Soap Films & Foam Physics
A 2D foam simulation based on Voronoi tessellation. Click to add bubbles and watch von Neumann coarsening: cells with fewer than 6 sides shrink, cells with more than 6 sides grow, and hexagonal cells remain stable.
About this lab
Plateau’s laws
In 1873, the Belgian physicist Joseph Plateau described the rules that govern soap film geometry, now called Plateau’s laws. In 2D foam (which is what you see when you look down at bubbles in a dish), these reduce to two rules: (1) three films always meet at a single vertex, and (2) the three angles at each vertex are all 120°. These rules are consequences of surface tension minimization — soap films locally minimize their total length, and three lines meeting at 120° is the configuration that achieves this. Any other arrangement is unstable and will spontaneously rearrange.
Von Neumann’s coarsening law
In 1952, John von Neumann derived a remarkably simple law for how bubbles in a 2D foam change size over time. If a bubble has n sides, its area changes at a rate proportional to (n − 6):
dA/dt = k(n - 6)
where k is a constant depending on surface tension and film mobility. Bubbles with fewer than 6 sides shrink. Bubbles with more than 6 sides grow. Hexagonal bubbles (n = 6) are the only stable shape — their area does not change. This is why foams coarsen over time: small bubbles vanish, large bubbles grow, and the average bubble size increases. The law is exact for ideal 2D foams and is a consequence of the 120° angle rule combined with the Gauss-Bonnet theorem.
T1 topological transitions
As a shrinking bubble loses area, some of its edges get very short. When an edge shrinks to zero length, a T1 transition occurs: the four cells meeting at what was an edge rearrange their connectivity. Two cells that were neighbors become separated, and two cells that were separated become neighbors. The edge effectively rotates 90°. This is the fundamental topological change in foam dynamics. When a bubble shrinks to nothing (a T2 transition), it vanishes entirely, and its neighbors fill the space.
Why hexagons?
The honeycomb conjecture, proved by Thomas Hales in 1999, states that the regular hexagonal tiling is the most efficient way to partition the plane into equal-area cells with minimum total perimeter. This is why natural foams tend toward hexagonal order, why basalt columns are hexagonal, and why bees build hexagonal honeycombs. The 120° angles of Plateau’s laws are exactly the interior angles of a regular hexagon, making the honeycomb the unique configuration that satisfies both Plateau’s laws and uniform cell size simultaneously.
The Weaire-Phelan structure
In three dimensions, the equivalent question — what partition of space into equal-volume cells minimizes total surface area? — is much harder. Lord Kelvin conjectured in 1887 that the answer was truncated octahedra. In 1993, Denis Weaire and Robert Phelan found a structure with 0.3% less surface area, using two different cell shapes (an irregular dodecahedron and a tetrakaidecahedron with hexagonal faces). The Weaire-Phelan structure inspired the design of the Beijing National Aquatics Center (“Water Cube”) for the 2008 Olympics. Whether it is truly optimal remains an open problem.