Discrete geometry · convex polyhedra · reciprocal diagrams

Dürer’s Edge-Unfolding Conjecture

Collaboration beta

Does every convex polyhedron have a spanning tree of its actual edges that lets the surface unfold into the plane without overlapping face interiors?

\forall P \in P_{convex} \exists T \subseteq E (P) : (V (P), T) is a spanning tree \land \forall F \neq F', relint (U_{P, T} (F)) \cap relint (U_{P, T} (F')) = \emptyset

Listed inThe Open Problems Project, Problem 9

Known results and sources

Dürer’s open question asks whether every convex polyhedron can be cut along an actual-edge spanning tree into a planar development with disjoint face interiors. This tetrahedron illustrates the kind of solid in the question; no unfolding is depicted.

Research problem

Exact mathematical statement

Let $P$ be a convex polyhedron, let $T$ be a spanning tree in its actual edge graph, and let $U_{P,T}$ be the planar development after cutting along $T$ . Dürer’s conjecture states

\forall P\in\mathcal P_{\mathrm{convex}}\;\exists T\subseteq E(P): \ (V(P),T)\text{ is a spanning tree and }\forall F\ne F',\;\operatorname{relint}(U_{P,T}(F))\cap\operatorname{relint}(U_{P,T}(F'))=\varnothing.

Boundary contact is allowed; intersections of face interiors are not.

Problem infographic

Problem at a glance

A forest-green scientific plate shows a convex tetrahedron with four vertices and six actual edges, highlighting a three-edge gold tree from one apex to the other vertices. A descriptive arrow opens it into a planar net of exactly four triangular faces with disjoint interiors. Smaller glyphs contrast allowed boundary contact with forbidden face-interior overlap, while a gold panel marks the general question as open. — Dürer’s edge-unfolding conjecture asks whether every convex polyhedron has a spanning tree in its actual edge graph such that cutting along the tree and developing the surface in the plane produces pairwise-disjoint face interiors. Boundary contact is allowed. The tetrahedral net shows one successful unfolding.

Current mathematical picture

Where work on Dürer’s Edge-Unfolding Conjecture stands

Open conjecture

The retained sixth-pass packet leaves Dürer’s full edge-unfolding conjecture open while closing coordinate-uniform octahedral coverage and therefore every triangular reciprocal type with at most three interior vertices. It preserves the scoped retractions and refutations for unrestricted degree-three substitution, the universal pointwise forest, the supporting-edge shadow lemma, and one-edge cycle exchange. At four interior vertices, an embedded deterministic verifier reports five underlying maximal planar graphs and sixteen boundary-preserving types: eight are universally harmless by retained leaf-compatible refinements, while types 4, 5, 7, 8, 13, 14, 15, and 16 remain unresolved by the current certificates and are not counterexamples.

Leading routeExact reciprocal strict-gap search

Fix a rational triangular reciprocal geometry, compute exact hinge slacks and reduced forest margins, seek a universal Farkas forest first, then decide exact finite forest coverage or find a strict rational uncovered potential.

Route status · Active route

Useful failure26-vertex projected path trap as a cap

The construction cannot be used as an almost-flat cap because its rim is nonplanar; its pointwise DFS and adaptive-network data remain rerun-required diagnostics only.

Route status · Eliminated route

Main reductionClosedness sharpens finite-height continuation

Fixed-tree validity sets are closed and semialgebraic, reducing continuation to any endpoint-extending tree rather than a one-edge exchange.

Evidence posture · Reported reduction

Completed special caseEight four-interior types covered by leaf refinements

Retained leaf-compatible refinement arguments handle eight of the sixteen reported types; the other eight are unresolved by these certificates and are not classified as counterexamples.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve finite-height endpoint exclusion

At every lower endpoint above 1 of the valid-tree component containing the sufficiently stretched range, produce some spanning tree valid immediately below the endpoint.

Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected supporting details in the research record. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Dürer’s Edge-Unfolding Conjecture in numbers

3.8kretained lines of mathematical investigation3,739 in the current working snapshot

Argument development: 2,847 · 75%
Explored or eliminated routes: 246 · 6%
Computational analysis: 262 · 7%
Open obligations: 198 · 5%
Definitions and setup: 237 · 6%

24selected mapped statements11routes investigated12reported milestones7open questions5contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byG. C. ShephardProposed1975Status checkedAug 2, 2026

Source recordedAug 6, 2026

Recommended next task

Prove finite-height endpoint exclusion

At every lower endpoint above 1 of the valid-tree component containing the sufficiently stretched range, produce some spanning tree valid immediately below the endpoint.

Suggested move: Seek a global invariant preventing all trees valid at an endpoint from losing validity toward the same side; do not require a basis exchange.

