Net (polyhedron)


Net (polyhedron)
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). (This system of edges is sometimes denoted as the polyhedron's unfolding.) 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] If the restriction that the cuts be along polyhedron edges is relaxed to permit cuts through the interior of faces, then there are several known methods to cut and unfold a convex polyhedron to a planar polygon. For example, cutting along the cut locus of a point suffices.

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.

Contents

Higher dimensional polytope nets

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

Tesseract2.svg
Tesseract
Truncated tesseract net.png
Truncated tesseract
Polychoron 24-cell net.png
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 1954 painting by Salvador Dali, Crucifixion (Corpus Hypercubus).

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.

See also

References

  1. ^ Shephard, Geoffrey (1975), "Convex Polytopes with Convex Nets", Math. Proc. Camb. Phil. Soc. 78 (3): 389–403, doi:10.1017/S0305004100051860, 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 

External links


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • NET — or Net may refer to: Net (device), fibers woven in a grid like structure Net (textile), any textile in which the warp and weft yarns are looped or knotted at their intersections New Jersey Nets, a basketball team The Net (1995 film), a film… …   Wikipedia

  • Polyhedron — Polyhedra redirects here. For the relational database system, see Polyhedra DBMS. For the game magazine, see Polyhedron (magazine). For the scientific journal, see Polyhedron (journal). Some Polyhedra Dodecahedron (Regular polyhedron) …   Wikipedia

  • Polyhedron model — A polyhedron model is a physical construction of a polyhedron, constructed from cardboard, plastic board, wood board or other panel material, or, less commonly, solid material.Since there are 75 uniform polyhedra, including the five regular… …   Wikipedia

  • Semiregular polyhedron — A semiregular polyhedron is a polyhedron with regular faces and a symmetry group which is transitive on its vertices. Or at least, that is what follows from Thorold Gosset s 1900 definition of the more general semiregular polytope. [Thorold… …   Wikipedia

  • List of Wenninger polyhedron models — This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models , by Magnus Wenninger.The book was written as a guide book to building polyhedra as physical models. It includes templates of face elements …   Wikipedia

  • List of mathematics articles (N) — NOTOC N N body problem N category N category number N connected space N dimensional sequential move puzzles N dimensional space N huge cardinal N jet N Mahlo cardinal N monoid N player game N set N skeleton N sphere N! conjecture Nabla symbol… …   Wikipedia

  • Schlegel diagram — In geometry, a Schlegel diagram is a special projection of a polytope down one dimension. It projects polyhedra to a plane figure, and polychora to 3 space. It is used as a visual aid in seeing the topological connectivity of the polytope… …   Wikipedia

  • Crucifixion (Corpus Hypercubus) — Artist Salvador Dalí Year 1954 Type oil on canvas Dimensions 194.3 cm × 123.8 cm (76.5 in × 48.7 in) …   Wikipedia

  • Regular polytope — In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j faces (for all 0≤ j ≤ n , where n is the dimension of the polytope) cells, faces and… …   Wikipedia

  • Polywell — The polywell is a plasma confinement concept that combines elements of inertial electrostatic confinement and magnetic confinement fusion, intended ultimately to produce fusion power. The name polywell is a portmanteau of polyhedron and potential …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.