Netzformen

Netzformen denotes the set of two-dimensional patterns that can be folded to form a given polyhedron. In geometry, a net (Netz) is a connected arrangement of polygonal faces laid out in the plane such that, by folding along the edges, the faces reconstruct the three-dimensional solid. Netzformen thus describe the distinct planar realizations of a polyhedron’s surface, counted up to planar congruence.

Not every pattern of polygons yields a valid net; some unfoldings cause overlaps when folded. For a cube, there are 11 non-congruent nets. For other polyhedra, the number of nets varies and can be large; researchers study enumeration and recognition of nets, often using computer algorithms to generate all possibilities without duplicates and to test foldability.

Netzformen have practical uses in education to teach spatial reasoning, and in design and manufacturing where flat templates are cut from sheet material and folded into boxes, packaging, or more complex structures. They also relate to origami and to computational geometry, where unfolding problems are examined within the broader study of polyhedral combinatorics and graph representations of solids. The concept connects the 2D patterns with their 3D counterparts and highlights how adjacency relationships between faces are preserved in unfolding.

See also: polyhedron net, unfolding, polyhedron, graph theory, computational geometry.