View source for Net (polyhedron)

{{Short description|Edge-joined polygons which fold into a polyhedron}}
[[File:Dodecahedron flat.svg|thumb|right|240px|A net of a [[regular dodecahedron]]]]
[[File:The 11 cubic nets.svg|thumb|right|240px|The eleven nets of a cube]]

In [[geometry]], a '''net''' of a [[polyhedron]] is an arrangement of non-overlapping [[Edge (geometry)|edge]]-joined [[polygon]]s in the [[plane (geometry)|plane]] which can be folded (along edges) to become the [[face (geometry)|face]]s of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and [[solid geometry]] in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard.<ref>{{citation|first=Magnus J.|last=Wenninger|author-link=Magnus Wenninger|title=Polyhedron Models|publisher=Cambridge University Press|year=1971}}</ref>

An early instance of polyhedral nets appears in the works of [[Albrecht Dürer]], whose 1525 book ''A Course in the Art of Measurement with Compass and Ruler'' (''Unterweysung der Messung mit dem Zyrkel und Rychtscheyd '') included nets for the [[Platonic solid]]s and several of the [[Archimedean solid]]s.<ref>{{citation|first=Albrecht|last=Dürer|author-link=Albrecht Dürer|title=Unterweysung der Messung mit dem Zyrkel und Rychtscheyd|location=Nürnberg|year=1525|url=https://archive.org/stream/albrechtdrersun01peltgoog#page/n142/mode/2up|pages=139–152|publisher=München, Süddeutsche Monatsheft}}. English translation with commentary in {{citation|first=Walter L.|last=Strauss|title=The Painter's Manual|location=New York|year=1977}}</ref><ref>Schreiber, Fischer, and Sternath claim that, earlier than Dürer, [[Leonardo da Vinci]] drew several nets for [[Luca Pacioli]]'s ''[[Divina proportione]]'', including a net for the regular dodecahedron. However, these cannot be found in [https://archive.org/details/diuinaproportion00paci online copies of the 1509 first printed edition of this work] nor in the 1498 [https://www.e-codices.unifr.ch/it/list/one/bge/le0210 Geneva ms 210], so this claim should be regarded as unverified. See: {{citation
 | last1 = Schreiber | first1 = Peter
 | last2 = Fischer | first2 = Gisela | author2-link = Gisela Fischer
 | last3 = Sternath | first3 = Maria Luise
 | date = July 2008
 | issue = 4
 | journal = Archive for History of Exact Sciences
 | jstor = 41134285
 | pages = 457–467
 | title = New light on the rediscovery of the Archimedean solids during the Renaissance
 | volume = 62| doi = 10.1007/s00407-008-0024-z
 }}</ref> These constructions were first called nets in 1543 by [[Augustin Hirschvogel]].<ref>{{citation|first= Michael|last=Friedman|publisher=Birkhäuser|year=2018|title=A History of Folding in Mathematics: Mathematizing the Margins|title-link=A History of Folding in Mathematics|series=Science Networks. Historical Studies|volume=59|isbn=978-3-319-72486-7|doi=10.1007/978-3-319-72487-4|page=8}}</ref>

==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>

==Shortest path==
The [[geodesic|shortest path]] over the surface between two points on the surface of a polyhedron corresponds to a straight line on a suitable net for the subset of faces touched by the path. The net has to be such that the straight line is fully within it, and one may have to consider several nets to see which gives the shortest path. For example, in the case of a [[cube]], if the points are on adjacent faces one candidate for the shortest path is the path crossing the common edge; the shortest path of this kind is found using a net where the two faces are also adjacent. Other candidates for the shortest path are through the surface of a third face adjacent to both (of which there are two), and corresponding nets can be used to find the shortest path in each category.<ref>{{citation|title=How to Fold It: The Mathematics of Linkages, Origami and Polyhedra|first=Joseph|last=O’Rourke|publisher=Cambridge University Press|year=2011|isbn=9781139498548|pages=115–116|url=https://books.google.com/books?id=EbwNKD0xkUwC&pg=PA115}}</ref>

[[The spider and the fly problem]] is a [[recreational mathematics]] puzzle which involves finding the shortest path between two points on a cuboid.

==Higher-dimensional polytope nets==
[[File:tesseract2.svg|thumb|upright|The [[Polycube#Octacube and hypercube unfoldings|Dalí cross]], one of the 261 nets of the [[tesseract]]]]
A net of a [[4-polytope]], a four-dimensional [[polytope]], is composed of polyhedral [[cell (geometry) |cells]] that are connected by their faces and all occupy the same three-dimensional space, just as the polygon faces of a net of a polyhedron are connected by their edges and all occupy the same plane. The net of the tesseract, the four-dimensional [[hypercube]], is used prominently in a painting by [[Salvador Dalí]], ''[[Crucifixion (Corpus Hypercubus)]]'' (1954).<ref>{{citation |last=Kemp |first=Martin |author-link=Martin Kemp (art historian) |title=Dali's dimensions |journal=[[Nature (journal) |Nature]] |volume=391 |issue= 6662 |page=27 |date=1 January 1998 |bibcode=1998Natur.391...27K |s2cid=5317132 |doi=10.1038/34063 |doi-access=free}}</ref> The same tesseract net is central to the plot of the short story [["—And He Built a Crooked House—"]] by [[Robert A. Heinlein]].<ref>{{citation |last=Henderson |first=Linda Dalrymple |author-link=Linda Dalrymple Henderson |editor-last=Emmer |editor-first=Michele |contribution=Science Fiction, Art, and the Fourth Dimension |date=November 2014 |doi=10.1007/978-3-319-01231-5_7 |pages=69–84 |publisher=Springer International Publishing |title=Imagine Math 3: Between Culture and Mathematics|isbn=978-3-319-01230-8 }}</ref>

The number of combinatorially distinct nets of <math>n</math>-dimensional [[hypercube]]s can be found by representing these nets as a tree on <math>2n</math> nodes describing the pattern by which pairs of faces of the hypercube are glued together to form a net, together with a [[perfect matching]] on the [[complement graph]] of the tree describing the pairs of faces that are opposite each other on the folded hypercube. Using this representation, the number of different unfoldings for hypercubes of dimensions 2, 3, 4, ... have been counted as
:1, 11, 261, 9694, 502110, 33064966, 2642657228, ... {{OEIS|id=A091159}}

==See also==
*[[Cardboard modeling]]
*[[Common net]]
*[[Paper model]]
*[[UV mapping]]

==References==
{{reflist}}

== External links ==
*{{Mathworld|urlname=Net|title=Net|mode=cs2}}
*{{Mathworld|urlname=Unfolding|title=Unfolding|mode=cs2}}
*[https://web.archive.org/web/20110717184828/http://weimholt.com/andrew/polytope.shtml Regular 4d Polytope Foldouts]
*[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html Editable Printable Polyhedral Nets with an Interactive 3D View]
*[http://www.korthalsaltes.com/ Paper Models of Polyhedra]
*[http://celeriac.net/unfolder/ Unfolder] for [[Blender (software)|Blender]]
*[http://www.cs.mcgill.ca/~fukuda/download/mathematica/UnfoldPolytope.tar.Z Unfolding] package for [[Mathematica]]

{{Mathematics of paper folding}}

[[Category:Types of polygons]]
[[Category:Polyhedra]]
[[Category:4-polytopes]]
[[Category:Spanning tree]]