Math 2.0

Convex and discrete geometry Polytopes and polyhedra Conjecture

Dürer's conjecture

Builds on Net of a polyhedronConvex polytope

Statement

Every three-dimensional convex polytope in ℝ3\R^3 has a net. 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.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Convex polytope, concept, no evidence levelConvex polytopeconceptNet of a polyhedron, concept, no evidence levelNet of a polyhedronconceptDürer's conjecture, conjecture, no evidence levelDürer's conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Convex polytope concept
  2. Net of a polyhedron concept
  3. Dürer's conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Net_(polyhedron)
About this page
TagPEDBPRSV9V
URIm2://PEDBPRSV9V
Kindconjecture
WritingW0 stub
MSC52B10, 52B05
StatusThis copy of the site has no build data. Run m2 status PEDBPRSV9V.
AboutNet of a polyhedron
UsesNet of a polyhedron, Convex polytope

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