Applications of Three Dimensional Extremal Length, I: Tiling of a Topological Cube

TL;DR

This paper introduces a method to tile a topological cube with cubes based on a triangulation, using a variational approach related to extremal length, ensuring boundary and combinatorial preservation.

Contribution

It presents a novel approach to cube tiling of topological cubes using a discrete extremal length framework, preserving boundary and combinatorial structure.

Findings

Existence of a unique cube tiling under certain conditions

Boundary vertices correspond to cube corners in the tiling

Cube sizes derived from a variational extremal length problem

Abstract

Given a triangulation of a closed topological cube, we show that (under some technical condition) there is an essentially unique tiling of a rectangular parallelepiped by cubes, indexed by the vertices of the triangulation. Moreover, i - the combinatorics is preserved, and ii- the boundary is preserved: vertices corresponding to the cubes at the corners of the rectangular parallelepiped are at the corners of the topological cube. Also, 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 three dimensional Euclidean space.