Net (polyhedron)

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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
, a net of a
polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a Three-dimensional space, three-dimensional shape with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. A ...
is an arrangement of non-overlapping
edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...
-joined
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the tw ...
s in the
plane Plane(s) most often refers to: * Aero- or airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft that is propelled forward by thrust from a jet engine, Propeller (aircraft), propeller, or rocket engine. Airplanes co ...
which can be folded (along edges) to become the
face The face is the front of an animal's head that features the eyes Eyes are organs of the visual system. They provide living organisms with vision, the ability to receive and process visual detail, as well as enabling several photo respon ...
s of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and
solid geometry In mathematics, solid geometry or stereometry is the traditional name for the geometry of Three-dimensional space, three-dimensional, Euclidean spaces (i.e., 3D geometry). Stereometry deals with the measurements of volumes of various solid fig ...
in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard. An early instance of polyhedral nets appears in the works of
Albrecht Dürer Albrecht Dürer (; ; hu, Ajtósi Adalbert; 21 May 1471 – 6 April 1528),Müller, Peter O. (1993) ''Substantiv-Derivation in Den Schriften Albrecht Dürers'', Walter de Gruyter. . sometimes spelled in English as Durer (without an Umlaut (linguis ...
, 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 In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s and several of the
Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex polytope, convex Uniform polyhedron, uniform polyhedra composed of regular polygons meeting in identical vertex (geometry), vertices, e ...
s. These constructions were first called nets in 1543 by Augustin Hirschvogel.

# Existence and uniqueness

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 In the mathematics, mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree (graph theory), tree which includes all of the Vertex (graph theory), vertices of ''G''. In general, a graph ...
of the polyhedron, but cutting some spanning trees may cause the polyhedron to self-overlap when unfolded, rather than forming a net. 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. 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 In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of ge ...
and such that the sum of these defects is exactly 4, then there necessarily exists exactly one polyhedron that can be folded from it; this is
Alexandrov's uniqueness theorem The Alexandrov uniqueness theorem is a rigidity (mathematics), rigidity theorem in mathematics, describing three-dimensional convex polyhedron, convex polyhedra in terms of the distances between points on their surfaces. It implies that convex pol ...
. 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. In 1975, G. C. Shephard asked whether every convex polyhedron has at least one net, or simple edge-unfolding. This question, which is also known as Dürer's conjecture, or Dürer's unfolding problem, remains unanswered. 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. In 2014 Mohammad Ghomi showed that every convex polyhedron admits a net after an
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angle In Eucli ...
. Furthermore, in 2019 Barvinok and Ghomi showed that a generalization of Dürer's conjecture fails for ''pseudo edges'', i.e., a network of geodesics which connect vertices of the polyhedron and form a graph with convex faces. A related open question asks whether every net of a convex polyhedron has a blooming, a continuous non-self-intersecting motion from its flat to its folded state that keeps each face flat throughout the motion.

# Shortest path

The
shortest path In graph theory, the shortest path problem is the problem of finding a path (graph theory), path between two vertex (graph theory), vertices (or nodes) in a Graph (discrete mathematics), graph such that the sum of the Glossary of graph theory te ...
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 In geometry, a cube is a three-dimensional space, three-dimensional solid object bounded by six square (geometry), square faces, Facet (geometry), facets or sides, with three meeting at each vertex (geometry), vertex. Viewed from a corner it i ...
, 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. The spider and the fly problem is a
recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
puzzle which involves finding the shortest path between two points on a cuboid.

# Higher-dimensional polytope nets

A net of a
4-polytope In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of ...
, a four-dimensional
polytope In elementary geometry, a polytope is a geometric object with Flat (geometry), flat sides (''Face (geometry), faces''). Polytopes are the generalization of three-dimensional polyhedron, polyhedra to any number of dimensions. Polytopes may exist ...
, is composed of polyhedral 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 In geometry, a hypercube is an N-dimensional space, ''n''-dimensional analogue of a Square (geometry), square () and a cube (). It is a Closed set, closed, Compact space, compact, Convex polytope, convex figure whose 1-N-skeleton, skeleton consis ...
, is used prominently in a painting by
Salvador Dalí Salvador Domingo Felipe Jacinto Dalí i Domènech, Marquess of Dalí of Púbol (; ; ; 11 May 190423 January 1989) was a Spanish Surrealism, surrealist artist renowned for his technical skill, precise draftsmanship, and the striking and bizarr ...
, '' Crucifixion (Corpus Hypercubus)'' (1954). The same tesseract net is central to the plot of the short story
"—And He Built a Crooked House—" '—And He Built a Crooked House—' is a science fiction short story by American writer Robert A. Heinlein, first published in ''Astounding Science Fiction'' in February 1941. It was reprinted in the anthology ''Fantasia Mathematica'' (Clifton ...
by Robert A. Heinlein. The number of combinatorially distinct nets of $n$-dimensional
hypercube In geometry, a hypercube is an N-dimensional space, ''n''-dimensional analogue of a Square (geometry), square () and a cube (). It is a Closed set, closed, Compact space, compact, Convex polytope, convex figure whose 1-N-skeleton, skeleton consis ...
s can be found by representing these nets as a tree on $2n$ nodes describing the pattern by which pairs of faces of the hypercube are glued together to form a net, together with a
perfect matching In graph theory In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph the ...
on the
complement graph In the mathematical field of graph theory, the complement or inverse of a Graph (discrete mathematics), graph is a graph on the same Vertex (graph theory), vertices such that two distinct vertices of are adjacent if and only if they are not ad ...
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

*
Paper model Paper models, also called card models or papercraft, are models constructed mainly from sheets of heavy paper Paper is a thin sheet material produced by mechanically or chemically processing cellulose fibres derived from wood, Textile, ...
* Cardboard modeling *
UV mapping UV mapping is the 3D modeling In 3D computer graphics, 3D modeling is the process of developing a mathematical coordinate-based representation of any surface of an object (inanimate or living) in three dimensions via specialized software ...