What would count as progress

Control remote face collisions and prove the endpoint-extension statement for the affine family.
Combine it with the exact external affine-stretching start theorem.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

27 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Active routeExact reciprocal strict-gap search

Route status · Active route

Active routeFour-interior reciprocal classification

Enumerate boundary-preserving triangular types, use only leaf-compatible degree-three reductions, and run exact Farkas and active-forest analysis on every surviving rational geometry.

Route status · Active route

Active routeAdmissible-root-core chord discharge

Use the forced legal exit-or-chord theorem, planarity, positive equilibrium, and actual-area identities to turn an outermost chord into a strictly improving sequence rather than one arbitrary cycle exchange.

Route status · Active route

Active routeFinite-height endpoint exclusion

Treat full finite-height continuation as a separate route: prove that some valid tree extends below every lower endpoint of the stretched component, without requiring a one-edge exchange.

Route status · Active route

Explored alternatives

Other routes

7 recorded

Completed history · source-reportedCoordinate-uniform octahedral Farkas coverage

Theorem 62.4 closes coordinate-uniform octahedral coverage with six forest templates. This former route is recorded as completed history and as a regression test; active reciprocal work begins at four interior vertices.

Route status · Completed history · source-reported

Narrowed routeDegree-three reciprocal recursion

The local affine-boundary star discharge survives only as leaf refinement. Arbitrary-depth recursive deletion remains unavailable until external-child margins are controlled.

Route status · Narrowed route

Eliminated route26-vertex projected path trap as a cap

The construction cannot be used as an almost-flat cap because its rim is nonplanar; its pointwise DFS and adaptive-network data remain rerun-required diagnostics only.

Route status · Eliminated route

Browse 4 more explored routes

Not yet justifiedPointwise failure to common mass obstruction

The route is eliminated on the 26-vertex minimal conflict because adaptive networks expose the quantifier gap; future negative work must search the actual fixed-mass forest-cone union directly.

Route status · Not yet justified

Refuted routeConvex supporting-edge shadow

The convex-block shadow theorem is refuted. Only the stronger sliced dual-shadow condition or the affine-boundary patch theorem remains viable.

Route status · Refuted route

Route held in reserveFlat-cap and four-cap source audit

Retrieve the exact imported statements before converting any internal strict reciprocal certificate into a claimed finite-height cap overlap or closed-polyhedron counterexample.

Route status · Route held in reserve

Narrowed routeEight-type four-interior reciprocal frontier

Use the canonical edge lists for the eight unresolved types, first looking for exact relative-Farkas forest covers and then for strict rational uncovered potentials.

Route status · Narrowed route

Route statements and reductions

Statements the next route can inspect and build on

Route statementStrict reciprocal-gap reduction

A rational strict potential outside every reduced forest cone for a genuine reciprocal geometry, followed by translation, rational offset completion, exact cap verification, and the audited flat-cap and four-cap bridges, would give a counterexample to Dürer’s conjecture.

Source-reported route statement · dependencies incomplete

Route statementFixed-reciprocal linearization

For fixed rational reciprocal geometry, every raw forest margin factors into a positive parent-hinge amplitude and a rational linear reduced margin in the reciprocal potential.

Source-reported route statement

Route statementSafe leaf refinement

Stellar subdivision of a coarse reciprocal triangle preserves a certified monotone forest when the coarse triangle is a leaf; the refined cells can be gathered to the child incident to the old exit side without changing ancestor margins.

Source-reported route statement

Route statementRoot-core exit-or-chord theorem

In a positive-equilibrium ambient graph, a monotone packet tree with nonempty admissible-root core has either a legal complete-packet exit from the support or a legal non-tree chord into the forced part of the support.

Source-reported route statement

Route statementFixed-tree validity sets are closed and semialgebraic

For a fixed spanning edge tree under affine stretching and the face-interior overlap convention, the set of parameters with a valid unfolding is a finite union of closed intervals and points.

Source-reported route statement

Route statementFinite-height endpoint-exclusion reduction

Given the sufficiently stretched starting theorem, a full affine-continuation proof follows if every lower endpoint above 1 of the valid-tree component containing all sufficiently large parameters has some tree valid immediately below it; the replacement need not be a one-edge exchange.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

7 featured tasks

Prove finite-height endpoint exclusion

At every lower endpoint above 1 of the valid-tree component containing the sufficiently stretched range, produce some spanning tree valid immediately below the endpoint.

Suggested move: Seek a global invariant preventing all trees valid at an endpoint from losing validity toward the same side; do not require a basis exchange.

Ready to work on

Find or exclude a strict reciprocal gap

For a rational irreducible reciprocal geometry, decide exactly whether the strict regular cone contains a potential outside every reduced forest polyhedron.

Suggested move: Run universal relative-Farkas tests first, then exact active-forest cover or strict-gap analysis with leaf witnesses excluded and strict hinge bounds enforced.

Ready to work on

Classify four-interior triangular reciprocal types

Enumerate boundary-preserving four-interior triangular reciprocal types and decide their universal forest coverage or strict gaps without unsafe degree-three deletion.

