Editing Net (polyhedron) (section)

==Existence and uniqueness==
[[File:4-hex octahedron.svg|thumb|Four hexagons that, when glued to form a regular octahedron as depicted, produce folds across three of the diagonals of each hexagon. The edges between the hexagons remain unfolded.]]
Many different nets can exist for a given polyhedron, depending on the choices of which edges are joined and which are separated. The edges that are cut from a convex polyhedron to form a net must form a [[spanning tree]] of the polyhedron, but cutting some spanning trees may cause the polyhedron to self-overlap when unfolded, rather than forming a net.<ref name=gfa/> Conversely, a given net may fold into more than one different convex polyhedron, depending on the angles at which its edges are folded and the choice of which edges to glue together.<ref>{{citation|url=https://www.ams.org/featurecolumn/archive/nets.html|title=Nets: A Tool for Representing Polyhedra in Two Dimensions|first=Joseph|last=Malkevitch|work=Feature Columns|publisher=[[American Mathematical Society]]|access-date=2014-05-14}}</ref> If a net is given together with a pattern for gluing its edges together, such that each vertex of the resulting shape has positive [[angular defect]] and such that the sum of these defects is exactly 4{{pi}}, then there necessarily exists exactly one polyhedron that can be folded from it; this is [[Alexandrov's uniqueness theorem]]. However, the polyhedron formed in this way may have different faces than the ones specified as part of the net: some of the net polygons may have folds across them, and some of the edges between net polygons may remain unfolded. Additionally, the same net may have multiple valid gluing patterns, leading to different folded polyhedra.<ref>{{citation | last1 = Demaine | first1 = Erik D. | author1-link = Erik Demaine | last2 = Demaine | first2 = Martin L. | author2-link = Martin Demaine | last3 = Lubiw | first3 = Anna | author3-link = Anna Lubiw | last4 = O'Rourke | first4 = Joseph | author4-link= Joseph O'Rourke (professor)|arxiv = cs.CG/0107024 | doi = 10.1007/s003730200005 | issue = 1| journal = [[Graphs and Combinatorics]]| mr = 1892436| pages = 93–104 | title = Enumerating foldings and unfoldings between polygons and polytopes| volume = 18 | year = 2002| s2cid = 1489 }}</ref>

{{unsolved|mathematics|Does every convex polyhedron have a simple edge unfolding?}}
{{anchor|1=Shephard's conjecture}}In 1975, [[Geoffrey Colin Shephard|G. C. Shephard]] asked whether every convex polyhedron has at least one net, or simple edge-unfolding.<ref>{{citation| last = Shephard | first = G. C. | author-link = Geoffrey Colin Shephard| issue = 3| journal = [[Mathematical Proceedings of the Cambridge Philosophical Society]]| mr = 0390915| pages = 389–403| title = Convex polytopes with convex nets| volume = 78| year = 1975 |doi=10.1017/s0305004100051860| bibcode = 1975MPCPS..78..389S| s2cid = 122287769 }}</ref> This question, which is also known as Dürer's conjecture, or Dürer's unfolding problem, remains unanswered.<ref>{{Mathworld|urlname=ShephardsConjecture|title=Shephard's Conjecture|mode=cs2}}</ref><ref>{{citation|url=http://www.openproblemgarden.org/op/d_urers_conjecture|title=Dürer's conjecture|work=Open Problem Garden|first=D. |last=Moskovich|date=June 4, 2012}}</ref><ref>{{citation|last=Ghomi|first=Mohammad|date=2018-01-01|title=Dürer's Unfolding Problem for Convex Polyhedra|journal=Notices of the American Mathematical Society|volume=65|issue=1|pages=25–27|doi=10.1090/noti1609|doi-access=free}}</ref> There exist non-convex polyhedra that do not have nets, and it is possible to subdivide the faces of every convex polyhedron (for instance along a [[cut locus]]) so that the set of subdivided faces has a net.<ref name=gfa>{{citation|title=Geometric Folding Algorithms: Linkages, Origami, Polyhedra|title-link=Geometric Folding Algorithms|first1=Erik D.|last1=Demaine|author1-link=Erik Demaine|first2=Joseph|last2=O'Rourke|author2-link=Joseph O'Rourke (professor)|publisher=Cambridge University Press|year=2007|contribution=Chapter 22. Edge Unfolding of Polyhedra|pages=306–338}}</ref> In 2014 [[Mohammad Ghomi]]  showed that every convex polyhedron admits a net after an [[affine transformation]].<ref>{{citation|arxiv=1305.3231|last1=Ghomi|first1=Mohammad|title=Affine unfoldings of convex polyhedra|journal=[[Geometry & Topology]]|date=2014|volume=18|issue=5|pages=3055–3090|bibcode=2013arXiv1305.3231G|doi=10.2140/gt.2014.18.3055|s2cid=16827957}}</ref> Furthermore, in 2019 Barvinok and Ghomi showed that a generalization of Dürer's conjecture fails for ''pseudo edges'',<ref>{{citation|last1=Barvinok|first1=Nicholas|last2=Ghomi|first2=Mohammad|date=2019-04-03|title=Pseudo-Edge Unfoldings of Convex Polyhedra|journal=[[Discrete & Computational Geometry]]|volume=64|issue=3|pages=671–689|doi=10.1007/s00454-019-00082-1|issn=0179-5376|arxiv=1709.04944|s2cid=37547025}}</ref> i.e., a network of geodesics which connect vertices of the polyhedron and form a graph with convex faces.

[[File:Net of dodecahedron.gif|thumb|[[Blooming (geometry)|Blooming]] a [[regular dodecahedron]]]]
A related open question asks whether every net of a convex polyhedron has a [[Blooming (geometry)|blooming]], a continuous non-self-intersecting motion from its flat to its folded state that keeps each face flat throughout the motion.<ref>{{citation
 | last1 = Miller | first1 = Ezra
 | last2 = Pak | first2 = Igor | authorlink2=Igor Pak
 | doi = 10.1007/s00454-008-9052-3
 | issue = 1–3
 | journal = [[Discrete & Computational Geometry]]
 | mr = 2383765
 | pages = 339–388
 | title = Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings
 | volume = 39
 | year = 2008| doi-access = free
 }}</ref>