Random cellular froths in spaces of any dimension and curvature

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Abstract
Froth is a random partition of a D-dimensional space by cells. This assembly of cells obeys two fundamental laws: Euler's relation and the condition of maximum vertex figure, imposed by geometry and by topological stability, respectively. These two conditions generate a set of relations between the variables that fully characterize the system topologically. The number of degrees of freedom of the system and a set of useful independent variables, the 'even valences', have been found. The influence of the space dimension and curvature on the range of variability of these valences is discussed and, up to D=5, the regions in valence space corresponding to differently curved froths are calculated explicitly.