Veritasium - The Infinite Pattern That Never Repeats

Short Summary

Key Points

• Kepler's conjecture (1611), that hexagonal close packing is the most efficient way to stack spheres, was only formally proven in 2017, about 400 years later.
• Penrose tilings use two shapes, a thick and a thin rhombus, or kites and darts, that tile the plane aperiodically, meaning the pattern never repeats.
• Hao Wang's conjecture that tilings are always periodic if possible was disproved by his student Robert Berger, who found 20,426 tiles that tile only non-periodically.
• Penrose tilings show near-fivefold symmetry, and the ratio of kites to darts approaches the golden ratio (about 1.618), suggesting the pattern cannot be periodic.
• Quasicrystals, theoretically modeled by Paul Steinhardt's team and independently discovered experimentally by Dan Shechtman, were a real material with fivefold symmetry that defied the long-held belief about crystals, and Shechtman won the 2011 Nobel Prize in Chemistry.

Details Worth Noting

• Platonic solids are the five regular polyhedra (tetrahedron, cube, octahedron, dodecahedron, icosahedron); Kepler used them as spacers between six known planetary spheres in his early solar system model.
• Kepler's 1611 pamphlet De Nive Sexangula (On the Six-Cornered Snowflake) asked why snowflakes always have six corners, and he speculated about water molecules stacking into hexagonal crystals.
• Penrose's kites-and-darts and thick-and-thin rhombus versions are aperiodic, and the video notes that two copies of a Penrose pattern overlaid produce a moiré pattern, with only small regions matching at any rotation.
• Quasicrystals were initially dismissed; Linus Pauling, a double Nobel Prize winner, said, "There are no quasicrystals, only quasi-scientists," and quasicrystals have since been explored for applications including non-stick cookware, electrical insulation, and ultra-durable steel.

Bottom Line

Some patterns can extend forever without repeating, and the video shows that such patterns, once thought impossible, exist both in mathematics and in real materials.