The angles a film has no choice about

A liquid’s surface has a tension, and the important thing about it for what follows is what it is not. It is not a skin with a grain, it has no preferred direction, and it pulls equally hard whichever way it is asked.

That single property fixes the geometry of every foam there has ever been, and it does so without any number from the liquid entering.

Two vectors sums, and two angles #

Where films meet, the tensions pulling along each of them must sum to zero, or the junction would accelerate. Each pull has the same magnitude, because the tension is the same in every film.

Three equal vectors in a plane sum to zero only when they are $120^\circ$ apart. There is no other arrangement: the condition $1 + 2\cos\theta = 0$ has one solution.

Four equal vectors in space sum to zero only in the tetrahedral arrangement, at $\arccos(-\tfrac13) = 109.4712^\circ$. Again, $1 + 3\cos\varphi = 0$ has one solution.

Those are the two junctions, and they are the only two. Plateau’s laws, stated from experiment in 1873 and proved as a theorem about area-minimising surfaces a century later, say exactly this: films meet three at a time along lines, and those lines meet four at a time at points, at those two angles and no others.

What is not allowed is worth naming. Four films meeting along a line is impossible, because four equal coplanar vectors could balance at many angles and the arrangement is unstable to any of them; such a junction, if it is somehow made, splits into two three-film lines within milliseconds. Three lines meeting at a point is impossible for the same reason.

That is why blowing bubbles into a container gives a structure with a very definite local geometry however chaotic the process, and why a foam’s junctions can be used to identify it as a foam without measuring anything.

There is a way of stating the whole first half in one sentence, and it is worth having because it explains why the result is so strong. The angles are what they are because the tension is isotropic and equal in every film, and for no other reason. No property of the liquid enters, no length enters, and nothing about how the foam was made enters. Anything that breaks one of those two conditions breaks the angles — a film under a shear that has not relaxed, or a junction between films of different surfactants — and where they hold, the angles hold exactly.

What the angles cost the shapes #

Requiring $120^\circ$ everywhere is a severe constraint on what a foam can look like, and it is the reason cells are not the shapes anybody would guess.

The pressure difference across a curved film falls as the radius grows, and that is what makes a foam’s cells curved rather than polyhedral. Two bubbles of different sizes cannot share a flat wall: the smaller one is at the higher pressure, so the film between them bulges into the larger. Only cells of exactly equal size can meet across flat faces — which is why an ideal foam is a problem about equal cells and a real one never quite is.

In two dimensions, a foam of cells all the same size can meet the requirement exactly, with hexagons: the angles of a regular hexagon are $120^\circ$. So a monodisperse two-dimensional foam is a honeycomb, and it is a honeycomb because of the tension balance rather than for any reason about efficiency.

Once the cells differ in size the hexagons cannot survive. A cell with more pressure inside it than its neighbour bulges into it — the smaller bubble has the higher pressure — so the walls curve, and a curved wall meeting two others at $120^\circ$ means the cell has fewer or more than six sides — the same competition that makes a small bubble feed a large one, read as a geometry. Cells with fewer than six sides have convex walls and shrink; cells with more grow. That is the whole of foam coarsening, and it follows from the angles rather than from anything about drainage.

In three dimensions nothing meets the requirement exactly. There is no polyhedron that tiles space with all its dihedral angles at $109.47^\circ$, so a three-dimensional foam is always slightly strained, and finding the arrangement that minimises the strain is a genuinely hard problem that was open for a century.

The problem the angles made hard for a century #

The three-dimensional version of the honeycomb question is a good example of a problem that is easy to state, obviously answerable, and was not answered for a hundred years.

What makes the least-area question hard is that the space of possibilities is not a list. Counting arrangements works when the objects are discrete and countable; here the candidates form a continuum of shapes, each deformable into its neighbours, and a search over them is a search over a function space rather than over a set. That is why the problem resisted for a century after it was posed and why the answer, when it came, came with a computer attached.

