The triple intersection property, three dimensional extremal length, and tiling of a topological cube

Let T be a triangulation of a closed topological cube Q, and let V be the set of vertices of T . Further assume that the triangulation satisfies a technical condition which we call the triple intersection property (see Definition 3.6). Then there is an essentially unique tiling C = { C v : v ∈ V } of a rectangular parallelepiped R by cubes, such that for every edge ( u , v ) of T the corresponding cubes C v , C u have nonempty intersection, and such that the vertices corresponding to the cubes at the corners of R are at the corners of Q. Moreover, the sizes of the cubes are obtained as a solution of a variational problem which is a discrete version of the notion of extremal length in R 3 .