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Net of a dodecahedron

When the Earth is mapped on a polyhedron, its net is a flat world map, e.g. the Dymaxion map using the regular icosahedron and a few subdivisions of the triakis icosahedron.

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 models of polyhedra to be constructed from material such as thin cardboard.

It is a long-standing open question whether or not every convex polyhedron P (one without "dents" - in other words, all dihedral angles between the edges are ≤ 180 degrees) has a net: whether the surface P may be cut along edges and unfolded flat to a planar polygon (without overlap). The problem was first explicitly posed in a paper by Shephard. ^[1] The history of and progress on this question is discussed in Part III of Geometric Folding Algorithms. ^[2]

Also, the shortest path over the surface between two points on the surface of a polyhedron corresponds to a straight line on a suitable net. The net has to be such that the straight line is fully within it, and one may have to consider several nets to see which gives the shortest path. For example, in the case of a cube, if the points are on adjacent faces one candidate for the shortest path is the path crossing the common edge; the shortest path of this kind is found using a net where the two faces are also adjacent. Other candidates for the shortest path are through the surface of a third face adjacent to both (of which there are two), and corresponding nets can be used to find the shortest path in each category.

The geometric concept of a net can be extended to higher dimensions.

tesseract

Truncated tesseract

24-cell

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 tesseract, the four-dimensional cube, is used prominently in a 1955 painting by Salvador Dali, Corpus Hypercubus.

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1. ^ Shephard, Geoffrey (1975), "Convex Polytopes with Convex Nets", Math. Proc. Camb. Phil. Soc. 78: 389-403, ISSN 1469-8064
2. ^ Demaine, Erik & O'Rourke, Joseph (July 2007), Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, ISBN 978-0-521-85757-4, <http://www.gfalop.org>

• Eric W. Weisstein, Net at MathWorld.
• Nets: A Tool for Representing Polyhedra in Two Dimensions
• Nuts About Nets!
• Regular 4d Polytope Foldouts