Unfolding Polycubes with Orthogonally Convex Layers

TL;DR

This paper introduces an edge unfolding algorithm for polycubes with orthogonally convex layers, enabling their flattening into a single, non-overlapping planar piece, which advances geometric unfolding techniques.

Contribution

The paper presents the first edge unfolding algorithm specifically designed for polycubes with orthogonally convex layers, expanding unfolding methods to a new class of polyhedra.

Findings

Successfully unfolds any polycube with orthogonally convex layers

Ensures the unfolded shape is a single, non-overlapping planar piece

Uses only edge cuts along cube edges

Abstract

A polycube is an orthogonal polyhedron composed of unit cubes glued together along entire faces, and homeomorphic to a sphere. A layer of a polycube refers to the portion lying between two horizontal cross-sections spaced one unit apart. We present an unfolding algorithm that flattens any polycube with orthogonally convex layers into a single, non-overlapping planar piece. The algorithm makes cuts only along cube edges-that is, it is an edge unfolding.