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In
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
, a net of a
polyhedron In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
is an arrangement of non-overlapping edge-joined
polygon In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

polygon
s in the
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
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 An organ is a group of tissues with similar functions. Plant life and animal life rely on many organs that co-exist in organ systems. A given organ's ti ...
s of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and
solid geometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
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 (without an umlaut), was a German pain ...

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 In three-dimensional space, a Platonic solid is a Regular polyhedron, regular, Convex set, convex polyhedron. It is constructed by Congruence (geometry), congruent (identical in shape and size), regular polygon, regular (all angles equal and all sid ...
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, ...
s. These constructions were first called nets in 1543 by
Augustin Hirschvogel Augustin Hirschvogel (1503 – February 1553) was a German artist, mathematician, and cartographer known primarily for his etchings. His thirty-five small landscape etchings, made between 1545 and 1549, assured him a place in the Danube School, a ...
.


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 m ...

spanning tree
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 Angular may refer to: Anatomy * Angular artery, the terminal part of the facial artery * Angular bone, a large bone in the lower jaw of amphibians and reptiles * Angular incisure, a small anatomical notch on the stomach * Angular gyrus, a region o ...
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 Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...
. 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 Bloom or blooming may refer to: Science and technology Biology * Bloom, one or more flowers on a flowering plant * Algal bloom, a rapid increase or accumulation in the population of algae in an aquatic system * Jellyfish bloom, a collective noun fo ...
, 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 ter ...

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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

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. The spider and the fly problem is a
recreational mathematics Recreational mathematics is mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometr ...
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 (geom ...
, a four-dimensional
polytope In elementary geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...
, is composed of polyhedral
cells Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Closed spaces * Monastic cell, a small room, hut, or cave in which a monk or religious recluse lives * Prison cell, a room used to hold peopl ...
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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

hypercube
, is used prominently in a painting by
Salvador Dalí Salvador Domingo Felipe Jacinto Dalí i Domènech, 1st Marquess of Dalí of Púbol (; ; ; 11 May 190423 January 1989) was a Spanish surrealist Surrealism was a cultural movement A cultural movement is a change in the way a number of differe ...
, ''
Crucifixion (Corpus Hypercubus) ''Crucifixion (Corpus Hypercubus)'' is a 1954 oil-on-canvas Oil painting is the process of painting with pigments with a medium of drying oil as the Binder (material), binder. Commonly used drying oils include linseed oil, poppy seed oil ...

Crucifixion (Corpus Hypercubus)
'' (1954). The same tesseract net is central to the plot of the short story "—And He Built a Crooked House—" by
Robert A. Heinlein Robert Anson Heinlein (; July 7, 1907 – May 8, 1988) was an American science fiction File:Imagination 195808.jpg, Space exploration, as predicted in August 1958 in the science fiction magazine ''Imagination (magazine), Imagination.'' Sc ...

Robert A. Heinlein
. The number of combinatorially distinct nets of n-dimensional
hypercube In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

hypercube
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, a perfect matching in a graph is a Matching (graph theory), matching that covers every vertex of the graph. More formally, given a graph ''G'' = (''V'', ''E''), a perfect matching in ''G'' is a subset ''M'' of ''E'', such that ever ...
on the
complement graph In graph theory, the complement or inverse of a 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 adjacent in . That is, to generate the complement of a graph ...

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


See also

*
Paper model Paper is a thin sheet material produced by mechanically or chemically processing cellulose fibres derived from wood, Textile, rags, poaceae, grasses or other vegetable sources in water, draining the water through fine mesh leaving the fibre e ...
* Cardboard modeling *
UV mapping UV mapping is the 3D modeling process of projecting a 2D image to a 3D model's surface for texture mapping Texture mapping is a method for defining high frequency detail Detail(s) or The Detail(s) may refer to: Film and television * ''Deta ...


References


External links

* *
Regular 4d Polytope Foldouts
* ttp://www.korthalsaltes.com/ Paper Models of Polyhedrabr>Unfolder
for
Blender A blender (sometimes called a mixer or liquidiser in British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone subs ...

Unfolding
package for
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning Machine learning (ML) is the study of computer algorithms that can improve automatically through experi ...

Mathematica
{{Mathematics of paper folding Types of polygons Polyhedra 4-polytopes Spanning tree