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noun

An open fabric woven of strands that are interlaced and knotted at usually regular intervals: mesh, netting, network, web. See thing.

verb

To gain possession of, especially after a struggle or chase: capture, catch, get, secure, take. Informal bag. Slang nail. See get/lose.

The sum that remains following all permissible deductions, including charges, expenses, discounts, commissions, or taxes.

Net assets, for example, are what remain after an individual subtracts the amount owed to creditors from his or her assets. Net pay is the salary an individual actually receives after deductions such as income tax and Social Security payments.

Devoid of anything extraneous; free from all deductions, such as charges, expenses, taxes; remaining after expenses.

• Cases, Boxes, and Bags - net: mesh of interwoven threads or ropes, used esp. to catch birds, fish, or insects
• Fabrics and Cloth - net: open-mesh fabric twisted or knotted at uniform intervals
• Accounting - net: money free from all charges, costs, and expenses
• Basketball
• Hockey
• Tennis
• Fishing Equipment and Tackle - net: any of various mesh devices with handles, used to snare fish in water for landing or boating
• Intrigues, Plots, and Corruption - net: entrapping device or circumstance

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.

For any given polyhedron that is unfolded into a net, many different equivalent nets can be formed depending on the choices of which edges are joined and which are separated.

Contents 1 Existence Theorem 2 Shortest Path 3 Higher dimensional polytope nets 4 See also 5 References 6 External links

Existence Theorem

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.

Shortest Path

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.

Higher dimensional polytope nets

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 1954 painting by Salvador Dalí, 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.

Wikimedia Commons has media related to: Polyhedral net

See also

• Paper model
• Cardboard modeling
• UV mapping

References

1. ^ Shephard, Geoffrey (1975), "Convex Polytopes with Convex Nets", Mathematical Proceedings of the Cambridge Philosophical Society 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

• Weisstein, Eric W., "Net" from MathWorld.
• Weisstein, Eric W., "Unfolding" from MathWorld.
• Weisstein, Eric W., "Shephard's Conjecture" from 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

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Dansk (Danish)
1.
n. - net, garn, snare
v. tr. - indbringe netto, tjene netto, fange i sit garn, sætte net om

idioms:

• cast one's net wider    ekspandere
• fall through the net    falde gennem nettet
• net curtain    netgardin
• slip through the net    slippe gennem nettet
• the net    Internet

2.
adj. - netto, netto-
n. - net, garn
v. tr. - tjene netto

idioms:

• net profit    nettooverskud

Nederlands (Dutch)
net, (net)kooi, netbal, valkuil, (computer) netwerk, (benaming voor) het Internet, netmateriaal, netto, netto-, uiteindelijk (resultaat/effect), netten maken, vangen, af-/bedekken, scoren, breien, een ruitjespatroon aanbrengen, opbrengen, winnen

Français (French)
1.
n. - (Sport) filet, (fig) piège, étau, (Télécom) réseau, (Tex) (voile) de tulle
v. tr. - prendre au filet, (fig) attraper (un criminel), tendre des filets dans (une rivière), (Sport) envoyer la balle dans le filet (au tennis)

idioms:

• fall through the net    passer à travers les filets
• net curtain    voilage
• slip through the net    passer à travers les mailles du filet
• the Net    Internet

2.
adj. - (Fin, Comm) net (un revenu, un profit), sèche (une perte), (gén) net (un résultat, un accroissement)
n. - revenu net, bénéfice net, poids net
v. tr. - rapporter/produire net, gagner/toucher net (qn)

idioms:

• net profit    bénéfice net

Deutsch (German)
1.
n. - Netz
v. - mit einem Netz bedecken od. fangen, einfangen, ins Netz schießen, zu einem Netz verknüpfen, mit Netzen abfischen, Netze auswerfen

idioms:

• fall through the net    (von den zuständigen Stellen) nicht erreicht werden
• net curtain    Tüllgardine
• slip through the net    entkommen
• the Net    das weltweite Computernetz

2.
adj. - Netto-
n. - Reingewinn, Nettoeinkommen
v. - rein oder netto einbringen

idioms:

• net profit    Reingewinn

Ελληνική (Greek)
n. - δίχτυ, δίκτυο, φιλέ(ς), (οικον.) καθαρό κέρδος
v. - πιάνω ή περιβάλλω με δίχτυ, (οικον.) αποκομίζω ή αποφέρω καθαρά, καθαρίζω, (καθομ.) στέλνω την μπάλα στα δίχτυα
adj. - αμιγής, καθαρός (κν. νέτος)

idioms:

