Common Edge-Unzippings for Tetrahedra

It is shown that there are examples of distinct polyhedra, each with a Hamiltonian path of edges, which when cut, unfolds the surfaces to a common net. In particular, it is established for infinite classes of triples of tetrahedra.

💡 Research Summary

The paper investigates a special case of polyhedral unfolding, focusing on “edge‑unzipping” – a Hamiltonian path consisting solely of edges that, when cut, flattens a convex polyhedron into a simple planar polygon (a net). The author proves that there exist infinitely many triples of distinct tetrahedra that share a common net obtained by cutting along such an edge‑unzipping path.

Main Result (Theorem 1).
Consider a convex equilateral hexagon whose side lengths are all 1, each interior angle α_i satisfies π/3 < α_i < π, and any pair of angles is linearly independent over the rationals (i.e., no rational linear relation aπ + bα_i = α_j). Under these conditions the hexagon serves as a common edge‑unzipping net for three non‑congruent tetrahedra.

Construction.
The hexagon is folded by a perimeter‑halving operation: opposite vertices are paired, and the two halves of the perimeter are glued together. Because the hexagon is equilateral, the glued edges have matching lengths, and Alexandrov’s existence theorem guarantees a unique convex polyhedron. The curvature is non‑zero at exactly four points, so the resulting polyhedron is a tetrahedron. By choosing one of the three possible opposite‑vertex pairings (0‑3, 1‑4, 2‑5) we obtain three different tetrahedra, each possessing a Hamiltonian path of three unit‑length edges (the three consecutive edges of the hexagon) that, when cut, yields the original hexagon.

Distinctness (Lemma 1).
The curvature at the four vertices of each tetrahedron can be expressed in terms of the hexagon angles. If two tetrahedra had identical curvature sets, the angles would satisfy a rational linear relation, contradicting the imposed ℚ‑linear independence. Hence the three tetrahedra are pairwise non‑congruent.

Edge‑unzipping verification.
Lemma 2 shows that in any tetrahedron every shortest path between two vertices coincides with an edge, because the tetrahedron’s 1‑skeleton is the complete graph K₄. Therefore, if the three unit‑length edges of the hexagon are shortest paths on the folded surface, they must be edges of the corresponding tetrahedron.

Shortest‑path property (Lemma 3).
The author proves that each of the three unit edges of the hexagon is indeed a shortest geodesic on the glued manifold. The proof uses the “fat” angle condition (α_i > π/3). If a diagonal were shorter than 1, the sum of angles around its endpoints would exceed π, forcing some interior angle of the hexagon to exceed π and violating convexity. Moreover, disks of radius 1 centered at the edge endpoints cannot contain any other vertex; otherwise an angle smaller than π/3 would appear. Even when a disk “overhangs” into a copy of the hexagon on the glued surface, the overhang width is bounded by 1 − √3⁄2, and any vertex entering this region would again force an angle < π/3. Hence the three edges are shortest paths, and by Lemma 2 they are edges of the tetrahedra.

Corollary 1.
The angle constraints define a 5‑dimensional open set in ℝ⁵ (the sixth angle is determined by the sum 4π). The ℚ‑linear independence condition excludes only countably many 4‑dimensional hyperplanes, which have measure zero. Consequently, there are uncountably many hexagons satisfying the theorem’s hypotheses.

Generalization and Conjecture.
The construction extends to any even n ≥ 4: an equilateral convex n‑gon can be folded by perimeter halving to produce n/2 distinct (n‑2)‑vertex polyhedra sharing the same net (Proposition 1). For regular n‑gons these are the well‑known “pita polyhedra.” The author conjectures that for all strictly convex equilateral n‑gons the corresponding zipper paths are also edge‑paths, but proving this would require new tools to decide when a geodesic on an Alexandrov‑glued surface coincides with an edge of the resulting polyhedron.

Significance.
This work provides the first explicit infinite family of polyhedral “net pairs” where the shared net arises from cutting along edge‑only Hamiltonian paths. It clarifies the relationship between curvature, shortest‑path geometry, and edge structure in the special case of tetrahedra, and opens a line of inquiry into higher‑vertex polyhedra where similar phenomena may occur. The paper also highlights the scarcity of existing techniques for verifying edge‑unzipping beyond tetrahedra, suggesting a fruitful direction for future research in geometric folding and unfolding.