Sizing the White Whale

TL;DR

This paper introduces a novel convex hull free computational framework leveraging zonotope symmetry to efficiently generate all vertices of a high-dimensional Minkowski sum called the White Whale, revealing complex combinatorial structures.

Contribution

The paper presents a new method for vertex enumeration of zonotopes that avoids convex hull computations by exploiting symmetry, demonstrated on the White Whale in nine dimensions.

Findings

Generated all vertices of the 9D White Whale zonotope.

Computed the number of edges up to dimension 9.

Identified vertices with exponential degree growth.

Abstract

We propose a computational, convex hull free framework that takes advantage of the combinatorial structure of a zonotope, as for example its symmetry group, to orbitwise generate all canonical representatives of its vertices. We illustrate the proposed framework by generating all the 1 955 230 985 997 140 vertices of the $9$-dimensional White Whale. We also compute the number of edges of this zonotope up to dimension $9$ and exhibit a family of vertices whose degree is exponential in the dimension. The White Whale is the Minkowski sum of all the $2^d-1$ non-zero $0/1$-valued $d$-dimensional vectors. The central hyperplane arrangement dual to the White Whale, made up of the hyperplanes normal to these vectors, is called the resonance arrangement and has been studied in various contexts including algebraic geometry, mathematical physics, economics, psychometrics, and representation theory.