Kelvin asked in 1887 what arrangement of equal-volume cells fills space with the least total area, and proposed one: a slightly curved truncated octahedron, whose faces meet at very nearly the required angles. His structure held the record for a hundred and six years.

In 1993 Weaire and Phelan found a better one, by about a third of a per cent, using a computer package written to model foams. It uses cells of two different shapes with the same volume, and its total area is $0.3$ per cent below Kelvin’s — a margin that no amount of looking at Kelvin’s structure would have suggested was available.

Two things about that are worth carrying. The improvement is small enough that it is not visible and large enough that it is definitely there; and it was found by a method that did not exist when the question was asked, which is the usual reason a hundred-year-old geometry problem finally moves.

Neither structure is proved optimal, and nobody expects a proof soon. The two-dimensional version — that the hexagonal honeycomb is the least-perimeter partition of the plane into equal areas — was conjectured by the ancients, assumed by everybody, and proved only in 1999.

A number makes the constraint’s severity concrete. In two dimensions the requirement is that three walls meet at $120^\circ$; a square lattice of cells has walls meeting at $90^\circ$ and is therefore not merely a worse arrangement but an impossible one — a square foam rearranges into a hexagonal one within a fraction of a second of being made, and the rearrangement is a topological event rather than a relaxation.

The film between two rings #

The second half of the subject is about what happens when the boundary is fixed and the film has to find its own shape.

Dip two coaxial rings into soap solution and pull them apart. The film that spans them is a surface of revolution whose area is stationary, and solving that condition gives a catenary rotated about the axis — a catenoid, the only minimal surface of revolution there is apart from the plane.

Its equation, for rings of radius one separated by $2h$, is $a\cosh(h/a) = 1$. That has two roots, one root, or none, and the whole behaviour of the film is contained in that fact.

Two roots means two catenoids. The one with the larger $a$ is fat and stable; the one with the smaller $a$ is narrow-waisted and is a saddle point rather than a minimum, so it is never seen. As the rings separate the two solutions approach each other, and at $h = 0.6627$ ring radii they merge and vanish.

There is no catenoid past that separation. The film does not thin gradually and break; the shape it has been holding stops existing, and it snaps.

Being wrong and surviving #

The interesting number is not the one where the film breaks. It is the one where the film stops being the best answer and does not break.

The alternative to the catenoid is the pair of flat discs — the Goldschmidt solution, which is not connected but which does span the boundary. Its area is $2\pi r^2$ regardless of separation.

The catenoid’s area grows with separation and crosses the discs’ at $h = 0.5293$. From there to $0.6627$ the film is larger in area than the alternative, and it persists.

The reason is that “minimal surface” means stationary rather than least. Getting from the catenoid to the two discs requires the film to pinch through zero radius at the waist, and every intermediate configuration has a larger area than either end. There is a barrier, and the film has no way over it.

That gap of $25$ per cent is the whole of what metastability means, made concrete: the film is not in the lowest state and cannot reach the lowest state, so it stays where it is until the state it is in stops existing.

The same shape of argument runs wherever a system has to deform continuously between two configurations. A new phase has to climb a barrier before it can grow; a supercooled liquid stays liquid; a supersaturated vapour stays a vapour. In each case the winning state is available and the route to it is not.

What is being minimised, and why area #

Nothing so far has said why a film seeks least area, and the reason is worth stating because it separates two things that are often run together.

Surface tension is an energy per unit area. A film of area $A$ costs $2\gamma A$ — twice, because a film has two surfaces — so minimising energy means minimising area, and the shape a film adopts is the answer to a purely geometrical question with the material appearing only as an overall factor.

That is why the angles have no liquid in them. Multiply $\gamma$ by ten and every energy in the problem multiplies by ten, and every shape is unchanged.

