Net (polyhedron)
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geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a net of a
polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
is an arrangement of non-overlapping edge-joined
polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
s in the plane which can be folded (along edges) to become the
face The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect th ...
s of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and
solid geometry Solid geometry or stereometry is the geometry of Three-dimensional space, three-dimensional Euclidean space (3D space). A solid figure is the region (mathematics), region of 3D space bounded by a two-dimensional closed surface; for example, a ...
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 ( , ;; 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 or Duerer, was a German painter, Old master prin ...
, 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 The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They ...
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 mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is no ...
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, the angular defect or simply defect (also called deficit or deficiency) is the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the ''exces ...
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. 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 angles. More general ...
. 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 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 A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. 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 upright=1.3, Isometric projection and net of naive (1) and optimal (2) solutions of the spider and the fly problem The spider and the fly problem is a recreational mathematics problem with an unintuitive solution, asking for a shortest path or g ...
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, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: Vertex (geometry), vertices, Edge (geo ...
, a four-dimensional
polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
, 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 analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
, 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), known as Salvador Dalí ( ; ; ), was a Spanish Surrealism, surrealist artist renowned for his technical skill, precise draftsmanship, ...
, ''
Crucifixion (Corpus Hypercubus) ''Crucifixion (Corpus Hypercubus)'' is a 1954 oil-on-canvas painting by Salvador Dalí. A nontraditional, surrealism, surrealist Crucifixion in art, portrayal of the Crucifixion, it depicts Christ on a polyhedron net of a tesseract (hypercube). ...
'' (1954). The same tesseract net is central to the plot of the short story "—And He Built a Crooked House—" by Robert A. Heinlein. The number of combinatorially distinct nets of n-dimensional
hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
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 on the
complement graph In the mathematical field of graph theory, the complement or inverse of a graph is a graph on the same vertices such that two distinct vertices of are adjacent if and only if they are not adjacent in . That is, to generate the complement of ...
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, ...


See also

* Cardboard modeling * Common net * Paper model * UV mapping


References


External links

* *
Regular 4d Polytope Foldouts
* ttp://www.korthalsaltes.com/ Paper Models of Polyhedrabr>Unfolder
for Blender
Unfolding
package for
Mathematica Wolfram (previously known as Mathematica and Wolfram Mathematica) is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network ...
{{Mathematics of paper folding Types of polygons Polyhedra 4-polytopes Spanning tree