Coherence and enumeration of tilings of 3-zonotopes

Abstract

A rhombohedral tiling of a d-zonotope Z is said to be coherent if it may be obtained by projecting the "top faces" of some (d + 1 )-zonotope onto Z. We classify those 3-zonotopes with five or fewer distinct zones which have all rhombohedral tilings coherent, and give concise enumeration formulas for the tilings of the zonotopes in each class. This enumeration relies in equal parts on the theory of oriented matroids and the theory of discriminantal arrangements of hyperplanes.