---
tag: PEDBPRSV9V
kind: conjecture
name: Dürer's conjecture
msc: [52B10, 52B05]
uses: [FXJ66Z6PQY, 300K6CB1KH]
about: [FXJ66Z6PQY]
sources:
  - url: https://en.wikipedia.org/wiki/Net_(polyhedron)
history:
  posed_by: "Geoffrey Colin Shephard"
  first_stated: "1975"
---

## Statement

Every three-dimensional [[convex polytope|300K6CB1KH]] in $\R^3$ has a [[net|FXJ66Z6PQY]].
That is, some spanning tree of its edge graph gives an edge unfolding without overlap.

## Remarks

The conjecture is open.
It is named after Albrecht Dürer, whose book on geometry drew nets of many polyhedra, but it was first stated as a question by Shephard.
Cuts may also be allowed to run across faces and not only along edges.
Then every convex polyhedron can be unfolded without overlap, for example by the source unfolding or the star unfolding.
Convexity matters: there are non-convex polyhedra whose edge unfoldings all overlap.
For a given convex polyhedron, some spanning trees may give overlapping unfoldings, so the question is whether at least one good tree always exists.

## Explanations

### L1 human

Can every convex solid with flat faces be cut along some of its edges and flattened into a single piece of paper with no part lying on top of another?
For the cube and all the familiar solids the answer is yes.
Nobody knows whether it is true for every convex solid, and no counterexample has been found.