Updated 21 June 2019

How would bubbles pack together, to give the least possible amount of surface film between them? This question has not been answered yet.

In the nineteenth century Lord Kelvin considered the simplest case of identically shaped bubbles. He discovered a packing of 14-sided bubbles or tetrakaidecahedra, based on the truncated octahedron but with slightly curved faces. The faces have to curve a little so that they meet at the correct angle (known as the tetrahdral angle) to minimise their surface energy.

Truncated octahedron |
A group of Kelvin bubbles |

Kelvin conjectured that this was the most efficient solution, but he could neither prove nor disprove his conjecture.

A hundred years went by before Denis Weaire and Robert Phelan disproved the Kelvin conjecture by finding a better solution. They asked the slightly less strict question, how would equal sized, *but not necessarily identically shaped*, bubbles pack? The best answer they could find was a mixture of 12- and 14-sided bubbles, also having slightly curved surfaces.

In the Weaire-Phelan packing, two pentagonal dodecahedra (12-sided) and six tetrakaidecahedra (of a rather different shape from Kelvin's) form a translation unit with a lattice periodicity which is simple cubic. The illustration shows the dodecahedra as wire frames and the tetrakaidecahedra as solid. The dodecahedra do not touch each other, but are entirely surrounded by tetrakaidecahedra.

Dodecahedron Tetrakaidecahedron |
Translation unit |

The dodecahedron has pentagonal faces with sides of unequal length, so it is "pentagonal" to distinguish it from the rhombic dodecahedron and "irregular" to distinguish it from the regular (Platonic) dodecahedron.

Another way to visualise the packing is to note that the hexagonal faces of the tetrakaidecahedra are truly flat and meet to stack them in long rods. A set of rods lies parallel to each of the three orthogonal axes of space. These three sets of rods interlace, touching on their eight smaller pentagonal faces but leaving dodecahedral voids between the larger pentagons.

More about bubble packings can be found in Weaire, D., "Froths, Foams and Heady Geometry", *New Scientist*, 21 May 1994.

Since then no better solution has been found, though at least one has come very close. Actual Weaire-Phelan bubbles also resisted experimental demonstration for more than a decade, but a foam was eventually made. Proof of the optimal solution remains as elusive as ever, we still do not know if this is it.

We can flatten out all the bubble faces to form conventional polyhedra. These faces lie orthogonal to the lines joining adjacent cell centroids, and to retain equal cell volumes some vertices must move very slightly. The geometry of the resulting spacefilling is in all other respects identical to Weaire and Phelan's: the differences are, in any case, barely noticeable.

The tetrakaidecahedron may now be seen an example of the Goldberg polyhedron, which helps to distinguish it from the one associated with Kelvin, the truncated octahedron. The dodecahedron has the same symmetry seen in crystals of iron pyrites and so is sometimes called a pyritohedron, although its angles are different.

Vertex coordinates and printable nets for card models of the two polyhedra are given below. The coordinate scale gives a lattice period, i.e. the size of the repeating pattern, of 20.

x y z 3.14980 0 6.29961 -3.14980 0 6.29961 4.19974 4.19974 4.19974 0 6.29961 3.14980 -4.19974 4.19974 4.19974 -4.19974 -4.19974 4.19974 0 -6.29961 3.14980 4.19974 -4.19974 4.19974 6.29961 3.14980 0 -6.29961 3.14980 0 -6.29961 -3.14980 0 6.29961 -3.14980 0 4.19974 4.19974 -4.19974 0 6.29961 -3.14980 -4.19974 4.19974 -4.19974 -4.19974 -4.19974 -4.19974 0 -6.29961 -3.14980 4.19974 -4.19974 -4.19974 3.14980 0 -6.29961 -3.14980 0 -6.29961 |
x y z 3.14980 3.70039 5 -3.14980 3.70039 5 -5 0 5 -3.14980 -3.70039 5 3.14980 -3.70039 5 5 0 5 4.19974 5.80026 0.80026 -4.19974 5.80026 0.80026 -6.85020 0 1.29961 -4.19974 -5.80026 0.80026 4.19974 -5.80026 0.80026 6.85020 0 1.29961 5.80026 4.19974 -0.80026 0 6.85020 -1.29961 -5.80026 4.19974 -0.80026 -5.80026 -4.19974 -0.80026 0 -6.85020 -1.29961 5.80026 -4.19974 -0.80026 3.70039 3.14980 -5 0 5 -5 -3.70039 3.14980 -5 -3.70039 -3.14980 -5 0 -5 -5 3.70039 -3.14980 -5 |

The polyhedral spacefilling is related to what crystallographers call a Frank-Kasper phase. These phases are characterised by the arrangement of atoms (here represented by the cell centroids) at the vertices of planar arrays of triangles and hexagons.

If you place a small sphere at each atom in the array and expand them all together until they meet to form a boundary, these boundaries will be flat faces. If you keep expanding the spheres until no intervening space is left, the flat faces all join up and the spheres become polyhedra known as Voronoi cells. The Weaire-Phelan polyhedra are the Voronoi cells of an A15 Frank-Kasper phase.

Equally expanding spheres meet in a plane surface (shaded). |

Since it has flat faces, the polyheral spacefilling is better suited to making card models than the real bubbles are. The scale of the nets provided above, printed on A4 sheets, gives a lattice period of 12.5 cms (5 ins).

The polyhedra are also easier to design for 3D printing than the bubbles! Tip: it can be worth making them hollow with a thick wall, so as to use less material. This saves cost and time, especially if you are printing off large numbers, and also makes them lighter.