What question did this study set out to answer?

The aim is to demonstrate that solid polycubes can be transformed into 2D nets without overlaps.

June 30, 2026Open Access

Unfolding Solid Polycubes

Key Points

The aim is to demonstrate that solid polycubes can be transformed into 2D nets without overlaps.
Utilized mathematical induction to establish the unfolding process.
Defined a perfect net to facilitate consistent gluing of edges throughout unfolding.
Maintained edge connections to avoid overlaps during the induction steps.
Any solid polycube can be edge-unfolded into a 2D net without refinement.
The configuration ensures designated edges remain at the far sides, allowing systematic gluing.
No overlaps occur at any stage of the induction process.

Abstract

A solid polycube is a face-connected set of unit cubes, where, unlike classical definitions for polycubes, shared faces between adjacent cubes are not removed. We prove by mathematical induction that any such polycube can be edge-unfolded into a 2D net without refinement. Our proof relies on the concept of a perfect net, defined as a net in which two designated edges are placed on the far left and far right sides. This configuration enables consistent gluing of nets in a single direction throughout the induction process, thereby guaranteeing that no overlaps occur at any step.

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