Dynamical partitions of space in any dimension

TL;DR

This paper provides a comprehensive statistical framework for analyzing topologically stable cellular partitions in any dimension, introducing new invariants and construction methods, and comparing them with known high-dimensional structures.

Contribution

It introduces a complete statistical description of stable cellular partitions in arbitrary dimensions, including invariants and construction procedures, expanding understanding of high-dimensional space partitions.

Findings

Derived average structural properties using a sequence of variables.

Presented a method to construct space partitions via cell-division and coalescence.

Compared homogeneous partitions with Voronoi and sphere packing structures.

Abstract

Topologically stable cellular partitions of D dimensional spaces are studied. A complete statistical description of the average structural properties of such partition is given in term of a sequence of D/2-1 (or (D-1)/2) variables for D even (or odd). These variables are the average coordination numbers of the 2k-dimensional polytopes (2k < D) which make the cellular structure. A procedure to built D dimensional space partitions trough cell-division and cell-coalescence transformations is presented. Classes of structures which are invariant under these transformations are found and the average properties of such structures are illustrated. Homogeneous partitions are constructed and compared with the known structures obtained by Voronoi partitions and sphere packings in high dimensions.