Kelvin tetrakaidecahedron

The Kelvin tetrakaidecahedron and its construction by truncation of an octahedron by a cube. The edges of the tetrakaidecahedron are one third as long as the edges of the octahedron. 

Of the 14 faces, 6 have four edges and 8 have six edges. The average number of edges per face is (6 x 4 + 8 x 6)/14 = 5 /7 This is very close to the results of experiments on ft brass, vegetable cells, and soap bubbles, as shown in Figure 1.9. For the Kelvin tetrakaidecahedron the ratio of surface area to that of a sphere of the same volume is 1.099. Most other shapes have much higher ratios. 

Waire and Phelan report that space filling is 0.3% more efficient with an array of of six polyhedra with 14 faces and two polyhedra with 12 faces than with the Kelvin tetrakaidecahedron. (This calculation allows faces in each to be curved.) The 14-faced polyhedra have 12 pentagonal and 2 hexagonal faces, while the 12-faced polyhedra have distorted pentagons for faces. The average number of faces per polyhedra = (6 x 14 + 2 x 12)/8 = 13.5. 

At high shear rates in some systems, the onions become large and very monodisperse in size, and they then order into a macrocrystalline packing. At rest, it is clear that the onions are not spherical, but polyhedral, because they must fill space. In the perfectly ordered macrocrystalline state, the typical shape of the space-filling onions appears to be that of the Kelvin tetrakaidecahedron, which is a model structure for liquid foams (see Section 9.5.1). These well-defined MLVs might be important as encapsulants in the pharmaceutical or cosmetics industries (Roux and Diat 1992). 

These local stmctural rules make it impossible to constmct a regular, periodic, polyhedral foam from a single polyhedron. No known polyhedral shape that can be packed to fiU space simultaneously satisfies the intersection rules required of both the films and the borders. There is thus no ideal stmcture that can serve as a convenient mathematical idealization of polyhedral foam stmcture. Lord Kelvin considered this problem, and his minimal tetrakaidecahedron is considered the periodic stmcture of polyhedra that most nearly satisfies the mechanical constraints. 

Support for this postulation came from work done on the shape of the ideal foam cell [32-40]. Ross and co-worker [34,35] proposed three minimal geometric structures, i.e. those which will subdivide space with minimum parti-tional area. These were the pentagonal dodecahedron, the minimal tet-rakaidecahedron, originally suggested by Thomson (Lord Kelvin), and the P-tetrakaidecahedron (Fig. 4). 

The pentagonal dodecahedron, however, is not entirely space-filling, i.e. a close-packed array of such figures has a number of interstitial voids. On the other hand, Kelvin s tetrakaidecahedron and the P-tetrakaidecahedron are. The latter requires 4% more surface area, so a system of such figures would spontaneously rearrange to the more stable array of Kelvin cells. Thus, it would seem that Kelvin s tetrakaidecahedron is the ideal candidate nevertheless, this is not observed in real systems Pentagonal faces are shown on foam cells. These... 

Fig. 4. Regular pentagonal dodecahedron (RPD) (a), Kelvin s minimal tetrakaidecahedron (Kelvin s cell) (b) and /J-tetrakaidecahedron (c)...

Princen and Levinson [6] and Reinelt and Kraynik [7] give the following values for the volume and surface area of Kelvin s tetrakaidecahedron A = 26.5094a2 and A = 26.7419a2, respectively. Compared to the orthic polyhedron the calculated surface here is decreased by is 0.18% [6] and 0.159% [7]. 

Shear stress for three-dimensional foams using the Kelvin s tetrakaidecahedron model is given in [29], The value of Young s modulus (modulus of extension) was calculated to be... 

Similarly, the grain/pore geometry of the intermediate stage should be reasonably approximated by Kelvin s tetrakaidecahedron with three coordinate cylindrical pores along its edges. 

It was repeatedly proposed to use Kelvin s tetrakaidecahedron (that is, minimal truncated octahedron) [381, 407, 479] with eight hexagonal and six quadrangular faces as the polyhedral model of a foam cell and of a cell of any three-dimensional biological tissue. Note, however, that it was statistically shown [195] that Kelvin s tetrakaidecahedron is encountered in biological tissues among other tetrakaidecahedral cells only in 10% of the cases. 

Figure 12 (a) Planar tetrakaidecahedron (or truncated octahedron) (b) Kelvin s minimal tetrakaidecahedron (bcc). 

What foam structure will minimize energy, which is just the total surface area of all of the films This is the Kelvin s problem. The solution of the problem in 2D was conjectured by him to be the honeybee s comb structure. This conjecture was proven recently by Thomas Hales for infinite structure or for finite structures with periodic boundary conditions. Besides this, only the N = 2 case (the double-bubble problem) has been solved in 2D and 3D. Cases for N larger or equal to 3 in 3D have been studied only partially. Concerning 3D infinite structures, Kelvin came up with the body-centered cubic structure, which he called tetrakaidecahedron. However, recently an alternative structure with a lower energy was computed by Weaire and Phalen. This has a more complicated structure with two different kinds of cells (see Figure 2.15). 

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