Discrete Extremal Length and Cube Tilings in Finite Dimensions

TL;DR

This paper investigates the limitations of extending discrete extremal length and cube tilings from two to higher dimensions, revealing that certain conditions prevent their existence in three dimensions.

Contribution

It demonstrates that the triple intersection property is too restrictive for cube tilings in three dimensions and generalizes extremal length results beyond three dimensions.

Findings

Triple intersection property is too strong for 3D cube tilings.

Discrete extremal length generalizes beyond three dimensions.

Certain necessary conditions for cube tilings are introduced.

Abstract

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.