Random Surfaces from Stacking Cubes: A Visual Journey (web app slides; 1920×1080)

Abstract. How does a corner of a crystal (like sugar cube) get its rounded shape? This seemingly simple question leads to beautiful mathematics at the intersection of probability and geometry.

We explore random lozenge tilings — ways of covering regions with diamond-shaped tiles chosen uniformly at random from astronomically many possibilities. These tilings can be viewed as discrete 3D surfaces built from unit cubes, and the central question is: what does a “typical” random surface look like? The answer reveals a striking phenomenon: as the system grows large, randomness gives way to order, and the random surface concentrates around a deterministic “limit shape,” with sharp boundaries separating frozen crystalline regions from disordered liquid regions — a phase transition you can see with your eyes.

This probabilistic story connects to surprising areas of mathematics: algebraic combinatorics (symmetric functions), algebraic geometry (limit shapes are algebraic curves), statistical mechanics (exactly solvable models), and operations research (Markov chain Monte Carlo, coupling from the past for perfect sampling). At the same time, it provides a rich source of visual inspiration which, naturally, can even be 3D printed.

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