In geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which 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.
Many different nets can exist for a given polyhedron, depending on the choices of which edges are joined and which are separated.
Higher-dimensional polytope nets
The geometric concept of a net can be extended to higher dimensions.
For example, a net of a polychoron, or four-dimensional polytope, is composed of polyhedral cells 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 above net of the tesseract, the four-dimensional hypercube, is used prominently in a painting by Salvador Dalí, Crucifixion (Corpus Hypercubus) (1954).
Whether or not every four-dimensional polytope may be cut along the two-dimensional faces shared by its three-dimensional facets, and unfolded into 3D to a single nonoverlapping polyhedron (as in the above unfolding of the tesseract), remains unknown, as does the corresponding question in higher dimensions. However, it is known to be possible for every convex uniform polychoron.
|Wikimedia Commons has media related to Polyhedral net.|
- Weisstein, Eric W., "Net", MathWorld.
- Weisstein, Eric W., "Unfolding", MathWorld.
- Weisstein, Eric W., "Shephard's Conjecture", MathWorld.
- Nets: A Tool for Representing Polyhedra in Two Dimensions
- Nuts About Nets!, George Olshevsky.
- Regular 4d Polytope Foldouts
- Editable Printable Polyhedral Nets with an Interactive 3D View
- Paper Models of Polyhedra
- Unfolder for Blender
- Unfolding package for Mathematica