Unfoldings of the Cube

TL;DR

This paper investigates the number of distinct ways to unfold a cube into the plane by slicing along edges, using graph theory and algebraic techniques to count unique unfoldings.

Contribution

It introduces a novel approach combining graph theory, algebra, and geometry to count and classify cube unfoldings, simplifying complex calculations.

Findings

Number of cube unfoldings computed using spanning trees

Application of Burnside's lemma to count incongruent unfoldings

Elementary techniques enable hand calculations for the problem

Abstract

Just how many different connected shapes result from slicing a cube along some of its edges and unfolding it into the plane? In this article we answer this question by viewing the cube both as a surface and as a graph of vertices and edges. This dual perspective invites an interplay of geometric, algebraic, and combinatorial techniques. The initial observation is that a cutting pattern which unfolds the cubical surface corresponds to a spanning tree of the cube graph. The Matrix-Tree theorem can be used to calculate the number of spanning trees in a connected graph, and thus allows us to compute the number of ways to unfold the cube. Since two or more spanning trees may yield the same unfolding shape, Burnside's lemma is required to count the number of incongruent unfoldings. Such a count can be an arduous task. Here we employ a combination of elementary algebraic and geometric techniques to bring the problem within the range of simple hand calculations.