The Polyhedral Model of Foams

Preliminary remarks. Models of the foam cell. The polyhedral shape of foam cells is the limit shape as the foam multiplicity grows infinitely. At the same time, this is a rather convenient structural model for actual foam with finite multiplicity. A polyhedron constructed of liquid films must satisfy the following two rules, stated by Plateau [9, 379, 407]  

Three films must meet along one edge making equal dihedral angles (of 120°). 

Three edges must meet at one node making equal angles (of 109°26 16 ) between each other. 

It is also natural to require that the number of faces (films) F, edges (the Plateau borders) B, and nodes N shall satisfy the fundamental topological Euler theorem about polyhedra [101,479]  

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. 

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Actual foam contains bubbles whose shape is intermediate between spheres and polyhedra. Such foam is said to be cellular [214, 280]. The distinction between the cellular and polyhedral kinds of foam is rather conventional and is determined by very low moisture contents (of the order of some tenth of per cent). Nevertheless, the polyhedral model of foam cells is used rather frequently [38,125,244,438,480],... 

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