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A dodecahedron being folded and unfolded into one of its nets.

A net or unfolding of a polyhedron is a way to unfold its faces to lie in a 2-dimensional plane, cutting the original polyhedron only along edges. Nets can be formalized a number of ways and extended to cover polytopes of ranks other than 3.

Polyhedral nets are of practical concern since they can be cut out of paper and folded into a paper model of a polyhedron. Nets can also help to visualize certain aspects of polytopes in 3 or 4 dimensions.

Definitions[edit | edit source]

An unfolding of a (non-uniform) dodecagonal prism where faces overlap.[1]

While the idea of a net is simple, there are some subtleties to consider when constructing a definition.

Geometric[edit | edit source]

A net, N, of a concrete polytope in n-dimensional Euclidean space, P, is a closed set with a connected interior in n-1-dimensional hyperplane, such that there is a continuous surjective map from N to the surface of P, which is isometric and 1-to-1 on the interiors of facets of P.

Two nets are the same if there is a isometry of the hyperplane that maps between them.

Discrete[edit | edit source]

A net is a tree (a connected graph without cycles) with a node for each facet of the polytope, and edges corresponding to ridges of the polytope, such that edges only connect facets which the corresponding ridge is incident on.

Two nets are the same if there is a symmetry of the polytope which maps between them. This is distinct from graph isomorphism, as two nets can correspond to isomorphic graphs without being the same.

Comparison of definitions[edit | edit source]

The two definitions are not equivalent. The discrete definition allows for nets of many polytopes that do not have geometric nets, such as abstract polytopes, maniplexes and maps. The discrete definition does allow nets which overlap themselves when unfolded geometrically, while the geometric definition does not.

Bloomings[edit | edit source]

A blooming is a continuous transformation in n-dimensional space from a rank-n polytope to a net without bending or intersecting the facets.

External links[edit | edit source]

References[edit | edit source]

Bibliography[edit | edit source]

  • Buekenhout, F.; Parker, M. (1998). "The number of nets of the regular convex polytopes in dimension ≤4". Discrete Mathematics.