Unfolding Genus-2 Orthogonal Polyhedra with Linear Refinement

TL;DR

This paper proves that all orthogonal polyhedra with genus up to 2 can be unfolded into a flat net with a linear number of cuts, advancing the understanding of polyhedron unfolding beyond genus-0.

Contribution

It introduces an unfolding method for genus-2 orthogonal polyhedra using a linear number of cuts, the first to extend unfolding results beyond genus-0.

Findings

Unfolding for genus-2 polyhedra achieved with linear cuts

Existence of at most 2 special leaves in the unfolding tree

Unfolding genus-3 and higher remains an open challenge

Abstract

We show that every orthogonal polyhedron of genus at most 2 can be unfolded without overlap while using only a linear number of orthogonal cuts (parallel to the polyhedron edges). This is the first result on unfolding general orthogonal polyhedra beyond genus-0. Our unfolding algorithm relies on the existence of at most 2 special leaves in what we call the "unfolding tree" (which ties back to the genus), so unfolding polyhedra of genus 3 and beyond requires new techniques.