In

Regular 4d Polytope Foldouts

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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 ...

, 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 ...

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 ...

, 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 aspanning 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 ...

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

Theshortest 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 ...

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 ...

, 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 a4-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 ...

, 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 ...

'' (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 ...

.
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 ...

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 ...

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

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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 ...

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