Combinatorics of slices of cubes

TL;DR

This work addresses the challenge of classifying all combinatorial types of hyperplane slices of the cube $C_d=[-1,1]^d$ up to $d=6$ (and central slices up to $d=7$), providing exact counts and certified numerics. The authors develop two complementary algorithms—an exact cell-decomposition approach and a certified numerical method based on critical points—to enumerate slice types, with substantial symmetry reductions that leverage the cube’s group of signed permutations. They connect central-slice types to NP-equivalence classes of self-dual threshold functions and investigate vertex counts, including a proven upper bound attained by central slices, along with conjectures about vertex counts and the structure of maximal-vertex slices. The paper also extends the study to graphs of slices via color classes, reproduces classical counts, and delivers a comprehensive online repository of code and data, enabling reproducibility and further exploration in discrete geometry and algebraic statistics.

Abstract

We present a complete computational classification of the combinatorial types of hyperplane sections, or slices, of the regular cube up to dimension six. For each dimension, we determine the exact number of distinct combinatorial types. When restricted to slices through the origin, our computations extend to dimension seven. The classification combines combinatorial, algebraic, and numerical techniques, with all results certified. Beyond enumeration, we analyze the distribution of types by number of vertices, establish new theoretical results about the combinatorics of slices of cubes, and propose conjectures motivated by our computational findings.

Figures (11)

• Figure 1: The cube $C_3$ and all its combinatorial types of slices: a purple triangle, a green quadrilateral, a yellow pentagon, a red hexagon.
• Figure 2: The slices $C_3\cap u_1^\perp$ (left) and $C_3\cap u_2^\perp$ (right) from \ref{['ex:hexagonal_slices']}.
• Figure 3: The slices $C_3\cap H_1$ (left) and $C_3\cap H_2$ (right) from \ref{['ex:quadrilateral_slices']}.
• Figure 4: Number of combinatorial types of slices of $C_4$ by number of vertices.
• Figure 5: Number of combinatorial types of affine slices of $C_5$ and $C_6$ by number of vertices.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Corollary 2.3
• Conjecture 1
• Theorem 2.4
• proof