# P3094: Dürer’s edge-unfolding problem

- ID: `P3094`
- Reference: `durers-edge-unfolding-problem`
- Page: https://theoremdb.org/statements/P3094
- Record maturity: Reviewed problem with recorded work

## Problem

Let \(P\) be the boundary of a convex three-dimensional polytope and let \(G(P)\) be its edge graph. Must there exist a spanning tree \(T\subseteq G(P)\) such that cutting \(P\) along \(T\) and isometrically developing the remaining disk into the plane produces one polygon whose face interiors are pairwise disjoint?

### Context

Known frontier: Every convex polyhedron becomes edge-unfoldable after a suitable affine transformation, and every nested three-prismatoid has a nonoverlapping edge unfolding; recent work also constructs highly overlapping edge unfoldings.

Open boundary: Prove a nonoverlapping spanning-tree edge unfolding exists for every original convex three-polytope, or exhibit one polytope for which every such unfolding overlaps.

### Problem setup

- **Definition (edge unfolding).** A planar development obtained by cutting a spanning tree of the original edge graph and rotating faces rigidly about uncut edges.
- **Definition (nonoverlapping).** Distinct face interiors have disjoint planar images; shared boundary points are allowed.
- **Remark.** An edge unfolding cuts only original polytope edges. The surviving hinges flatten the connected surface into one planar polygon. The question asks whether every convex polyhedron has at least one cut tree that avoids face-interior overlap.

### What counts as a solution

- For a proof, construct or establish a valid spanning tree for every convex three-polytope and prove injectivity on face interiors.
- For a disproof, give explicit coordinates and faces for one convex polytope and exhaust or structurally exclude every spanning-tree edge unfolding.

## Status

OPEN as checked on 2026-08-01. Strongest checked neighboring result: Every convex polyhedron becomes edge-unfoldable after a suitable affine transformation, and every nested three-prismatoid has a nonoverlapping edge unfolding; recent work also constructs highly overlapping edge unfoldings. Exact unresolved remainder: Prove a nonoverlapping spanning-tree edge unfolding exists for every original convex three-polytope, or exhibit one polytope for which every such unfolding overlaps. [1](#reference-1) [2](#reference-2) [3](#reference-3)

## Work

### Evidence for the current status

**Claim 1 (Current status and exact unresolved remainder).** OPEN as checked on 2026-08-01. Strongest checked neighboring result: Every convex polyhedron becomes edge-unfoldable after a suitable affine transformation, and every nested three-prismatoid has a nonoverlapping edge unfolding; recent work also constructs highly overlapping edge unfoldings. Exact unresolved remainder: Prove a nonoverlapping spanning-tree edge unfolding exists for every original convex three-polytope, or exhibit one polytope for which every such unfolding overlaps.

The problem was checked as open on 2026-08-01.

The strongest neighboring result found in the cited sources is: Every convex polyhedron becomes edge-unfoldable after a suitable affine transformation, and every nested three-prismatoid has a nonoverlapping edge unfolding; recent work also constructs highly overlapping edge unfoldings.

The exact unresolved remainder is: Prove a nonoverlapping spanning-tree edge unfolding exists for every original convex three-polytope, or exhibit one polytope for which every such unfolding overlaps.

A complete resolution must meet the following acceptance conditions:
- For a proof, construct or establish a valid spanning tree for every convex three-polytope and prove injectivity on face interiors.
- For a disproof, give explicit coordinates and faces for one convex polytope and exhaust or structurally exclude every spanning-tree edge unfolding.