It is also why soap films were used as analogue computers. Before the numerical era, the shape of a minimal surface spanning an awkward boundary was found by bending a wire and dipping it — and the same apparatus solves other problems that reduce to the same equation, including the torsion of a shaft of arbitrary cross-section, where the shape of a film over the section gives the stress distribution directly — one equation, two subjects wearing one coat.

What a film does that a calculation cannot #

There is one more reason the subject stayed experimental for so long, and it is about the difference between finding a solution and finding the right one.

A landscape with a barrier in it explains why the film and the calculation agree about which answers exist and disagree about which one appears. A numerical search that walks downhill finds whichever minimum it started nearest; a soap film does exactly the same thing, settling into the local minimum it happens to reach. Neither is finding the global answer, and the fact that they usually agree is a statement about the landscape rather than about either method.

A wire frame with a complicated shape usually has several minimal surfaces spanning it — a cube frame has more than a dozen, differing in how the internal films are arranged, and dipping the same frame twice can give different ones. Every one of them satisfies the equations; only one has the least area.

That is not a defect of the soap film. It is a property of the problem, and a numerical minimiser has exactly the same difficulty: started from a different guess, it converges to a different surface. Where the film is better than the computation is that it explores physically — the surface it settles into is one the geometry can actually be reached, which is not something a numerical scheme guarantees.

Where the film is worse is that it cannot be asked which of its answers is smallest. Plateau’s own programme was to catalogue the solutions by dipping frames, and the cataloguing was reliable while the ranking was not.

The modern practice uses both. A film shows which topologies exist and a computation refines and ranks them, which is roughly how the Weaire–Phelan structure was arrived at and how the double bubble’s shape was settled.

The man who could not see them #

The laws on this page are entirely about angles — about what a junction looks like — and the person who established them spent the last forty years of his working life unable to see anything at all.

Joseph Plateau was a Belgian physicist who, as a student in 1829, stared directly at the sun for twenty-five seconds in the course of an investigation into afterimages. His sight declined over the following years and he was completely blind by the mid-1840s. Whether the sun caused it is now doubted — an inflammatory eye disease is the more likely explanation — but the story was his own account and the timing is what it is.

He went on working for another four decades. His son, his son-in-law and his colleagues built the wire frames, dipped them, manipulated the films and described what they saw; Plateau directed the experiments, formulated the questions and worked out the theory. The two-volume treatise of 1873 that founded the subject was assembled that way.

It contains a great deal more than the junction laws. The instability that breaks a falling stream into drops is in it, and carries his name; so is the systematic catalogue of the surfaces that span various wire frames, which is the work that established how many solutions such a frame typically has.

It is worth noticing what kind of results those are. They are geometrical, they are stated as angles and shapes, and they were arrived at by someone who received all of it as spoken description. That is possible because the content is a relation rather than an appearance: three equal vectors summing to zero is a statement that can be reasoned about without being looked at, and the experiment’s role is to confirm that the films really do obey it.

The laws remained empirical for a century. Proving that the two junctions Plateau described are the only singularities an area-minimising surface can have took machinery — geometric measure theory — that did not exist until the 1960s, and the theorem was established by Jean Taylor in 1976. A hundred and three years between the observation and the proof, on a question that had never been in serious doubt.

Plateau’s other lasting invention has nothing to do with liquids. In 1832 he built a spinning disc with slots and a sequence of drawings around its rim which, viewed in a mirror, produced the illusion of continuous motion — one of the direct ancestors of cinema, and made by the same person, before the blindness, in the course of the same investigation into how the eye retains an image.

The surfaces that are used as structures #

The remark that soap films were used as analogue computers is understated: they were used to design buildings, and some of those buildings are still standing.

A membrane roof — a fabric or cable net held in tension over a set of masts and anchors — has to be in uniform tension everywhere, or it wrinkles where the tension falls and tears where it concentrates. A surface in uniform tension with no pressure across it is a surface of zero mean curvature, which is a minimal surface. So the problem of finding the shape of such a roof is exactly the problem of finding the minimal surface spanning a given boundary, and there is no analytic answer for any boundary an architect would choose.

