TITLE: Do bubbles between plates approximate Voronoi diagrams?
QUESTION [33 upvotes]: For example, soap bubbles:

                  


Image from UPenn:
"A 2-dimensional foam of wet soap bubbles squashed between glass plates, after 10 hours of coarsening by the diffusion of gas from smaller to larger bubbles."

I suspect the answer is No, even though, for example, the average number
of sides of a cell is $6$ just as it is for Voronoi diagrams (which is just
a consequence of Euler's formula).
Is there a characterization of the graphs (or dual graphs) realized by
such planar soap bubbles?

REPLY [16 votes]: There are two nice connections to generalizations of Voronoi diagrams that I'm aware of.  
Moukarzel showed that 2D soap bubbles are sectional multiplicative Voronoi partition (SMVP), i.e. 2D slices (sections) of a 3D generalized (multiplicative) Voronoi diagram where each source $f_i$ has a weight $a_i$ so that a cell $\Omega_i$ in the diagram consists of all points $x$ such that $d(x,i)/a_i<d(x,j)/a_j$ for all $j\neq i$.  Note that this is not a bijective correspondence.  First, not all SMVP's are realizable as 2D foams, and second, there isn't a unique SMVP associated to a foam, if I recall correctly there's a 3D space of them, as one can translate the sources of the SMVP in two dimensions and set a global scale factor.

Second, Eppstein showed that the graphs formed by 2D bubbles are precisely the 3-regular bridgeless planar multigraphs.  That these conditions are necessary is not too hard to see from Plateau's laws for foam equilibrium.  To show that any such multigraph can be realized as an equilibrium 2D bubble, Eppstein uses a construction involving a kind of Voronoi diagram in the Poincaré disk model of the hyperbolic plane.  

(Ref. 12 is the very related "Planar Lombardi Drawings for Subcubic Graphs" also by David Eppstein)
I asked Eppstein about the connection between the two constructions in a comment here but never had a chance to work it out in detail for myself.