Kelvin’s tetrakaidecahedron

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... 

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. 

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

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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