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 based on the truncated octahedron but with slightly curved faces and conjectured that this was the most efficient solution. A hundred years went by before Weaire and Phelan found a better one. They asked the slightly less strict question, how would *equal sized*, but not necessarily identically shaped, bubbles pack? The best answer they came up with was a mixture of 12- and 14-sided bubbles having slightly curved surfaces. We still do not know is this is the optimal solution.

In the Weaire-Phelan packing, two irregular pentagonal dodecahedra (12-sided) and six tetrakaidecahedra (14-sided) 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.

The tetrakaidecahedron is also called the "Goldberg polyhedron" to distinguish it from Lord Kelvin's tetrakaidecahedron (which packs on its own and is approximated by the regular truncated octahedron). 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 meet to stack them in 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.

We can flatten out the faces to form proper 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.

This spacefilling is an example of 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.

By popular request, here are vertex coordinates for the two polyhedra:

## Dodecahedron3.1498 0 6.2996 -3.1498 0 6.2996 4.1997 4.1997 4.1997 0 6.2996 3.1498 -4.1997 4.1997 4.1997 -4.1997 -4.1997 4.1997 0 -6.2996 3.1498 4.1997 -4.1997 4.1997 6.2996 3.1498 0 -6.2996 3.1498 0 -6.2996 -3.1498 0 6.2996 -3.1498 0 4.1997 4.1997 -4.1997 0 6.2996 -3.1498 -4.1997 4.1997 -4.1997 -4.1997 -4.1997 -4.1997 0 -6.2996 -3.1498 4.1997 -4.1997 -4.1997 3.1498 0 -6.2996 -3.1498 0 -6.2996 |
## Tetrakaidecahedron3.14980 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 |

Since it has flat faces, the Frank-Kasper phase is better suited to making card models. Printable nets for these are now avaiable for download. The scale of this model gives a lattice period, i.e. the size of the repeating pattern, of 12.5 cms (5 ins).

Updated 4 March 2019