2DEntfaltungen

2DEntfaltungen, or two-dimensional unfoldings, refer to methods for representing a surface or solid in the plane. In geometry, the term is most commonly associated with nets: a two-dimensional arrangement of polygons connected along edges that can be folded to form a three-dimensional polyhedron. For example, a cube has 11 distinct nets, and general results show that every convex polyhedron possesses at least one net, although counting all nets is a complex combinatorial problem.

Unfolding of curved surfaces extends this idea to non-planar geometries. Cylinders and cones can be developed into flat patterns with minimal distortion—a cylinder becomes a rectangle, while a cone yields a sector-based pattern. By contrast, developing a sphere onto a plane necessarily introduces distortion. Developments of curved surfaces are foundational in cartography and garment patternmaking, where three-dimensional shapes are represented on two-dimensional materials or maps.

Beyond classical geometry, two-dimensional unfoldings appear in computer graphics, material fabrication, and origami design, where flat templates are used to produce three-dimensional objects. In these contexts, algorithms for automatic unfolding seek to minimize distortion, preserve connectivity, or optimize the number of pieces required for assembly.

History and usage: The study of polyhedral nets and geometric unfoldings has roots in classical geometry and was advanced in the modern era through computational and combinatorial approaches. The term Entfaltungen is common in German-language sources for these flattening processes.

See also: net (polyhedron), development (geometry), unfolding (differential geometry).