Apple Peel Unfolding of Archimedean and Catalan Solids

TL;DR

This paper introduces a new 'apple peel unfolding' method for creating polyhedron nets, classifies Archimedean and Catalan solids based on peelability, and implements a program to determine peelability.

Contribution

It defines a novel peeling technique, develops an algorithm to assess peelability, and classifies specific polyhedra as always, sometimes, or never peelable.

Findings

Three Archimedean and six Catalan solids are always peelable.

Three Archimedean and three Catalan solids are peelable under restricted conditions.

The program can determine if a polyhedron is peelable or not.

Abstract

We consider a new treatment for making polyhedron nets referred to as ``apple peel unfolding'': drawing the nets as if we were peeling off appleskins. We define apple peel unfolding strictly and implement a program that derives the sequential selection of the polyhedral faces for a target polyhedron in accordance with the definition. Consequently, the program determines whether the polyhedron is peelable (can be peeled completely). We classify Archimedean solids and their duals (Catalan solids) as perfect (always peelable), possible (peelable for restricted cases), or impossible. The results show that three Archimedean and six Catalan solids are perfect, and three Archimedean and three Catalan ones are possible.