GPT 6 Astra Ultra attempt on Math-1039

Edge-Unfolding Convex Polyhedra

Definitions and mathematical statement

Let $P\subset\mathbb R^3$ be a bounded convex polyhedron with nonempty interior. Its boundary consists of finitely many planar polygonal faces, meeting along its original edges and vertices. An edge-cut tree is a spanning tree of the vertex-edge graph of $P$: it contains every vertex, is connected, and has no cycle.

Cut the boundary surface along all edges of such a tree, duplicating the cut edges on the resulting boundary. The remaining surface is a topological disk made of the original polygonal faces, still attached along the uncut edges. An edge unfolding places this disk into $\mathbb R^2$, isometrically on each face, with the two incident faces of each uncut edge attached along their common image. It can be obtained by successively rotating faces about uncut edges until the surface is flat.

The conjecture states that every $P$ admits a choice of edge-cut tree whose unfolded faces have pairwise disjoint interiors. This is the usual nonoverlap condition for a net; contacts along face boundaries are allowed. The flat result must be one connected piece containing every face exactly once. Cutting through the interior of a face, adding new vertices to supply extra cuts, or separating the faces into several independent pieces is not allowed.

The problem is about existence of the final planar net. It does not require that a physical motion from the original polyhedron to that net avoid every intermediate collision. Convexity is imposed on the original three-dimensional solid, not on the planar net, which may be nonconvex.

Short English statement

Can every convex polyhedron be cut only along an edge spanning tree and flattened into a single net with no overlapping face interiors?

Research attempt

Scope and literature check.

Demaine's original posting [S2] and the primary nonconvex examples in [A1] were checked on 1 October 2026. Nonconvex polyhedra can fail to edge-unfold, even when individual faces are convex; those results do not settle this convex-solid conjecture. The attempt below constructs a net for every right prism over a convex polygon and audits the edge-tree requirement explicitly. This is a familiar restricted family, not a new general theorem.

Construction of a connected net.

Let $Q$ be a convex $m$-gon with successive edge lengths $l_1,\ldots,l_m>0$, and let $P=Q\times[0,h]$, $h>0$. Assume no redundant collinear vertices. Put $s_0=0$ and $s_i=\sum_{j\le i}l_j$. Develop the lateral faces as the consecutive rectangles \[ R_i=[s_{i-1},s_i]\times[0,h]. \] They form one strip. Keep their consecutive common edges, but cut the final seam between $R_m$ and $R_1$. Attach an isometric copy of the bottom polygon along the edge $[0,l_1]\times\{0\}$, lying in $y\le0$. Attach the top polygon along $[0,l_1]\times\{h\}$, lying in $y\ge h$. Such placements exist because each attaching edge is a supporting edge of the convex polygon. The copies may extend horizontally beyond the strip; that is harmless. Their interiors lie in disjoint open half-planes, while the strip interiors have $0<y<h$. Within the strip, rectangles have disjoint interiors by their disjoint open horizontal intervals. Thus no two face interiors overlap, and all faces remain in one connected piece.

The cuts are an original-edge spanning tree.

There are $2m$ vertices and $3m$ original edges. Cut the one vertical seam, and cut every top and bottom polygon edge except the two used for attachment to $R_1$. The total number of cut edges is \[ 1+2(m-1)=2m-1. \] The remaining cut edges on each base form a path through all $m$ vertices, because they are a polygon boundary cycle with one edge removed. The cut vertical seam connects a vertex of one such path to its counterpart in the other. Hence the cuts are connected on all $2m$ vertices and have no cycle: they are exactly the required spanning tree. The uncut face-adjacency graph is the path of $m$ rectangles with one base attached to each side of $R_1$; it is also a tree. No face-interior cuts or added subdivision vertices occur.

A metric reason this does not prove the general case.

The two decisive facts are that each lateral face is a rectangle of the same height, and that unfolding consecutive faces places all their top edges on $y=h$ and all bottom edges on $y=0$. These follow from the right-prism geometry. They do not follow merely from a polyhedron having a belt of faces, or from the uncut dual graph being a tree. For an oblique prism, even the lateral parallelograms need not develop into a straight strip with two parallel bounding lines. For a general convex polyhedron, there need not be two distinguished base faces at all. A three-dimensional affine shear also does not preserve the planar isometries used for unfolding; applying such a shear to a successful spatial prism does not automatically shear its net consistently across every face.

Verification and first remaining gap.

At $m=3$ the construction has three rectangles, two triangles, and five cut edges on six vertices; at $m=4$ it recovers a valid rectangular-box net with seven cut edges on eight vertices. The proof checks interior separation by half-planes rather than by a drawing, so extremely thin bases and large or small heights introduce no exception. Boundary contacts are permitted by the definition. For arbitrary convex polyhedra one still needs a choice of edge tree together with a comparable global separation certificate for all the developed faces. This attempt gives neither an algorithm that always finds such a tree nor a counterexample.

Final status.

UNRESOLVED. Right prisms over arbitrary convex polygons are handled with an explicit net and edge-tree certificate; the full convex-polyhedron assertion remains beyond the argument. Completion estimate: 5%.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3094 (OPG-60037), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Edge-Unfolding Convex Polyhedra. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/edge_unfolding_convex_polyhedra.
• A1 M. Bern, E. D. Demaine, D. Eppstein, E. Kuo, A. Mantler and J. Snoeyink, Ununfoldable Polyhedra with Convex Faces, primary author manuscript, abstract and introduction checked 1 October 2026. https://erikdemaine.org/papers/Ununfoldable/paper.pdf.