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In geometry, a net of a polyhedron is an arrangement of non-overlapping edge-joined in the plane that can be folded (along edges) to become the faces 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.
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 and several of the Archimedean solids.. English translation with commentary in 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 online copies of the 1509 first printed edition of this work nor in the 1498 Geneva ms 210, so this claim should be regarded as unverified. See: These constructions were first called nets in 1543 by Augustin Hirschvogel.
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. 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.
The spider and the fly problem is a recreational mathematics puzzle which involves finding the shortest path between two points on a cuboid.
The number of combinatorially distinct nets of -dimensional can be found by representing these nets as a tree on 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
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