Frei Otto’s institute in Stuttgart solved it by dipping. Wire models of the intended boundary — the mast tops, the anchor points, the edge cables — were dipped in soap solution, and the resulting film was the roof’s shape.

The difficulty is that a soap film lasts seconds and a building has to be dimensioned to millimetres. Otto’s group developed stereo-photogrammetry for the purpose: photograph the film from two known positions, measure the images, and reconstruct the surface numerically before it breaks. The German pavilion at Expo 67 and the roof of the Munich Olympic stadium of 1972 were both formed that way, and the Munich roof was among the first buildings whose geometry was fixed by computation from a physical model rather than drawn.

The method has been superseded by numerical form-finding, which does the same minimisation directly and does not burst. What it established is the vocabulary: every tensile roof built since is a minimal surface or a deliberate departure from one, and the departures are understood as such.

The same surfaces turn up where nothing was designed at all. A family of minimal surfaces that repeat periodically in all three directions — the ones found by Schwarz in the 1860s and the gyroid found a century later — appear spontaneously in block copolymer melts, in the folded membranes of cell organelles, and in the chitin of some butterfly wing scales. In the last case the periodicity is a few hundred nanometres, which makes the structure a three-dimensional photonic crystal, and the colour of the wing is the band gap of a minimal surface that assembled itself.

Where the model runs out #

A real film has thickness, and the thickness is not passive. The two surfaces interact through the liquid between them, and below about a hundred nanometres that interaction becomes comparable with the surface energy — which is what stabilises the very thin black films and what makes a soap film’s lifetime a question about physical chemistry rather than about geometry. It is also where the attraction that needs no charge begins to matter, since the two surfaces feel one another through it.

Rupture is a competition of a different kind and nothing in the area argument predicts it. An instability selects a wavelength — the disturbance that grows fastest — and once a film has thinned to tens of nanometres the van der Waals attraction between its two surfaces takes over from surface tension entirely. The angles are exact and say nothing whatever about how long the film holding them will last.

Gravity is ignored and it is not always negligible. The comparison is between the surface energy and the weight of the liquid, which gives a length of a couple of millimetres for soap solution. A film much larger than that sags, drains, and thins at the top, and the shapes here are exact only for films small compared with that length or for the instant after they are made.

The junction angles are exact and the junctions are not lines. A real three-film junction is a Plateau border — a channel of liquid with three concave walls — of a width set by how much liquid the foam holds. The angles are unchanged, but the picture of a line has a thickness, and it is through those channels that a foam drains — at a rate the fourth power of their width decides.

And the catenoid analysis assumes the film has time to find its shape. Pull the rings apart quickly and the film is not in equilibrium at any moment; it can break before the critical separation, or survive past it briefly. Every number in the second half of this essay is a statement about a quasi-static process.

Comparing surface tension against gravity gives the length above which none of these shapes survives unchanged. Below the capillary length — about 2.7 mm for water — surface tension decides everything and the angles are exact; above it gravity drains the films, thins the vertical ones and thickens the horizontal ones, and the junctions bend. A foam in a glass obeys the rules at the top and sags out of them at the bottom.

The ladder from here #

Later rungs on this anchor: Plateau borders and foam drainage, where the geometry acquires a length; foam coarsening, in which the angles fix which cells grow and which shrink; the Kelvin problem and the Weaire–Phelan structure, which is the arrangement of equal-volume cells that comes closest to the impossible angles in three dimensions; and the double bubble, whose least-area shape for two prescribed volumes was conjectured from soap films and proved a century later.

The neighbouring ladders are the skin that is not a skin, where the tension is defined, the small bubble blows up the big one, which is the pressure that curves the walls, and the surface that pulls toward the stronger side, where a difference in tension rather than an equality is what does the work.