Suggested move: Apply leaf refinement only when a certified coarse forest makes the refined cell a leaf, then run boundary-leaf, universal-Farkas, and exact cover/gap passes.

Ready to work on

Resolve the eight remaining four-interior types

For types 4, 5, 7, 8, 13, 14, 15, and 16 of the sixth-pass table, produce exact universal forest certificates or a strict reciprocal gap; do not infer obstruction from failure of the current leaf certificates.

Suggested move: Start from the canonical edge sets and test exact relative-Farkas forest coverage before any wider enumeration.

Ready to work on

Build an independently checkable reciprocal engine

Generate exact geometry, slacks, forests, reduced margins, relative-Farkas or branch certificates, rational completion, actual primal reconstruction, zero-jump suppression, and deterministic hashes.

Suggested move: Implement an exact forest enumerator and independent verifier before trusting any new cover or strict-gap result.

Ready to work on

Discharge the root-core chord

Show that an outermost legal non-tree chord forced by the admissible-root core yields a finite monotonicity-preserving rotation sequence, an enlarged root core, a reduced revisit potential, or a legal exit.

Suggested move: Use an outermost chord and a strict potential involving root-core size, forced-gate deficit, or repeated departures; test against the exact one-exchange counterexample.

Prerequisites still open

Audit the external negative bridge

Retrieve and verify the exact flat-cap overlap theorem, one-common-flattening argument, actual-edge hypotheses, and four-cap assembly before a reciprocal gap is called a counterexample.

Suggested move: Bind the exact primary-source statements and hypotheses, including triangulation, genericity, boundary, pseudo-edge, seam, and uniformity conditions.

Blocked by the current route

Sourced mathematical context

The known mathematical landscape

Context collected Aug 2, 2026

Current statusOpen problem

It remains unknown whether every convex polyhedron has a nonoverlapping edge unfolding to one simple planar polygon. The conjecture concerns at least one successful edge-cut tree per polyhedron; the existence of overlapping unfoldings does not refute it.

[2][6][8]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026
PreprintA preprint constructed, for every requested thickness, a convex polyhedron having an edge unfolding with overlap of that thickness; this concerns existence of highly overlapping unfoldings and does not settle whether the same polyhedron has some nonoverlapping edge unfolding.[4]
2021
PreprintEdge unfoldings were constructed for prismatoids with sufficiently separated bases and for prismatoids with rectangular base and nonobtuse triangular side faces.[3]
2019
PreprintO'Rourke's survey organized twenty combinations of cut type and polyhedron class, with the original convex edge-unfolding cell still open.[2]
2017
PreprintBarvinok and Ghomi constructed a 340-vertex convex polyhedron with a pseudo-edge graph having no nonoverlapping unfolding, disproving the pseudo-edge generalization but not the original edge conjecture.[1]

Mathematical neighborhood

Related results and reusable starting points

Current focusDürer's edge-unfolding problem

Weaker or relaxed formanycut unfolding of convex polyhedra

Allowing cuts along general surface curves rather than only original edges yields a nonoverlapping unfolding for every convex polyhedron.

[2]

Stronger or generalized formedge-unzipping of convex polyhedra

Requiring the cut tree to be a Hamiltonian path is false in general.

[2]

Stronger or generalized formpseudo-edge unfoldings of convex polyhedra

The version allowing a prescribed convex geodesic pseudo-edge graph has counterexamples.

[1]

Solved special casetall or rectangular-base prismatoids

Specified subclasses of prismatoids admit nonoverlapping edge unfoldings.

[3]

Later mathematical changes

What changed after the initial research map

Later recorded revisions that changed the mathematics, without inventing a date or an AI attribution.

Sixth-pass corrections and four-interior frontierThe sixth-pass packet closes the octahedral layer, preserves four scoped retractions or refutations, and reports a five-graph, sixteen-type four-interior census with eight harmless and eight unresolved types.

Changed the research frontierLater mathematical revision

Research stage 13

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected supporting details in the research record. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

12 mapped milestonesretained argument map

Browse all 12 mapped stages

stage 1Reciprocal curvature and forest reduction
stage 2Packet purification structure
stage 326-vertex pointwise trap and adaptive escape reported
stage 4Distinct 26-vertex cap interpretation withdrawn
stage 5Explicit adaptive-network inequalities retained
stage 6Fixed-reciprocal problem becomes exact and linear
stage 7Convex supporting-edge shadow theorem retracted
stage 8Affine-boundary patches motivate degree-three recursion
stage 9One rational octahedral cone exactly covered
stage 10Leaf refinement replaces unrestricted degree-three deletion
stage 11Finite-height continuation reduced to endpoint exclusion
stage 12Fourth-pass restart frontier

Reciprocal curvature and forest reductionThe current work binds almost-flat curvature to reciprocal-cell areas and formulates the obstruction through exact weighted forest margins.