Printable (PDF) nets for models: - Dodecahedron - Tetrakaidecahedron - |
How would bubbles pack together, to give the least possible amount of surface film between them? This question has not been answered yet; a packing found by Weaire and Phelan is the nearest we have come. They looked for answers to the slightly simpler question, how would equal sized (but not necessarily identically shaped) bubbles pack? The best answer they came up with, which might or might not be the best possible one, was a mixture of 12- and 14-sided bubbles having slightly curved surfaces.
In the 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.
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: - Dodecahedron - Tetrakaidecahedron - The scale of this model gives a lattice period, i.e. the size of the repeating pattern, of 12.5 cms (5 ins).
Updated 18 Feb 2006