Apple Peel Unfolding of Archimedean and Catalan Solids

📅 2026-04-17

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🤖 AI Summary

This study investigates whether Archimedean solids and their duals, the Catalan solids, can be unfolded into non-overlapping planar nets via a continuous “apple-peeling” path—traversing adjacent faces without gaps or overlaps. We formalize this unfolding paradigm for the first time, introducing rigorous face-selection rules and developing an automated verification algorithm grounded in geometric and topological analysis. Applying this framework, we systematically classify the unfoldability of all Archimedean and Catalan solids: three Archimedean and six Catalan solids admit perfect apple-peeling unfoldings; three Archimedean and three Catalan solids are unfoldable only under restricted conditions; and the remaining solids cannot be unfolded in this manner.

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📝 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.

Problem

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apple peel unfolding

Archimedean solids

Catalan solids

polyhedron nets

peelability

Innovation

Methods, ideas, or system contributions that make the work stand out.

apple peel unfolding

polyhedron nets

Archimedean solids