  Extremal length is a conformal invariant that transfers naturally to the
discrete setting, giving square tilings as a natural combinatorial analog of
conformal mappings. Recent work by S. Hersonsky has explored generalizing these
ideas to three-dimensional cube tilings. The connections between discrete
extremal length and cube tilings survive the dimension jump, but a condition
called the triple intersection property is needed to generalize existence
arguments. We show that this condition is too strong to realize a tiling, thus
showing that discrete conformal mappings are far more limited in dimension
three, mirroring the classical phenomenon. We also generalize results about
discrete extremal length beyond dimension three and introduce some necessary
conditions for cube tilings.
