Unfolding

An unfolding is the cutting along edges and flattening out of a polyhedron to form a net. Determining how to unfold a polyhedron into a net is tricky. For example, cuts cannot be made along all edges that surround a face or the face will completely separate. Furthermore, for a polyhedron with no coplanar faces, at least one edge cut must be made from each vertex or else the polyhedron will not flatten. In fact, the edges that must be cut corresponds to a special kind of graph called a spanning tree of the skeleton of the polyhedron (Malkevitch).

In 1987, K. Fukuda conjectured that no convex polyhedra admit a self-overlapping unfolding. The top figure above shows a counterexample to the conjecture found by M. Namiki. An unfoldable tetrahedron was also subsequently found (bottom figure above). Another nonregular convex polyhedra admitting an overlapping unfolding was found by G. Valette (shown in Buekenhout and Parker 1998).

Examples of different polyhedra that can be constructed from the same net are not difficult to construct, but Fukuda conjectured that every convex polyhedron can be uniquely constructed from any of its unfoldings. The counterexample shown above was found by T. Matsui.

The question of whether every convex polyhedron admits a self-unoverlapping unfolding (Shephard 1975) is still unsettled (Malkevitch). Shephard's conjecture states (and most mathematicians believe) that the answer is yes.

Cuts other than along edges of the polyhedron can also be considered. For example, a star unfolding is a way of unfolding a polyhedron by cutting along shortest paths on the surface of it. Aronov and Rourke (1992) showed that every convex 3-dimensional polyhedron has a star unfolding (Malkevitch).

One way to restrict the choice of cuts is to require the surface to open along a single path, producing one continuous strip. In a zipper unfolding, this path is a cut path, and an edge-cut zipper unfolding is a Hamiltonian unfolding. Every Platonic solid and Archimedean solid has such an unfolding (Demaine et al. 2010). A spiral unfolding adds the geometric condition that the strip wind around the polyhedron. Every Platonic solid and Archimedean solid also has a nonself-intersecting spiral unfolding (O'Rourke 2015).

Apple-peel fold-outs are spiral strips inspired by the shape left when an apple is peeled. Kimura et al. (2015) constructed apple-peel fold-outs for regular four-dimensional polytopes. Yoshino and Chaidee (2026) subsequently formulated apple-peel unfolding as a greedy traversal beginning with an ordered pair of adjacent faces or cells. Their spiral rule selects the sharpest clockwise turn using a signed determinant, while their zonal rule selects the candidate having greatest coordinate along the peeling axis. Completing such a traversal does not guarantee a valid net, since the resulting strip may self-intersect.