• cast one's net wider    επεκτείνω τη δράση μου
• fall through the net    διαφεύγω από τον κλοιό
• net curtain    δικτυωτή κουρτίνα
• net profit    (οικον.) καθαρό κέρδος
• slip through the net    διαφεύγω από τον κλοιό
• the net    το Διαδίκτυο

Italiano (Italian)
rete, netto, realizzare

idioms:

• cast one's net wider    espandersi
• fall through the net    svincolarsi
• net curtain    tenda a rete
• net profit    reddito netto
• slip through the net    sfuggire dalla rete
• the net    Internet

Português (Portuguese)
n. - rede (f), malha (f), armadilha (f)
v. - lançar a rede, cobrir ou proteger com rede, enredar, fazer rede
adj. - líquido, livre

idioms:

• cast one's net wider    lançar a rede a uma maior distância
• fall through the net    ter problemas econômicos, por não ser amparado pelo programa de assistência social
• net curtain    cortina de rede
• net profit    lucro líquido
• slip through the net    passar pela rede
• the net    o ponto principal

Русский (Russian)
сетка, сеть, тюль, нетто, чистая прибыль, ловить сетью/сачком, заработать

idioms:

• cast one's net wider    собирать без разбора
• fall through the net    проскользнуть сквозь сеть
• net curtain    тюлевая занавеска
• net profit    чистая прибыль
• slip through the net    проскользнуть сквозь сеть
• the net    система ловли

Español (Spanish)
1.
n. - red, malla, tul, redecilla, trampa, mosquitero, retículo
v. tr. - coger con red, pescar, tejer

idioms:

• fall through the net    no recibir la atención merecida, escabullirse
• net curtain    visillos
• slip through the net    escabullirse, escurrirse
• the Net    el Internet, el aro (en baloncesto)

2.
adj. - reticular, de punto de malla, cogido con red, hecho con red, puro, sin mezcla, neto, líquido
n. - neto, ganancia neta, peso neto, cantidad neta
v. tr. - ganar neto

idioms:

• net profit    beneficio neto

Svenska (Swedish)
n. - nät, hål, tyll, bur, målbur, grind (cricket)
v. - förtjäna, fånga, lägga ut/spänna/knyta nät
adj. - netto, egentlig

中文（简体）(Chinese (Simplified))
1. 网, 网状制品, 网状物, 罗网, 陷阱, 使净得, 使净赚, 得到, 使得到

idioms:

• cast one's net wider    扩大范围探索
• fall through the net    遗漏, 漏网之鱼
• net curtain    网眼帘
• net profit    纯利润, 纯利
• slip through the net    漏网之鱼, 遗漏
• the net    全球信息网

2. 净余的, 纯粹的, 净数, 净利, 净重, 净值, 使净得, 使净赚, 得到, 使得到

中文（繁體）(Chinese (Traditional))
1.
n. - 網, 網狀製品, 網狀物, 羅網, 陷阱
v. tr. - 使淨得, 使淨賺, 得到, 使得到

idioms:

• cast one's net wider    擴大範圍探索
• fall through the net    遺漏, 漏網之魚
• net curtain    網眼帘
• net profit    純利潤, 純利
• slip through the net    漏網之魚, 遺漏
• the net    全球資訊網

2.
adj. - 淨餘的, 純粹的
n. - 淨數, 淨利, 淨重, 淨值
v. tr. - 使淨得, 使淨賺, 得到, 使得到

한국어 (Korean)
1.
n. - 그물, 함정, 통신망, 거미줄
v. tr. - (노력의 결과)얻다, 그물로 잡다

idioms:

• cast one's net wider    그물을 던지다
• the net    ~망

2.
adj. - 최종의, 에누리 없는
n. - 정가, 정미, 순량
v. tr. - 순익을 올리다

日本語 (Japanese)
n. - 網, 網細工, 網状のもの, 情報網, 罠, 純量, 正味, 正価
v. - 網でとらえる, 網で覆う, ネットに引っ掛ける, 網に編む, 純益を上げる, 純益をもたらす, 網を張る
adj. - 正味の

idioms:

• net curtain    網カーテン
• net profit    純益
• the net    インターネット

العربيه (Arabic)
‏(الاسم) شبكه, شرك, مقدار صاف ( ربح او وزن) (فعل) يصيد بشبكه, يضرب الكرة فيصيب الشبكه ( في التنس), يربح ربحا صافيا (صفه) صاف ( وزن الخ), نهائي‏

עברית (Hebrew)
n. - ‮רשת, מכמורת, מלכודת, "אינטרנט", סל (בכדורסל)‬
v. tr. - ‮העלה ברשת, לכד, כיסה ברשת, רישת‬
adj. - ‮נטו, נקי (רווח)‬
n. - ‮הכנסה או רווח בניכוי ההוצאות‬
v. tr. - ‮הרוויח נטו‬

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