### Background and intake notes

- Original intake status: OPEN as checked on 2026-08-01. Strongest checked neighboring result: Every convex polyhedron becomes edge-unfoldable after a suitable affine transformation, and every nested three-prismatoid has a nonoverlapping edge unfolding; recent work also constructs highly overlapping edge unfoldings. Exact unresolved remainder: Prove a nonoverlapping spanning-tree edge unfolding exists for every original convex three-polytope, or exhibit one polytope for which every such unfolding overlaps.
- The release review checked 3 structured sources on 2026-08-01.
- Equivalent-formulation queries: Dürer edge unfolding every convex polyhedron open; convex 3-polytope spanning tree nonoverlapping net; Dürer problem affine unfolding pseudo-edge counterexample
- Strongest checked neighboring result: Every convex polyhedron becomes edge-unfoldable after a suitable affine transformation, and every nested three-prismatoid has a nonoverlapping edge unfolding; recent work also constructs highly overlapping edge unfoldings.
- Exact unresolved remainder: Prove a nonoverlapping spanning-tree edge unfolding exists for every original convex three-polytope, or exhibit one polytope for which every such unfolding overlaps.

### Other known results

- **Claim 2** (supported): Every convex polyhedron becomes edge-unfoldable after a suitable affine transformation, and every nested three-prismatoid has a nonoverlapping edge unfolding; recent work also constructs highly overlapping edge unfoldings. [1](#reference-1) [2](#reference-2) [3](#reference-3)

### Prior approaches

- **Route 1** (supported): The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Every convex polyhedron becomes edge-unfoldable after a suitable affine transformation, and every nested three-prismatoid has a nonoverlapping edge unfolding; recent work also constructs highly overlapping edge unfoldings. Unresolved remainder: Prove a nonoverlapping spanning-tree edge unfolding exists for every original convex three-polytope, or exhibit one polytope for which every such unfolding overlaps. [1](#reference-1) [2](#reference-2) [3](#reference-3)

### Open directions

- **Route 2** (reported): Prove a nonoverlapping spanning-tree edge unfolding exists for every original convex three-polytope, or exhibit one polytope for which every such unfolding overlaps.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `durers-edge-unfolding-problem`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Mohammad Ghomi, “Affine unfoldings of convex polyhedra”. Geometry & Topology 18(5) (2014), 3055-3090. DOI 10.2140/gt.2014.18.3055. abstract and main theorem https://doi.org/10.2140/gt.2014.18.3055
   - Also cited at M. Ghomi, “Affine unfoldings of convex polyhedra,” Geometry & Topology 18(5) (2014), 3055–3090. abstract and main theorem
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Proves that every convex polyhedron becomes edge-unfoldable after a suitable affine transformation, while leaving the original metric problem open.
   - Source used to assess the problem's recorded status.
   - For Dürer’s edge-unfolding problem: This is the dated publication status for the canonical target Dürer’s edge-unfolding problem.
   - Source named by the research packet.
2. <a id="reference-2"></a>Manuel Radons, “Edge-unfolding nested prismatoids”. Computational Geometry 116 (2024), 102033. DOI 10.1016/j.comgeo.2023.102033. abstract and main theorem https://doi.org/10.1016/j.comgeo.2023.102033
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Proves nonoverlapping edge unfoldings for every nested three-prismatoid.
   - Source used to assess the problem's recorded status.
   - For Dürer’s edge-unfolding problem: Proves nonoverlapping edge unfoldings for every nested three-prismatoid.
3. <a id="reference-3"></a>MIT CompGeom Group, H. A. Akitaya, E. D. Demaine, F. Frei, S. Langerman, A. Lubiw, and J. O’Rourke, “Overlapping Unfoldings of Cones and Convex Polyhedra,” arXiv:2607.09606 (2026). abstract and introduction https://doi.org/10.48550/arXiv.2607.09606
   - preprint; primary source; arXiv:2607.09606, checked 2026-08-01; checked 2026-08-01
   - Source use: original_summary
   - Frames Dürer’s problem as unresolved and constructs edge unfoldings with arbitrarily high overlap thickness, which does not settle existence of a nonoverlapping tree.
   - Source used to assess the problem's recorded status.
   - For Dürer’s edge-unfolding problem: Frames Dürer’s problem as unresolved and constructs edge unfoldings with arbitrarily high overlap thickness, which does not settle existence of a nonoverlapping tree.