, how it bends, whether it's brittle or ductile, its band gap: all written in the geometry of the voids between atoms.
Every material property you've ever measured is a consequence of empty space shaped by two numbers.